McGraw-Hill Ryerson Functions |}Functions In previous mathematics courses, you have studied linear relations and some non-linear relations. In this chapter, you will learn what distingu will represent functions in a variety of forms, identify the domain and range of functions, and investigate the behaviour of graphs of functions. Your understanding of the quadratic function will be extended and you will learn how to determine the intersection ofa linear function and a quadratic function. You will apply your {fo situations, including how to model the knowledge of quadratics to real arch of the support of a bridge. shes some relations as functions. You By the end of this chapter, you wi © explain the meaning of the term function and distinguish a function from a relation that is not a function. through investigation of linear and quacratic relations using a variety of representations © represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions © explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions f(x) = x, F(x) = 8, f(x) = Vx, and f(x) = 4; describe the domain and range of a function appropriately; and explain any restrictions on the domain and range in contexts arising from real-world applications © determine the number of zeros of a quadratic function, using a variety of strategies © determine the maximum of minimum value ofa quadratic function whose equation is given nthe form f(x) = ax? + bx + c, using an algebraic method solve problems involving quadratic functions erising from real-world applications and represented using function notation determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding. quadratic equation and a point on the function solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically verify, through investigation with and without technology, that Vab = Va x V'b, 0 0 and b>, and use this relationship to simplify radicals, and radical expressions obtained by adding, subtracting, and multip|eye Re) 411 Refer to the Prerequisite Skills Appendix on pages 478 to 495 for examples of the topics and further practice. Graphs and Lines 1. Greph each linear relation. ay by y=-jxts 9 2x-sy+12=0 dy=6 x1 2. Netermine the equation in the form y = mx + b for each linear relation. 3. Determine the equation in the form y = mx + b for the line passing through ‘each pair of points. a) (0, 8) and (4, 3) b) (—3, 13) and (2, ~2) 9) (4, -1) and (12, 9) 4, Graph each pair of linear relations to find their point of intersection. a) y=2x+4andy=-x+1 b) y= 4x-Sandy=—2x+5 ©) 3x ~ Sy = ~4 and -2x + 3y=2 2 MHR + Functions 11 + Chapter 1 5. Use an algebraic method to find the point of intersection of each pair of lines. a) y= 3x45 ax-y byy=x+4 yrox-4 Oo x-2y=7 2x ~ 3y = 13 ‘Work With Polynomials 6, Expand and simplify each expression. a) (x + 2) b) (n+ 3)(n— 8) 9 jun 4 d) 3x + 3)(x — 2) @) {k- Mk + 1) f) 3x ~ a)fex+ 3) 7. Factor completely. a) x + 2x15 b) + 6x+9 ©) On? — 25 d) -x8 =x +42 e) 3f + 61+3 f) —5x* + 40x - 80 8, Identify if each quadratic expression is a perfect square trinomial. For the perfect square trinomials, write the factored form. a) x — 6x +12 by x! — 12x + 36, Q 2x bax td @) x + 18x +9 e)xttaxsa f) ant + an +9 9. What value of k makes each quadratic expression a perfect square trinomial? a) x4 ex k by x8 — 10x +k Qe axtk @) t+ 1ax +k ee t5xtk te -tixtk gt txek hy) x —3x+k 10. Factor out the rational coefficient of the xterm in each, Le 3, Bye a) ox Ox by Gxt + 5x ° - ~ ax 4) ae + 9xQuadratic Relations a} . For each quadratic relation, state 1) the coordinates of the vortex i) the equation of the axis of symmetry l) the direction of opening iv) the y-intercept Then, sketch a graph of the relation. a) y= 20+ 3 13. Comploto the square to express each 5 2 y= 3 3h 1 quadratic relation in the form y= a(x ~ h)? + k. Then, give the Determine the equation of the quadrati Each graph has the same shape as y = x, ay=x+4xt1 b) y= x ~ 10x 14. Without graphing, predict how the graphs of the equations in each pair will differ. Explain your reasoning. a) y= (x + 5 and y= (x +5)'+2 by y= x? - 4x +3 and y= x' ~ 4x 15. Use Technology Verify your answers to question 14 by graphing the two equations using a graphing calculator. Chapter Problem Andrea has a co-op placement at an actuarial firm. Actuarial science applies mathematical and statistical methods to ass: risk for insurance providers and financial institutions. Andrea’s assignments include collecting numerical data and developing equations for these businesses. ‘Throughout the chapter, you will be looking at a variety of tasks that Andrea has been given in her co-op placement. o Prerequisite Skills * MHR. 3:Functions, Domain, and Range When mathematicians and scientists recognize a relationship between items in the world around thom, they try to model the relationship with an equation. The concept of devoloping an equation is used in other fields too. Economists predict the growth of sectors of the economy using equations. Pollsters try to predict the outcome of an election using equations, Does the value of one measured quantity guarantee a unique value for the second related quantity? This quostion defines the difference between a relation and a function. relation + an identified pattern between two variables that may be represented as ordered pairs, a table of values, a graph, or an equation function + arelation in which each value of the independent variable (the first coordinate) corresponds to exactly one value of the dependent variable (the second coordinate) sir) —E—E—E——E—E_———— > How can you tell if a relation is a function? Data on summer jobs are collected from somo students in a grade 11 class. Some analysis is done to look for patterns in the data. A: Neil’s Time Worked and Amount Earned, by Week Eck era ek) 20 190 18 m 26 247 2 209 Ee [ 24 228 10 95 14 126 B: Number of Weeks Worked and Amount Earned by 10 Different Students Number of weeks Worked Total Amount Earned ($) 10 195 8 675 6 520 480 1100 1400 975 1200 1580 1740 4 MHR + Functions 11 + Chapter 11, Graph the given sets of data, Tools Describe any trends in the two graphs. I oa) 3. From the graph of the data in table A, can you predict how much * graphing calculator Neil would earn if he worked 28 h one week? 4, From the graph of the data in table B, can you predict the amount that a student who worked for 8 weeks would earn? Reflect Which set of data is a function? Explain using the terms independent variable and dependent variable. sige. = — = =— loos : * grid paper How can you make connections between equations, graphs, and functions? Method 1: Use Pencil and Paper The first Investigate illustrated that one value for the independent variable can be associated with more than one different value for the dependent variable. Any relation that has this property is not a function. In this Investigate, you will look at how this concept can be related to the ‘oquation for a relation. 