Question 1:
Graph- Question 1
30
20
10
x axis
0
-15 -10 -5 0 5 10 15
-10
-20
-30
y axis
y=2x y=2x+4 y=2x+6
x y=2x y=2x+4 y=2x+6
-10 -20 -16 -14
-9 -18 -14 -12
-8 -16 -12 -10
-7 -14 -10 -8
-6 -12 -8 -6
-5 -10 -6 -4
-4 -8 -4 -2
-3 -6 -2 0
-2 -4 0 2
-1 -2 2 4
0 0 4 6
1 2 6 8
2 4 8 10
3 6 10 12
4 8 12 14
5 10 14 16
6 12 16 18
7 14 18 20
8 16 20 22
9 18 22 24
10 20 24 26
Question 2)
The graph that I have created shows 3 linear, parallel lines. We know that they linear as it is a
straight line, and it’s parallel as those 3 lines will never meet. These 3 lines share similarities
but are also different from each other.
Similarities:
These lines share the same gradient, 2. We know this as the formula is y=mx+c, where m is
the gradient and c is the y-intercept. This means for every 1 across, you move 2 up. You can
also use the formula x2-x1/y2-y1 to calculate the slope of a line
All three lines don’t cross quadrant 4. Every line crosses quadrant 1,2 and 3.
Each have a positive slope and each point increases by 2 because of the gradient.
All three are linear and parallel. In the formula, all of their powers are equal to 1.
The y-intercepts are constants.
Differences:
They all have different y-intercepts. For line y=2x the y-intercept is (0,0) – it crosses through
the origin. For formula y=2x+4 the y-intercept is (0,4) and for the formula y=2x+6 the y-
intercept is (0,6). To find this, you can use the formula y=mx+c (c is the y-intercept) or see
where the lines crosses the y axis.
They have different x-intercepts. X intercept for y=2x is (0,0). X-intercept for y=2x+4 is -2
and for y=2x+6 is -3.
Question 3:
x y=2x y=2x-3 y=2x-5
-10 -20 -23 -25
-9 -18 -21 -23
-8 -16 -19 -21
-7 -14 -17 -19
-6 -12 -15 -17
-5 -10 -13 -15
-4 -8 -11 -13
-3 -6 -9 -11
-2 -4 -7 -9
-1 -2 -5 -7
0 0 -3 -5
1 2 -1 -3
2 4 1 -1
3 6 3 1
4 8 5 3
5 10 7 5
6 12 9 7
7 14 11 9
8 16 13 11
9 18 15 13
10 20 17 15
Graph 2- Question 3
25
20
15
10
0
x axis
-15 -10 -5 0 5 10 15
-5
-10
-15
-20
-25
-30
y axis
y=2x y=2x-3 y=2x-5
Question 4)
The graph that I have created shows 3 linear, parallel lines. We know that they linear as it is a
straight line, and it’s parallel as those 3 lines will never meet. These 3 lines share similarities
but are also different from each other.
Similarities:
These lines share the same gradient, 2. We know this as the formula is y=mx+c, where m is
the gradient and c is the y-intercept. This means for every 1 across, you move 2 up. You can
also use the formula x2-x1/y2-y1 to calculate the slope of a line
All three lines don’t cross quadrant 2. Every line crosses quadrant 1,3,4.
Each have a positive slope and each point increases by 2 because of the gradient.
All three are linear and parallel. In the formula, all of their powers are equal to 1.
The y-intercepts are constants.
Differences:
They all have different y-intercepts. For line y=2x the y-intercept is (0,0) – it crosses through
the origin. For formula y=2x-3 the y-intercept is (0,-3) and for the formula y=2x-5 the y-
intercept is (0,-5). To find this, you can use the formula y=mx+c (c is the y-intercept) or see
where the lines crosses the y axis.
They have different x-intercepts. X intercept for y=2x is (0,0). X-intercept for y=2x-3 is and
for y=2x+6 is -3.
Question 5:
x y=x-2 y=3x-2 y=5x-2
-10 -12 -32 -52
-9 -11 -29 -47
-8 -10 -26 -42
-7 -9 -23 -37
-6 -8 -20 -32
-5 -7 -17 -27
-4 -6 -14 -22
-3 -5 -11 -17
-2 -4 -8 -12
-1 -3 -5 -7
0 -2 -2 -2
1 -1 1 3
2 0 4 8
3 1 7 13
4 2 10 18
5 3 13 23
6 4 16 28
7 5 19 33
8 6 22 38
9 7 25 43
10 8 28 48
Graph- Question 5
60
40
20
0
x axis
-15 -10 -5 0 5 10 15
-20
-40
-60
y axis
y=x-2 y=3x-2 y=5x-2
Question 6:
Similarities:
All three lines have the same y-intercept. All three lines cross the y-intercept at (0,-2).
You can calculate the y-intercept in different ways. The first and easiest method is to
look at the formula y=mx+c. The c represents what the y-intercept is, and in this case all
three formulas have c represented as -2. You can also use the x and y intercept method.
This is when at the y-intercept, x= 0 and at the x-intercept, y=0.
