FUNCTIONS(GRAPHS OF FUNCTIONS) WORKSHEET
1. The diagram below shows the graph of the function y = f(x), defined for all x where b > a > 0. ,
1 Consider the function g(x) = f ( x a) b .
(a)
Find the largest possible domain of the function g.
(2)
IB Mathematics Higher Level Worksheet by Dileep
(b)
On the axes below, sketch the graph of y = g(x). On the graph, indicate any asymptotes and local maxima or minima, and write down their equations and coordinates.
(6) (Total 8 marks)
2.
The quadratic function f(x) = p + qx x2 has a maximum value of 5 when x = 3. (a) Find the value of p and the value of q.
(4)
(b)
The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.
(2) (Total 6 marks)
IB Mathematics Higher Level Worksheet by Dileep
3.
The diagram shows the graph of y = f(x). The graph has a horizontal asymptote at y = 2.
(a)
1 Sketch the graph of y = f ( x) .
(3)
(b)
Sketch the graph of y = x f(x).
(3) (Total 6 marks)
8x 4. Sketch the graph of f(x) = x + x 9 . Clearly mark the coordinates of the two maximum points and the two minimum points. Clearly mark and state the equations of the vertical asymptotes and the oblique asymptote.
(Total 7 marks)
2
IB Mathematics Higher Level Worksheet by Dileep
5.
a+ x The graph of y = b + cx is drawn below.
(a)
Find the value of a, the value of b and the value of c.
(4)
IB Mathematics Higher Level Worksheet by Dileep
(b)
b + cx Using the values of a, b and c found in part (a), sketch the graph of y = a + x on the axes below, showing clearly all intercepts and asymptotes.
(4) (Total 8 marks)
6.
(a)
Express the quadratic 3x2 6x + 5 in the form a(x + b)2 + c, where a, b, c
.
(3)
(b)
Describe a sequence of transformations that transforms the graph of y = x2 to the graph of y = 3x2 6x + 5.
(3) (Total 6 marks)
7.
The diagram shows the graphs of a linear function f and a quadratic function g.
5
IB Mathematics Higher Level Worksheet by Dileep
f On the same axes sketch the graph of g . Indicate clearly where the x-intercept and the asymptotes occur.
(Total 5 marks)
IB Mathematics Higher Level Worksheet by Dileep
x
8. (a) Let a > 0. Draw the graph of y =
a 2 for a x a on the grid below.
(2)
(b)
Find k such that
a x dx = k 2
a x dx 2 .
(5) (Total 7 marks)
9.
Let f be a function defined by f(x) = x arctan x, x (a) Find f(1) and f( 3 ).
(2)
(b)
Show that f(x) = f(x), for x
.
(2)
(c)
< f ( x) < x + 2 , for x Show that x 2
.
7
IB Mathematics Higher Level Worksheet by Dileep
(2)
(d)
Find expressions for f(x) and f(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.
(8)
(e)
Sketch a graph of f, showing clearly the asymptotes.
(3)
(f)
Justify that the inverse of f is defined for all x
and sketch its graph.
(3) (Total 20 marks)
10.
(a)
The graph of y = ln(x) is transformed into the graph of y = ln(2x + 1). Describe two transformations that are required to do this.
(2)
(b)
Solve ln(2x + 1) > 3 cos (x), x [0, 10].
(4) (Total 6 marks)
11.
5 Consider the graphs y = e and y = e sin 4x, for 0 x 4 .
x x
(a)
5 On the same set of axes draw, on graph paper, the graphs, for 0 x 4 . Use a scale of 1 cm to 8 on your x-axis and 5 cm to 1 unit on your y-axis.
(3)
IB Mathematics Higher Level Worksheet by Dileep
(b)
n Show that the x-intercepts of the graph y = ex sin 4x are 4 , n = 0, 1, 2, 3, 4, 5.
(3)
(c)
Find the x-coordinates of the points at which the graph of y = ex sin 4x meets the graph of y = ex. Give your answers in terms of .
(3)
(d)
(i)
Show that when the graph of y = ex sin 4x meets the graph of y = ex, their gradients are equal.
(ii)
Hence explain why these three meeting points are not local maxima of the graph y = ex sin 4x.
(6)
(e)
(i)
Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima
5 of y = e sin 4x for 0 x 4 . You do not need to show that they are maximum values, but the values should be simplified.
x
(ii)
Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r.
(7) (Total 22 marks)
12.
The real root of the equation x3 x + 4 = 0 is 1.796 to three decimal places. Determine the real root for each of the following. (a) (x 1)3 (x 1) + 4 = 0
(2)
IB Mathematics Higher Level Worksheet by Dileep
(b)
8x3 2x + 4 = 0
(3) (Total 5 marks)
13.
A tangent to the graph of y = ln x passes through the origin. (a) Sketch the graphs of y = ln x and the tangent on the same set of axes, and hence find the equation of the tangent.
(11)
(b)
x Use your sketch to explain why ln x e for x > 0.
(1)
(c)
Show that xe ex for x > 0.
(3)
(d)
Determine which is larger, e or e.
(2) (Total 17 marks)
14.
Find the values of k such that the equation x3 + x2 x + 2 = k has three distinct real solutions.
(Total 5 marks)
IB Mathematics Higher Level Worksheet by Dileep
10
15.
(a)
Sketch the curve f (x) = |1 + 3 sin (2x)|, for 0 x . Write down on the graph the values of the x and y intercepts.
(4)
(b)
By adding one suitable line to your sketch, find the number of solutions to the equation f (x) = 4( x).
(2) (Total 6 marks)
16.
A system of equations is given by cos x + cos y = 1.2 sin x + sin y = 1.4.
(a)
For each equation express y in terms of x.
(2)
IB Mathematics Higher Level Worksheet by Dileep
11
(b)
Hence solve the system for 0 < x < , 0 < y < .
(4) (Total 6 marks)
17.
The graph of y = f (x) for 2 x 8 is shown.
1 , ( ) On the set of axes provided, sketch the graph of y = f x clearly showing any asymptotes and indicating the coordinates of any local maxima or minima.
(Total 5 marks)
18.
Find the set of values of x for which
0.1x 2 2 x + 3 < log10 x.
12
IB Mathematics Higher Level Worksheet by Dileep
(Total 6 marks)
19.
(a)
Sketch the curve y = ln x cos x 0.1, 0 < x < 4 showing clearly the coordinates of the points of intersection with the x-axis and the coordinates of any local maxima and minima.
(5)
(b)
Find the values of x for which ln x > cos x + 0.1, 0 < x < 4.
(2) (Total 7 marks)
IB Mathematics Higher Level Worksheet by Dileep
13