1. Copy and complote the tables of values for the relations y = x* and xey ian = 39 3-3) 2 2 ‘. oO oO 1 1 2 2 3 3 2. Graph both relations on the same set of axes. 3. On the same set of axes, draw vertical lines with equations x = —3, x= -2,x=-1,x=1,x=2,andx= 3, Reflect Compare how the lines drawn in step 3 intersect each of the relations. Which relation is a function? Explain why. 1.1 Functions, Domain, and Range * MHR 5Tools Method 2: Use a Graphing Calculator * graphing calculator 1. Graph Y1 = x! Use the standard window settings. Refer to the Technology Appendix, pages 496 to 516, if you need help with eraphing equations. and select 1:ClrDraw to remove the vertical lino, 3, Is y = x*a function? Explain why or why not. 4. Graph x = y* by first solving the equation for y to obtain y © Enter ¥1 = (x)*0.5 and ¥2 = —(x)*0.5. 5. Repeat step 2. Is x = y* a function? Explain why or why not Example 1 SS _——SSSS==== Use the Vertical Line Test vertical line test Use the vertical line test to determine whether each relation is a + amethod of function. Justify your answer. determining whether a relation is a function a) b) ae = every vertical ine intersects the relation at only one point, then the relation is a function, F | I 5 6 MHR + Functions 11 » Chapter 1Solution a) This relation is a function, No vertical line can be drawn that will pass through more than one point on the line. b) This relation is a not function. An infinite number of vertical lines can be drawn that will pass through more than ‘one point on the curve, For example, the vertical line x = 6 passes through the points (6, 4) and (6, 0). ¢) This relation is a function. No vertical line can be drawn that will pass through more than one point on the curve. 4d) This relation is not a function. An infinite number of vertical lines can be drawn that will pass through more than one point on the circle. 1.1 Functions, Domain, and Range * MHR 7domain + the set of first coordinates of the ordered pairsina ‘elation range + the set of second coordinates cf the ordered pairsina relation Connections Brace brackets {} are used to denote a set of felated data points or values. real number + anumber in the set of all Integers, ‘terminating decimals, repeating dedmals, nnon-terminating decimals, and ‘non-repeating decimals, represented by the symbo For any relation, the set of values of the independent variable (often the x-values) is called the domain of the rolation. The set of the corresponding values of the dependent variable (often the y-values) is called the range of the relation. For a function, for each given element of the domain there must be exactly one element in the range, Example 2 —E—_——_—_—_—_—_—S= Determine the Domain and Range From Data Determine the damain and range of each relation. Use the domain and range to determine if the relation is a function, a) (-3, 4), (6, ~6), (2, 7), 6, 3), (6, -8)) b) The table shows the number Bees |e Number of children of each age at a sports camp, 7 5 12 6 5 7 2 @ 4 2 3 10 1 Solution a) domain (~3, —2, 5, 6}, range {—8, —6, 3, 4, 7} ‘This relation is not a function. The x-value x = 5 has two corresponding y-values, y = ~6 and y = 3, The domain has four elements but the range has five elements. So, one value in the domain must be associated with two values in the range. b) domain (4, 5, 6, 7, 8, 9, 10}, range {5, 8, 9, 11, 12, 14, 22) ‘This is a function because for each value in the domein there is exactly one value in the range. When the equation of a relation is given, the domain and range can be determined by analysing the allowable values from the sot of real numbers. 8 MHR Functions 11 + Chapter 1Example 3 SSE Determine the Domain and Range From Equations Determine the domain and the range for each relation. Sketch a graph of each. 1 xt ayy=2x-5 by =(x-1F +3 ay= @y=Ve-14+3 extt+y'=36 ‘Solution a) y = 2x ~ 5 isa linear relation. There are no restrictions on the values that can be chosen for x or y, Is set na demain {x €R} Read asthe domainisallreal iaconse ayo bers? expressing that xis any range ly ¢ RI} mes teal number Te symbol means “Is an element b) y = (x - 1) + 3 is a quadratic relation. Thore are no restrictions on the values that can be chosen for x, so the domain is all real numbers. domain (x € R} ‘The parabola has a minimum at its vertex (1, 3). All values of y are greater than or equal to 3. range (ye R, y= 3) | | | Fetes) ©) Division by zero is undefined, The expression in the denominator of = cannot be zero, So, x + 3 # 0, which means that x # ~3, All other values can be used for x. The vertical line x = ~3 is called an in asymptote. — ‘. Read as “the domain is all real numbers that are not potable heb domain {xe R, x # —3) or approaches more and equal 2 more closely but never For the range, there can never be a touches situation where the result of the division is zero, as 1 divided by a non-zero value can never result in an answer of 0. This function has another asymptote, the x-axis, Any real number except ~3 can be used for x and will result in all real numbers except 0 for the range, Use @ table of values or a graphing calculator to check this on the graph. range (ye R, y # 0} For exanple, forthe raph of y= the xeaxls and the y-axis are asymptotes, 1.1 Functions, Domain, and Range * MHR 9Connections In grade 10, you learnes that e + y?= is the equation of a crcl with centre the origin and radius r must be greater than or equal to zero, So, in Vx—143,x-120, orx= 1. 