WORKING OUT:
Y-intercept
X=0
Y= x-2 / Y=0-2 Y= -2
Y= 3x-2 / Y= 3x0-2= -2 Y=-2
Y=5x-2 / Y= 5x0-2= -2 Y=-2
All three lines don’t cross quadrant 2. Every line crosses quadrant 1,3,4.
Each have a positive slope but increase differently depending on their gradient
All three are linear and non-parallel. In the formula, all of their powers are equal to 1.
Differences:
Each line has a different gradient. They do not share the same gradient. The gradient for y=x-
2 is 1. The gradient for y=3x-2 is 3 and the gradient for y=5x-2 is 5. This means for rise/run,
y=5x-2 rises the most and y=x-2 rises the least. You can find this out by using the formula
y=mx+c where m is the gradient. You can also use the formula x2-x1/y2-y1 using the graph
to find the gradient.
Different x-intercepts. X intercept for y=x-2 is (2,0). X-intercept for y=3x-2 is (2/3,0) and for
y=5x-2 is (2/5,0).
Working out:
Y=x-2
0=x-2
X=2
Y=3x-2
0=3x-2
2=3x
X=2/3
Y=5x-2
0=5x-2
2=5x
X=2/5
Question 7:
The gradient for y=x-2 is 1. The gradient for y=3x-2 is 3 and the gradient for y=5x-2 is 5.
This means for rise/run, y=5x-2 rises the most and y=x-2 rises the least. You can find this out
by using the formula y=mx+c where m is the gradient. You can also use the formula x2-
x1/y2-y1 using the graph to find the gradient.
Question 8:
x y=-x+4 y=-2x+4 y=-4x+4
-10 14 24 44
-9 13 22 40
-8 12 20 36
-7 11 18 32
-6 10 16 28
-5 9 14 24
-4 8 12 20
-3 7 10 16
-2 6 8 12
-1 5 6 8
0 4 4 4
1 3 2 0
2 2 0 -4
3 1 -2 -8
4 0 -4 -12
5 -1 -6 -16
6 -2 -8 -20
7 -3 -10 -24
8 -4 -12 -28
9 -5 -14 -32
10 -6 -16 -36
Graph- Question 8
50
40
30
20
10
x-axis
0
-15 -10 -5 0 5 10 15
-10
-20
-30
-40
y-axis
y=-x+4 y=-2x+4 y=-4x+4
Question 9:
The graph that I have created shows 3 linear, non-parallel lines. We know that they linear as
it is a straight line, and it’s not parallel as those 3 lines converge and intersect each other.
These 3 lines share similarities but are also different from each other.
Similarities:
All three lines have the same y-intercept. All three lines cross the y-intercept at (0,4).
You can calculate the y-intercept in different ways. The first and easiest method is to
look at the formula y=mx+c. The c represents what the y-intercept is, and in this case all
three formulas have c represented as 4. You can also use the x and y intercept method.
This is when at the y-intercept, x= 0 and at the x-intercept, y=0.
WORKING OUT:
Y-intercept
X=0
Y= x+4 / Y=0+4 Y= 4
Y= 2x+4 / Y= 2x0+4= 4 Y=4
Y=4x+4/ Y= 4x0+4= 4 Y= 4
All three lines don’t cross quadrant 3. Every line crosses quadrant 1,2,4.
Each have a negative slope
All three are linear and non-parallel. In the formula, all of their powers are equal to 1. We
know that they are non-parallel as each line intersects at their y-intercepts. They all share a
common y-intercept.
All of their gradients are negative numbers - -1,-2, -4
Differences:
Each line has a different gradient. They do not share the same gradient. The gradient for -x+4
is -1. The gradient for y=-2x+4 is -2 and the gradient for y=-4x+4 is -4. You can find this out
by using the formula y=mx+c where m is the gradient. You can also use the formula x2-
x1/y2-y1 using the graph to find the gradient.
Different x-intercepts. X intercept for y=-x+4 is (4,0). X-intercept for y=-2x+4 is (2,0) and
for y=-4x+4 is (1,0).
Working out:
Y=-x+4
0=-x+4
-4=-x
X=(4,0)
Y=-2x+4
0=-2x+4
-4=-2x
X=2
Y=-4x+4
0=-4x+4
-4=- 4x
X=1
Question 10:
Each line has a different gradient. They do not share the same gradient. The gradient for -x+4
is -1. The gradient for y=-2x+4 is -2 and the gradient for y=-4x+4 is -4. You can find this out
by using the formula y=mx+c where m is the gradient. You can also use the formula x2-
x1/y2-y1 using the graph to find the gradient.
Question 11:
x y=x+5 y=-2x-3 y=x^2+5 y=-2x^2-3
-10 -5 17 105 -203
-9 -4 15 86 -165
-8 -3 13 69 -131
-7 -2 11 54 -101
-6 -1 9 41 -75
-5 0 7 30 -53
-4 1 5 21 -35
-3 2 3 14 -21
-2 3 1 9 -11
-1 4 -1 6 -5
0 5 -3 5 -3
1 6 -5 6 -5
2 7 -7 9 -11
3 8 -9 14 -21
4 9 -11 21 -35
5 10 -13 30 -53
6 11 -15 41 -75
7 12 -17 54 -101
8 13 -19 69 -131
9 14 -21 86 -165
10 15 -23 105 -203
Question 12:
There are a lot of different functions presented on this one graph. Every graph is unique and
share similarities with only some of the other graphs.