4) The expression under a radical sign y 6 domain (xe R, x= 1) | The value of the radical is alwaysO or greater and is added to 3 to give the | value of y. So, the y-values are always greater than or equal to 3. This gives | the range. range (ye R, y= 3) e) In x* + y? = 36, x must be less than or equal to 36, as must y?, since both x* and y? are always positive. So, the values for x and y are from ~6 to 6. domain (xe R, ~6 = x = 6) Read as “the domain is all real ‘numbers that are greater than ‘or equal to -G and less than or ‘equal to 67 range {ye R, Example 4 Determine the Domain and Range of an Area Function Amy volunteors to help enclose a rectangular area for a dog run behind the humane society. The run (a is bordored on one side A by the building wall. The society has 100 m of foncing available, a) Express the area function in terms of the width, b) Determine the domain and range for the area function 10 MHR + Functions 11 » Chapter 1 NiLiiLi was wvSolution Let x represent the width of the rectangular pen and 100 — 2x represent the length, both in metres. Let A represent the area, in. square metres. a) A(x) = x(100 — 23) Area = length x width = 2x! + 100x b) For the domain, x > 0, since there must be a width to enclose an area. For the length to be gr domain (x R, 0 x? 1.1 Functions, Domain, and Range * MHR 151.2 | } | = Functions and Function Notation The first instances of notation may have occurred when early humans attempted to show the concept of numbers. A jawbone in the Deutsches Museum in Munich, Germany, which has been dated at approximately 30 000 s.c.e., shows this early attempt. It has 55 equally spaced notches carved into it, arranged in groups of 5. This is one of the earliest pieces of evidence that show human interest in designing a notation for others to understand and convey the concept of a number system, In this section, you will extend the concept uf « function and formalize several notations that are used to represent a fun Tools Investigate SS _ >= Optional i + erid paver How can you use a function machine? Cs ‘A function machine generates ordered pairs by performing mathematical * graphing calculator operations on an input value. For each input value from the domain that enters a particular function machine, a unique output value in the range emerges. The output is determined by the rule of the defining function. If some values for x and their associated y-values are known, it is often possible to determine what function was used by the machine. Suppose you are told that a linear function machine has two steps. The first step involves a multiplication or a division, and the second step involves an addition or a subtraction, You are told that when the input value is x = 5, the output value is y = —9. As well, when the input value is x = 1, the output value is y = 3. Connections Tes not by accident that taper result te output we use the term digit. aes ‘is unknown: ma People started counting on thee fingers, or 7 resultofthe vio gts, When there were Le Sa sco" am ove too many items to count sunk ’ on 10 fingers items Such s stones sna 1. From the information above, what are the coordinates of two points pebbles were used. The may ome of the linear function’ cul, rom which we . . . von i Petia ioe cucu 2. Use the two points to determine the equation of the linear function in the form y = mx + b 16 MHR + Functions 11 » Chapter 13. Use the defining function to find a) yifxe (-3,-2, -1] b) xifye {-6, —15, —18} 4, Reflect Use the defining equation to describe the steps that are. performed by the function machine in generating the data. 5. Reflect At the start of this Investigate, you were told that the first step was a multiplication or a division and that the an addition or a subtraction a) What type of value for m would suggost that the first step is multiplication? a division? b) What type of value for b would suggest that the second step is an addition? a subtraction? 6. Make up your own linear function machine, Exchange two ordered pairs that your function machine generates with a partner. Determine the defining function for each other’s function machine ‘To write a function using function notation, the form f(x) = ... is used to indicate a function, f, with independent variable x. The notation (3) means the value obtained when x = 3 is substituted. f(3) is read as ‘fat 3” or “fof 3.” Letters other than fcan be used in function notation. Often, scientists and mathematicians will Use aletter related to the quantity being measured, For example, if the height is being measured as a function of time, express the function as (2). Example 1 SL ————SSSSSS== Find Values Using Function Notation For each function, determine f(—2), f(5), and #(3} a) fl) = 2x4 by f(x) = 3x — x +7 oft) = 11 4) fx) = 7% ‘Solution a) f(x) =2x-4 f(-2) = 2(—2) — 4 sunstitute x = -2, 1.2 Functions and Function Notation * MHR 17mapping diagram '* agraphical representation that felates the values in one set (the domain) to the values ina second set (the range) using diected arrows fram domain to range ») axt= x7 3(-2)P — (-2) +7 12+ 247 24 365% —5 +7 9) f(s) = 11 is a constant function. f(-2) = 11 #6) = 11 f(Z]=0 4 fe) 2(5) f(-2) = f(5) = ‘A mapping diagram is a reprosentation that can be used when the relation is given as a set of ordered pairs. In a mapping diagram, the domain values in one oval are joined to the range values in the other oval, using arrows. Ina mapping diagram, a relation is a function if there is exactly one arrow leading from each value in the domain, This indicates that each element in the domain corresponds to exactly one element in the range. {1>—# Aa 1 = sf ieis Ur ee Domain Range 18 MHR + Functions 11 + Chapter 1Example 2 Interpret Mapping Diagrams Use the mapping diagrams to 1) write the set of ordered pairs of the relation Ii) state if the relation is a function a = >» =} | 0 5 Range Domain Range Solution a) i) (21, ~5), (2, -1), (3, -5), (4, 0), (5, 0), (6, 5), (7, 8)) li) Since every value in the domain maps to exactly one value in the range, this relation is a function. b) i {(2, 1), (2, 2), (5, 3), (8, 7), (11, 5), (14, 4), (14, 6)) li) Since the values x = 2 and x = 14 both map to more than one value in the range, this relation is not a function. While mapping diagrams are useful in situations where the relations are given in ordered pair form, they are impractical when a function is written in function notation. For this reason, a second form of mapping has been developed. This form is referred to as mapping notation and is illustrated in the next example, ‘Example 3 —S——>>SS—_= Represent Functions Using Mapping Notation Write each function in mapping notation, a) f(x) = 3x*— ax +1 b) g(x) = ax +4 QMO = -4.98 - 4 4) P(x) = (500 — 2x)(300 + x) 1.2 Functions and Function Notation * MHR 19.Connections Great Slave Lake in the Northwest Territories Is the deepest lake in North America Atits deepest, itis 614m deep. Lake Ontario has an average depth of 196 mand is 235 m deep at its deepest. ‘Solution > 3x8 2x41 Read "fis a function that maps x to 3x®- 2x+1! b)gix3ax+4 Read "gis a function that maps x to 3x + 4" Qhit> 49 —4 Read “his a function that maps tto -49%°- 4" 4) P: x (500 — 2x)(300 + x) ReadPis a function that maps xto (500 ~ 2x)(300 + x)" Example 4 ——__— >= Solve a Problem Using Function Notation ‘The tempercture of the water at the surface of a deep lake is 22 °C ona warm summer's day. As Renaldo scuba dives to the depths of the lake, he finds that the temperature decreases by 1.5 °C for every 8 m he descends. a) Model the water temper ure at any depth using function notation. b) Use this function to determine the water temperature at a depth of 40 m. ©) At the bottom of the lake, the temperature is 5.5 °C. How deep is the lake? Solution a) Let d represent the depth, in metres, below the surface of the lake and T represent the temperature, in degrees Celsius, at this depth. T(d) = 22 —1.5[4) The temperature decreases by 15°C foreach, b) For a depth of 40 m, substitute d = 40. 7140) = 22 - 1.5{ 42) = 22-75 = 145 ‘The temperatw at a depth of 40 m is 14.5 * ©) Substitute 7{d) = 5.5 and solve for d. b= 22-1a(2) (2) = - 1.5d = 8 X 16.5 d= 88 The dopth of the lake is 88 m. 20 MHR + Functions 11 » Chapter 1LR sty ‘« In function notation, the symbol f(x) represents the dependent variable. It indicates that the function fis expressed in terms of the independent variable x. For example, y = 3x* — 5 is written as f(x) = 3x* — 5, ‘© Relations and functions given as ordered pairs can be represented using mapping diagrams. This involves using directed arrows from each value in an oval representing the domain to the corresponding | value or values In an oval representing the range. © Tn a mapping diagram, a relation is not a function when an element from the domain has two or more arrows leading to | different elements of the range. © Mapping notation can replace function notation. For example, f(x) = 3x* ~ 5 can be written as f: x > 3x? — 5. Communicate Your Understanding 1) Samuel missed the explanation of function notation. Explain how to answer the following. Given f(x) = x* + 5, find f(—2). (2 Michelle has written the function defined by y = 3t + 5t ~ 5 as f(x) = 3f + St — 5. Is she correct? Explain why or why not. t €3 A quadratic function has the same shape as y = x*. but it opens downward and has its vertex at (0, 3). Is each of the following a representation of the same function? Dsl) —x M43 Bfixooxt 43 1.2 Functions and Function Notation * MHR 21@ Practise For help with questions 1 to 4, refer to Example 1. 1. For eac and Af 2 a) fx) = Bx + 11 b) f(x) = 3a¢ + 2x +1 9 fle) = 2004+ 4? function, determine f(4), f(-5), @) fix) = -6 ) fy=t f) fo) = Vx45 2. Find the value of each function at x Sketch the graph of each function. a) fx) = 5x +4 b) Kix) = ax. 9 pix) =—4 d) g(x) = 11x° + 3x1 e) f(x) = (3x — 3)(2x + 2) f) hog = ~26 ~ axle - 7) 3. A linear function machine uses a function of the form f(x) = ax. Find the value of a for each given point on the function, and then write the defining equation of the function. a) (3, -12) b) (5, 15) 9 (12 4) (-3,3) 4, Give an example essnnganroneg of a linear function —nowewiing Dae and a constant ra, function, both in coats | tel function notation, Doseribo the somuntcatog similarities and the differences between the two functions. 22 MHR + Functions 11 + Chapter 1 For help with questions 5 to 6, refer to Example 2. 5. Show each set of data in a mapping diagram. a) {(1, 4), (2, 1), (3, -2), (4, —5), (5, -8), (6, —11), (7, -14), (8, -17)} b) {(—3, 4), (-2, -1), (-1, -4), (0, -5), (2, ~4), (2, 1} 9 ((-5, 6), (-4, 9), (3, 1), (-! (4, —2), (3, 8), (8, 8)} 4) (9, 9), (7, 9), (5, 9), 6. Determine if each relation in question 5 is a function. Justify your answer. 7. Write the ordered pairs associated with each mapping diagram. a) " — ww 2 f) a ee / 2 . Ay ie " ~~ | = es oman Range8. Determine whether each relation in question 7 is a function. Justify your answer. 9. What advantages do mapping diagrams have over a list of ordered pairs? For help with question 10, refer to Example 3. 10. Write cach function in mapping notation. a) flx) = =x +4 b) glx) = x? + 5x ©) sx) = Vax — 4 @) 1k) t 2k © Connect and Apply 11, Describe two different ways to determine if a relation is a function. 12, Use Technology If the output of a quadratic function machine gives data that fit an equation of the form fx) = ax? + bx + c, a graphing calculator can be used to determine the equation if at least three data points are given, Data aro given from such a function machine as follows: {(1, 4), (2, 11), (3, 24)}. a) Enter the values of the domain in La and the values of the range in 12. b) Plot the data. ©) Run quadratic regression to determine the quadratic equation that fits tiese data, Record the equation that results from this regression. 4) Use this function to determine the range values for the domain values x = ~3, x=0,and x= 5, Refer to the Technology Appendix. pages 496 to 516, if you need help with displaying data, quadratic regression or finding values. 16. 13. a) Complete a table Ressanng an Proving of values for the repressing stcng Tats relation Sasi $0) = Vand es at graph the data, = _ conmentaig b) Is this relation a function? Explain. ©) Could you have identified whether the relation was a function from the data in the table of values? Explain. For help with questions 14 to 16, refer to Example 4. 14, Rivers located near an ocean experience a large wave called a tidal bore due to the tides. The speed, v, in kilometres per hour, of the tidal bore in a river is a function of the depth, d, in metres, of the river. The function is uid) = 11.27Vd. a) Determine the domain and the range of this function. b) Make a table of values and graph the function. 15. The value, V, in dollars, of an n-year-old car is given by V(n) = 22.000 a) How much was this car worth when it ‘was first purchased? b) Determine the value of the car after i) 10 years ii) 12 years ©) How long would it take the car to depreciate to a value of $2000? 4) Is V(n) a function? Justify your answer, + 1000. ‘Tho amount, A, in dollars, that needs to be invested at an interest rate i to have $100 after 1 year is given by the relation Ali) = 00 . Note that i must be expressed in decimal form, a) Determine the domain and the range for this relation, b) Graph the relation. 9) How much money needs to be invested at 5% to give $100 after 1 year? 4d) What rate of interest is required if $90 is invested? 1.2 Functions and Function Notation * MHR 2317. Create a linear function machine and two points that are generated by the machine. ‘Trade points with a classmate to determine the function that generated the points. 18, Create a quadratic function machine of the form f(x) = ax? + b. Determine the coordinates of the y-intercept and of one other point that is generated by the machine. Trade points with a classmate to determine the funtion, 19. Chapter Problem While working at her co-op placement, Andrea is asked to work with a two-variable function to determine premiums for insurance policies. She is to calculate some values of the function and place them in the appropriate cell in a spreadsheet. The function is fla, 1) = 500 + 2n ~ 10r for a driver with a rating of r (related to the driver's record, 1 being a poor driver up to 5 being an excellent driver) and an age of n (from 40 to 45 years of age), For example, a 42-year- old driver with a rating of 4 would have a policy premium of fla2, 4) = 500 + 2(42) — 10(4) 500 + 84 ~ 40 = 544 This value has been placed in the spreadsheet for you. Help Andrea by copying and completing the spreadshoot, 24 MHR + Functions 11 » Chapter 1 Achievement Check 20. On Earth, the time, ¢, in seconds, taken for an object to fall from a height, h, in metres, to the ground is given by the formula 1h) = \) 3. On the moon, the formula changes to (h) = V7 a) Express each relation using mapping notation. b) Determine the domain and the range of each relation. ©) Is each relation a function? Explain. 4) Graph both relations on the same set of axes. Compare the graphs and describe any similarities or differences. €) Determine the difference between the time it takes for an object to fall from a height of 25 m on Earth and the time it takes on the moon. Justify your answer. @ Extend 21. The initial velocity, v, in kilometres per hour, of a skidding car can be determined from the length, d, in metros, of the skid mark made by using the relation v(d) = 12.6Vd + 8. a) Determine the domain and the range of the relation. b) Graph the relation. ©) [Is the relation a function? Justify your answer. 22, Math Contest Given f(x) = f(x +1) +3 and f(2) = 5, what is the value of f(8)? AS Bil c 20 D -13 23, Math Contest Civon fla) + 2g(x) = 12x* + 3x + 8 and 2flx) + 3g(x) = 18x? + 6x + 13. find the value of f(2) + g(3). 24, Math Contest f(x) is a linear function. Given f(f(3)) = 2 and f(f(2)) = 1, what is the value of f(0)? Al Bo ¢-05 D11.3 Maximum or Minimum of a Quadratic Function Some bridge arches are defined by quadratic functions, Engineers use these quadratic functions to determine the maximum height or the minimum clearance under the support of the bridge @ variety of points. They can give this information to the bridge builders. A quadratic function can be written in a number of forms. Each form has differont advantages. In all forms, a determines the direction of opening iy and the shape. + From the standard form, f(x) = ax‘ + bx + c, the y-intercept can be 2 identified as c. ae + From the factored form, f(x) = a(x ~ r(x ~ s), the x-intercepts can be a identified as rand s. * From the vertex form, y = a(x — h)? + k, the coordinates of the vertex can be identified as (h, K). If a is positive, the minimum value is k. If a is nogative, the maximum value is k Tools nsipg ti — SEE —— + graphing calculator fe How can you connect different forms of the same quadratic function? * grid paper 1. Graph each pair of functions. a) fx) = (x + 2)" + 3 and f(x) = x8 + ax + by f(x) = (x + 3)* — 4 and f(x) = x + 6x + Le) fx) = 2x ~ 3) + 4 and f(x) = 2x" — 12x + 22 4) fx) = 3(x — 1)? — 7 and fw) = 3x¢ — 6x — 4 2. Why are the graphs of the functions in each pair the same? write the first 3. How can you equation in each pair in the form of the second equi 4, How can you rewrite the second equation in each pair in the form of the first equation? 5. Reflect How can you use a graph to verify that two quadratic functions in difforent forms represent the same function? If you are using a graphing calculator, is it enough to observe that the graphs look the same on your screon? Explain, 1.3 Maximum or Minimum of a Quadratic Function * MHR 25.Connections Completing the square Is part of a process by which a quadratic function in standard form can be arranged into vertex form, y= ox AP + k You learned this technique ingrade 10. To convert a quadratic function from standard form to vertex form, you can use the technique of completing the square. €xample 1 SSE — —E—E—_E => Find the Vertex by Completing the Square Find the vertex of each function by completing the square. Is the vertex a minimum or a maximum? Explain. a) fix) =x + 5x +7 &) (x) = —Pe + ox +5 Solution a) f(x) — x8 + Sx +7 = x + ox t (3) — [3] + 7 aaanatr ne cottcen of x squared to make ‘the first tree terms a perfect square = («+ Sf - 8 trina Subtract he same amount. [3 so 1-2 nl ahd ‘the value of the function does not change. ‘The vertex is at [- 5, 3) ‘This is a minimum because a is 1, a positive value, so the parabola opens upward. by fx) = ~2 + 8x + 5 Factor aut the coefficient of doe - 12) +5 3 oe — 12x + 36 ~ 36) +5 Add and subact 6° = 36 to make a perfect square tinomia 21x — 6 - glx — 6 — 36] + 5 ~2er- oF + 2a +5 3 xpaq-24 2 gy Gx — 6 + 29 The vertex is at (6, 29). This is a maximum because the value of a is negative, indicating that the parabola opens downward. 26 MHR * Functions 11 + Chapter 1stges “Sr = — Tools * graphing calculator How can you use partial factoring to find a minimum or a maximum? or 1. Graph the function f(x) = 2x? + 4x. * grid paper 2. How many x-intercepts does this function have? 3. Use the x-intercepts to find the vertex of the parabola. 4. Graph the functions g(x) = 2x? + 4x + 2 and A(x) = 2x* + 4x +5 0n the same set of axes as fx). 5. How many x-intercepts does g(x) have? h(x)? 6. Describe how to find the vertex of the now parabolas, g(x) and A(x), based on the vortex of the original parabola, f(x). Reflect Using your answer from step 6, suggest a method that can be used to find the maximum or minimum of a parabola of the form f(x) = 20 + 4x + k for any value of k €xample 2 SE — ——E— SSS Use Partial Factoring to Find the Vertex of a Quadratic Function Find the vertex of the function y = 4x — 12x + 3 by partial factoring, Is the vertex a minimum or a maximum value? Explain. Solution Work with the function y = 4x* ~ 12x to find the x-coordinate of the vertex, since the x-coordinate of the vertex of y = 4x* ~ 12x + 3 will be the same. y= 4x(x — 3) For y = 0: Connections 0 = Ax(x ~ 3) Thevertex of a cquaseatic function 4x=Oorx-3=0 Use the zero principle If AB= 0, then either A= Or 8=0. | ion the line of symmetry, which is halfway between the intercepts, x=Oor x=3 These give the x-intercepts of the ‘unction y= 4x° - 12x. ‘The average of these two x-intercepts will give the x-coordinate of the vertex for y = 4x¢ = 12x and y = 4x* = 12x + 3: O+3_3 2 2 1.3 Maximum or Minimum of a Quadratic Function * MHR 27Connections dy part ecarng fx) = ox? + bx + kan be expressed as F Hacer +e Thistsa family of ee ren f To find the y-coordinate of the vertex, substitute x = 3 into y= 4x — 12x +3. al ™{a]+s (g]-18 +9 o- 149 ‘The vertex of the function y = 4x¢ — 12x + 3 is at (3 minimum, because the value of a is positive. Example 3 SSE EEE_=___ >= Solve a Problem Involving a Minimum or a Maximum Rachel and Ken are knitting scarves to sell at the craft show. The wool for each scarf costs $6. They were planning to sell the scarves for $10 each, the same as last year when they sold 40 scarves. However, they know that if they raise the price, they will be able to make more profit, even if they end up selling fewer scarves. They have been told that for every 50¢ increase in the price, they can expect to soll four fewer 6 What selling price will maximize their profit and what will the profit be? Solution Let x represent the number of 50¢ price changes. Since each scarf cost $6 and was sold for $10, the profit was $4 per scarf. ‘As they raise the price, their profit per scarf will bo (4 + 0.5x) for x changes to the price. They will sell 40 — 4x scarves when they make the price change. Profit = profit per scarf x number sold P(x) = (4 + 0.5x)(40 — 4x) = ~2xt + 4x + 160 ‘Method 1: Complete the Square to Determine the Vertex P(x) = -2(8 - 2x) + 160 -2(x? — 2x + 1-1) + 160 = -2(x— 1? +2 + 160 = -2(x ~ 18 + 162 ‘The maximum value of this quadratic function is 162 when x = 1. This means that they will make a maximum profit of $162 if they increase the price once. The selling price is 10 + 0.5(1) or $10.50, 28 MHR + Functions 11 » Chapter 1Method 2: Use Partial Factoring to Determine the Vertex Find the x-coordinate of the vertex of the function Q(x) = —2x* + 4x, knowing that the vertex of P(x) = —2x? + 4x + 160 has the same x-coordinate. QU) = -2x(x = 2) Substitute Q(x) = 0 to find the x-intercepts. 0 = ~2x(x ~ 2) -2x=Oorx-2=0 x=Oor x=2 ‘The x-coordinate of the vertex is x = 1 (the average of 0 and 2). P(t) = -2(1)? + 4(1) + 160 = 162 The vertex of this function is at (1, 162). This means that they will make a maximum profit of $162 if they increase the price once. The selling price is 10 + 0.5(1) or $10.50. Example 4 ENS —E—_————_ Connect Projectiles to Quadratic Functions Jamie throws a ball that will move through the air in a parabolic path due to gravity. The height, h, in metres, of the ball above the ground after t seconds can be modelled by the function h(t) = —4.9f + 40¢ + 1.5. a) Find the zeros of the function and interpret their meaning. b) Determine the time needed for the ball to reach its maximum height. ©) What is the maximum height of the ball? Solution a) * Use the window settings shown. * Graph Y1 = ~4.9x! + 40x + 1.5, Fed Pied poe NY1E-4. 9K2+4ORH1 * Press (2nd ) [CALC] to access the CALCULATE, menu. Connections The zeros of a function are the values of the Independent variable for which the function has value zero. They correspond to the >cintercepts of the graph of the function, 1.3 Maximum or Minimum of a Quadratic Function « MHR 29* Solect 2:zero to find the x-intercepts of the function. See the Use Technology feature at the end of this section for a TNspire™ CAS sang sir i cater aera) iSticomee vara The zeros are approximately ~0.037 and 8.2. The solution t = ~0.037 indicates when, in the past, the ball would have been thrown from ground level in order for it to follow the given path. The solution t = 8.2 indicates when the ball will return to the ground. The ball returns to the ground 8,2 s after Jamis threw it. b) The maximum is midway between the two zeros. So, find the average of the two solutions from part a) 0.037 + 8.2 2 ‘The ball will take approximately 4.1 s to reach its meximum hoight. = 4.0815 9 The maximum height can be found by substituting f = 4.1 into the function. H(Q) = 4.98 + a0t + 1.5 (4.1) = ~4.9(4.1) + 40(4.1) + 1.5 03.13 The ball will reach a maximum height of approximately 83.1 m. This solution can be verified using the sc maximum function on the graphing calculator, (80Wa327_veas.t32659, PeNen Ts 19 © The minimum or maximum value of a quadratic function occurs at the vertex of the parabola. © The vertex of a quadratic function can be found by — graphing ~ completing the square: for f(x) = a(x — h)® + k, the vertex is (h, K) b ~ partial factoring: for flx) = ax{x +b) + &; the x-coordinate of the vertex is 2 ‘© The sign of the coefficient a in the quadratic function fl) = ax? + bx + c or f(x) = a(x — h¥ + k determines y whether the vertex is a minimum or a maximum. if > 0, then the parabola opens upward and has a minimum, a0 Ifa <0, then the parabola opens downward and has a maximum, 30 MHR + Functions 11 » Chapter 1Communicate Your Understanding C1 In one step of completing the square, you divide the coefficient of x by 2 and square the result. Why? How aro the functions f(x) = 4x(x - 8), g(x) = ax(x— 3) + Z, and A(x) = 4x(x— 3) — 1 related? Explain using words and diagrams. €3. Ryan does not understand the concept of partial factoring to determine the vertex. Use the function y = 3x* — 9x ~ 17 to outline the technique for him. @ Practise For help with questions 1 and 2, rejer to Example 1. 1. Complete the square for each function. a) y= + 4x b) fix) = + 7x +11 ©) glx) =x - 3x41 d) y=x@-41x-4 e) fix) =x + 13K +2 f) y=x-9x-9 . Determine the vertex of each quadratic function by compli the vertex is a minimum or a maximum. a) fix) =x? + 10x + 6 ing the square. State if by fx) = 2x* + 12x + 16 ve) f(x) = -3x* + 6x +1 @) fx) = -x8 + 12x — 5 e) f(x) = 3 1) fl) = Bxt + Bx + For help with question 3, refer to Example 2. 3, Use partial factoring to determine the vertex of each function. State if the vertex is a minimum or a maximum, a) f(x) = 3x8 = 6x + 11 b) fx) = = 2x" + Bx — 3 ©) fle) = @) fl) = e) f(x) = 03x" - 3x +6 f) f(x) = -0.2x° ~ 2.8x — 5.4 4. Use Technology Use a graphing calculator to verify your answers to questions z and 3. © Connect and Apply For help with questions 5 and 6, refer to Example 3. V6. An electronics store sells an average of 60 entertainment systems per mon‘h at an average of $800 more than the cost price. For every $20 increase in the selling price, the store sells one fewer system. What amount over the cost price will maximize revenue? St Last year, a banquet hall charged $30 per person, and 60 people attended the hockey banquet dinner. This year, the hall's manager has said that for every 10 extra people that attend the banquet, they will decrease the price by $1.50 per person. What size group would maximize the profit for the hall this year? For help with question 7, renosingsodtovng refer to Example 4. mongol “Siew VE A ball is kicked into Feblem ssh the air and follows cciny ay a path described by h(t) = —4.9F + 6f + 0.6, where tis the time, in seconds, and h is the height, in metres, above the ground. Determine the maximum height of the ball, to the nearest tenth of a metre. 1.3 Maximum or Minimum of a Quadratic Function * MHR 318. The cost, C, in dollars, of fuel per month for Sanjay to operate his truck is given by lv) = 0.0029 — 0.48v + 142, whore v represents his average driving speed, in kilomotres per hour. Find the most efficient speed at which Sanjay should drive his track. 9. Arnold has 24 m of fencing to surround a garden, bounded on one side by the wall of his house, What are the dimensions of the largest rectangular garden that he can enclose? 10. The area shown is to be enclosed by 30 m of fencing. Find the dimensions that will maximize the enclosed area 11, The sum of two numbers is 10. What is the maximum product of these numbers? 12. A function models Reasoning and Previn tho offectivenoss of vet a TV commercial. : seetnegiete After n viewings, Problem ling the effectiveness, €, cometng Aeesiog ise Commencing a) Determine the range for the effectiveness and the domain of the number of viewings. Explain your answers for the domain and range. b) Use either completing the square or partial factoring to find the vertex. Is it a minimum or a maximum? Explain. ©) What conclusions can you make from this function? 4) Graph the function on a graphing calculator to verify your conclusions from part c), 13, All quadratic functions of the form y= 2x? + bx have some similar properties. a) Choose five different values of b and greph each function. b) What are the similar properties? ©) Determine the vertex of each parabola. 4) Find the relationship between the vertices of these parabolas. 32 MHR + Functions 11 + Chapter 1 e 14. 15. 7. 18. Extend A sheet of metal that is 30 cm wide and 6 m long 30" is to be used to make a rectangular cavestrough by bending the shoot along the dottad lines, Not to Scale. ‘What value of x maximizes the capacity of the eavestrough? A ball is thrown vertically upward with an initial velocity of v metres per second and is affected by gravity, g. The height, h, in metres, of the ball after f seconds is given by He) = Lee + vt a) Show that the ball will reach its v maximum height at t = b) Show that the maximum height of the ball will be 2g Math Contest Civen that x? = y" = 2, where x, y, and z are integers, how many different values of z are there for z < 10017 Ao B83 ¢4 Dw Math Contest A fiction of two var ables is defined as fix, y) = a8 + y+ 4x — by +7. What is the minimum value of this function? aA? B-13 ¢-6 DO Math Contest A dog's 15-m-long leash is attached. to a building, The leash is attached 10 m from one corner of the building, Assume that the sides of the building are long ‘enough that the dog cannot g0 around any of the other comers. The {greatest area that the dog can cover, in square metres, is Arson Bp A7OR 10m building Cis, D125"Use Technology Use a TI-Nspire™ CAS Graphing Calculator to Find the Tools Maximum or Minimum and the Zeros of a Quadratic Function + TNspire’™ CAS raphing calculator Jamie throws a ball that will move through the air in a parabolic path due to gravity. The height, A, in metres, of the ball above the ground after t secends can be modelled by the function A(t) = ~4.98 + 40t + 1.5 Connections a) Find the zeros of the function and interpret their meaning. aed on pee 29 is used to model the steps needed to fing the maximum or minimum and the zeros, ‘of a quadratic function using a TINspire™ CAS graphing calculator, b) Determine the time needed for the ball to reack its maximum height, ©) What is the maximum height of the ball? ‘Solution a) Turn on the TI-Nspire™ CAS graphing calculator. * Pross @ and select 6:New Document + Select 2:Add Graphs & Geometry. + Type ~4.9x° + 40x + 1.5 for function ft and press @. + Press @9. Select 4:Window. + Select 1:Window Settings. Set XMin to ~2, XMax to 10, Ymin to ~40, and YMax to 100. Tab down to OK and press @. + Press @ and select 6:Points & Lines. * Select 2: * Press @. * Press @ and then @ to grab the point. Use the cursor keys (the arrows on the NavPad) to move the point along the greph toward the left zero. When you reach the zero, “zero” will appear in a box. Read the coordinates of the zero. It occurs at a time of approximately —0.037 s. Similarly, you can find the right zero at a time of about 8.20 s. On. Move the cursor to the graph and press @. b) To find the maximum height of the ball, move the point toward the maximum on the graph. When you reach the maximum, “maximum” will appear the maximum, {t occurs approximately 4.08 s box. Read the coordinates of atime of 9) The maximum height of the ball is approximately 83.13 m. Use Technology: Use a TINspire™ CAS Graphing Calculator to Find the Maximum or Minimum and the Zeros of a Quadratic Function * MHR 33 ;Skills You Need: Working With Radicals The followers of the Gresk mathematician Pythagoras discovered values that did not correspond to any of the rational numbers. As a result, a new type of number needed to be defined to zepresent these values. These values are called irrational numbers. One type of irrational number irrational number is of the form Vn, where n is not a ‘+ anumber that cannot perfect square. Such numbers are sometimes referred to as radicals, be expressed in the farm 8, whee and b In this section, you will see how to use the operations of addition, ‘neo are integers and b + subtraction, and multiplication with radicals. Investigate ——S— LL L——SS== How do you multiply radicals? 1. Copy and complete the table, Whore necessary, use a scientific calculator to help you evaluate each expression, rounding to two. decimal pla Vax Va= V4 x4 ver x Ver= Ver xet V925 x W795 =) | V225 x 225 = VExv5 Vaxs VaxVn= [Vax Viz x V9= Viex9 ve3 x Viev= | V23x121= 2, What do you notice about the results in each row? 3. What conclusion n you make from your observations? Explain. 4, Reflect a) Make a general conclusion about an equivalent expression for Va x Vb. b) Do you think that this will be true for any values of a and b? Justify your answer, 34 MHR + Functions 11 + Chapter 1‘The number or expression under the radical sign is called the radicand. If the radicand is greater than or equal to zero and is not a perfect square, then the radical is an irrational number, An approximate value can be found using a calculator. In many situations, it is better to work with the exact value, so the radical form is kept. Uso the radical form when an approximate answer is not good enough and an exact answer is needed. Somotimes entire radicals can be simplified by removing perfect square factors. The resulting exprossion is called a mixed radical. €xample 1 —————SSS=== Change Entire Radicals to Mixed Radicals Express each radical as a mixed radical in simplest form. a) V50 by V27 9 V180 Solution Choose 25 2, nat 5 x 10, as 25 is a perfect square factor. (V2s)(V2) Use Vob = Vox Vo, ov2 by V27 =V9x3 =(VollV3) use Vad = Vax Vo. =3V3 9 Vi80 = V36 x 5 =(Va6 vs) Ve or V180 = V9x4x5 (vallvallvs) = (3)(2)V5_ =6V5 radicand + anumber or expression under a radical sign entire radical + aradicalin the form Vn, where n> 0, such as vas oVb, where a # 1 or-1 and b> 0, suchas 3V5 1.4 Skills You Need: Working With Radicals * MHR 35‘Adding and subtracting radicals works in the same way as adding and subtracting polynomials. You can only add like terms or, in this case, like radicals. For example, the terms in the expression 2V’3 + 5V7 do not have the same radical, so they cannot be added, but the terms in the expression 3V'5 + 6V/5 have a common radical, so they can be added: aVs + eV5 = aVi. ] example 2 ————————>SSSSS==== ‘Add or Subtract Radicals Simplify. a) 9V7 ~3V7 b) 4V3 ~ 2V27 9 5Ve +3Vi8 ao) Lae - 2ve + 250 Solution a) 9V7 ~ 3V7 = 6V7 by aV3 - 2V27 = 4V3 - 2V0x3 Simplify V27 first. =4V3 - 2V9 x V3 =4aVa -2x ava =4V3 -6V3 -2V3 0 5V8 + 8V18 = 5V4x2+3V9X2 — First simplify both radicals. sVaV2 + 3V9V2 5x 2Vv2+3x 3V2 = 10V2 + 9V2 = 19V2 a) LVa8 ~ 363 + 2V50 = AVERT - SVORT + 2VBKE = Lavi - AVav7 + BVi8Vi Lx ave 2x aves 2x sve 25 — v7 + Wa 27 + V5 or V7 4 10V2 36 MHR + Functions 11 » Chapter 1Example 3 a ———————__} Multiply Radicals Simplify fully. ay (2Va}laVe] by 2Val4 + 5V3) 9 -7vV2leve - 11) a) (V3 + 52-3) ey (2V2 + aVallev2 - ava) Solution ay (2ValaV6) = (2 x a)lV3 x Ve) Use the commutative property and the associative property. =6V3x6 Multbly coefficients and then multiply = Vis radieands. =6V9OX2 =6x3V2 = 18V2 fi Connections by 2Val4 + 5V3) = 2V3(4) + (2V3}[5V3) Use the cistrbutive property. Recall that =0V3 + 10V5 dle pone. = 8V3 + 1018) Reopoted omy aa + an radicals oN ™ 9 -7ValeVe - 11) 7Va)leVe) - (7V2)-11) = -42V16 + 77V2 (-42y(a) + 772 = 168 + 77V2 a) (V3 + s\l2 - V3) = Vata) + Val-Va) + 5(2) + s{-V3) =2V3 - Vo +10 -5V3 =2V3-3+10-5V3 =-3V3 +7 e) (2V2 + 8Va)l2v2 - 3V3) = ‘2v2) - (ava) (2) — 9(3) Connections Recall the difference of squares: (a byo- 0) = ob ‘The factors in part e) have the same pattern They are called conjugates. 27 ‘Simplify and collect ike terms. 1.4 Skills You Need: Working With Radicals * MHR_ 37°Example 4 SEE —E [SS Solve a Problem Using Radicals “A square-based pyramid has a height of 9 cm. ‘The volume of the pyramid is 1089 cm’, Find. Sem the exact side lergth of the square base, in ‘simplified form. Solution Let x represent the side length of the base. = } X area ofbase x height =, 1089 = (9) 1089 = 3x* Connections ——_— The answer 111V3 cm is exact. An approximate answer can be found using a calculator. To ‘the nearest hundredth, ‘the side length is 19.05 cm. “Only the postive rat is needed hecauice xs a lengthy ‘The exact side length of the square base of the pyramid is 11/3 cm. OTE icy © Va x Vb = Vab for a= 0 and b= 0. © An entire radical can be simplified to a mixed radical in simplest form by removing the largest porfect square from under the radical to form a mixed radical, For example, V50 = V25 X 2 =5V2 © Like radicals can be combined through addition and subtraction. For example, 3V7 + 2V7 =5V7. © Radicals can be multiplied using the distributive property. For example, 4V2(5V3 ~ 3) = 20V6 ~ 12/2 and (V2 - allv2 +1) = Va + V2 -sV2-3 =2-2V2-3 = -2V2- 3B MHR Functions 11 Chapter 1Communicate Your Understanding 1 Marc is asked to simplify the expression V3 ions are unlike, the terms cannot be combined. Is he correct? Explain why or why not. expré V75. He says that since the radical (2 Describe the steps needed to simplify the expression Val2V3 — 4V2), 3 Ann wants to simplify the radical V/108. She starts by prime factoring 108: 108 = 2X2X3X3X3 Rayanne looks for the greatest perfect square that will divide into 108 to produce a whole number. Kayanne finds that this value is 36, Explain why both techniques will result in the same solution. © Practise For help with question 1, refer to the Investigate. 1. Simplify. a) sav) ») Valsv2) 9 Vs(-2V7) Ay sVal-avs) e) 2ValaVva] A -6val-v11) For help with question 2, refer to Example 1. 2. Express each as a mixed radical in simplest form. A) V12 b) V2a2 9 V1a7, a) V20 e) V252 f) V302 For help with questions 3 and 4, refer to :xample 2. 3. Simplify. a) 2V3 - 5V3 + 4V3 b) 11V5 - 4V5 - 5V5 - 6V5 V7 -2V7+-V7 d) 2V2 - 8V5 + 3V2 +4V5 re) Ve ~ 4V2 + 3V6 - V2 Vt) 2V10 ~ V10 - 4V10 + V5 4, Add or subtract as indicated. a) 8V2 - 4V8 + V32, b) 4V18 + 3V'50 + V200 9 V20 - 4Vi2 - Vi125 + 2V3 @) 2V26 + V54 + V150 + 5V7 e) 5V3 ~ V72 + V243 + V8 ) Va4 + Vea + Vo9 + V198 For help with questions 5 to 7, refer to Example 3. 5. Expand and simplify. a) sVel2V3) by -2V2lavia) 9 8Vs(Vi0) 4) 3Vis(-2V3] e) 11ValsV3) Ay -2Vel2V6) 6. Expand. Simplify where possible, a) als - V5) b) Valsv2 + 4V3) 9 Valvé - V3) a) -2V5la + 2V5) e) aV2l2Ve + 3Vi2) f) 3V3l2v7 — 5V2) 7. Expand. Simplify where possible. a (V2 + sllVv2 +5) by (2V2 + allV2 4) 9 (V3 + avails + 5V2) d) (3 + 2V8lV5 - 5) w=) (1 + V5)la - V5) uy [4 = aV7) [V7 +1) 8. Simplify. a) 1V54 - 4Vis0 b) 220 + avai - V125 1 3 28 9 5V8 + 3V50 - 2Vi18 a) 2Vi25 - 2V203 - 3Va5 + 3V46 1.4 Skills You Need: Working With Radicals * MHR 39