Similarities:
Y-intercepts
Line y=x+5 and line y=x^2+5 share the same y-intercept. Using the formula y=mx+c (c
being the y-intercept) we can conclude that they both have the y intercept (0,5)
Line y=2x-3 and y=2x^2-3 both have the same y intercept (0,-3).
Quadratic and Linear Equations
Y=x+5 and y=-2x-3 are both linear equations. This is because the power is equal to 1.
Y=x^2+5 and y=-2x^2-3 are both quadratic equations. This is because the highest
power is equal to 2.
Parabolas
Y=x^2+5 and y=-2x^2-3 both form parabolas and do not cross the x-axis. A parabola is
a curve that is symmetrical and forms an almost U-shape.
Horizontal lines
Y=x+5 and y=-2x-3 are both horizontal lines that are not parallel.
Differences:
Quadrants
All 4 lines cross at different quadrants:
y= x+5 crosses quadrants 1,2 and 3
y=-2x-3 crosses quadrants 2,3 and 4
y=x^2+5 crosses quadrants 1 and 2
y=-2x^2-3 crosses quadrants 3 and 4
Slopes
y= x+5 and y=-2x-3 both have different slopes. y= x+5 has a positive slope while y=-2x-3
has a negative slope.
Parabola and horizontal lines
y=-2x^2-3 and y=x^2+5 are both quadratic functions and parabolas while y= x+5 and
y=-2x-3 are linear equations/functions and horizontal lines.
X-intercepts
y=x+5 and y=-2x-3 have different x-intercepts. y=x+5 has an x intercept of (-5,0) while
y=-2x-3 has a y intercept of (-3/2,0)
Y-intercepts
Both parabolas have different y intercepts. y=x^2+5 has a y intercept of (0,5) while y=
-2x^2-3 has a y intercept of (0,-3).
Question 13:
All linear functions take the form of y=mx+b or y=mx+c. Both coefficients m and b represent
functions. Now, m represents the gradient/rise over run. To calculate the gradient you use the
formula x2-x1/y2-y1. For example, for this graph the gradient with working out will be:
y=2x+4
30
25
20
15
10
0
0 2 4 6 8 10 12
We pick two points. (5,15) and (10,25). Then we use the formula to calculate the
gradient.
10-5/25-15 = 2
This means that the gradient is 2. We can verify this by looking at the formula. The
formula is y=2x+5. As I explained before, m (coefficient of x) is equal to 2. Therefore, m
is equal to the gradient.
The gradient is the slope and how steep the line is.
Coefficient b/c represent the y-intercept. This is the easiest to calculate and can be
calculated in two different ways. First of all, in the formula y=mx+c – the c represents
the y-intercept. The y-intercept is where the line crosses the y-axis and can easily be
identified by looking at the graph. The first way to calculate it, without a graph, is to
use the x and y intercept method. At the x intercept, y=0 (in the formula y=mx+c) and at
the y intercept, x is equal to 0 (in the formula y=mx+c).
Now, let’s use the example y=-5x+4. Using the formula to find the y-intercept we must
substitute x for 0.
Y=-5x0+4
Y=0+4
Y=4
Using that method we discovered that y=(0,4)
The other method is to look at the graph and the y-axis. Using this graph we will be able
to find the y-intercept.
y=-5x+4
30
25
20
15
10
x-axis
5
0
-5 -4 -3 -2 -1 -5 0 1 2 3 4 5
-10
-15
-20
y-axis
As you can see the line crosses the y-axis at 4. In this graph the y-intercept Is 4 and in
the formula we used y=-5x+4 (same formula in graph) we discovered that in the formula
y=mx+c, c is the y-intercept.
Question 14:
o 𝑦 = 2𝑥 + 5
o 𝑦 = −4𝑥 + 7
1. Y=2x+5
Well using my previous knowledge I know that the gradient is 2 and the y-intercept is 5.
I also know that this is will be a linear, horizontal line because of its power. I know that
it will be a positive slope as a positive gradient will produce a positive slope. I know that
it will also cross every quadrant except for number 4. This is because in the first graph
with the formula y=2x+4, the line did not cross quadrant 4. These formulas are very
similar but have different y intercepts. I can calculate the x-intercept by using the
formula (y is equal to 0 in the x-intercept)
0=2x+5
-5=2x
x-intercept = (-5/2,0)
2. Y=-4x+7
Using the formula y=mx+c I know that the gradient is -4 and the y intercept is (0,7). I
know that it will also be a linear, horizontal line because of its power. It will be a
negative slope because of its negative gradient. Using the equation y=-4x+4 in question 8
I know that the line will cross every quadrant except quadrant 3. The x-intercept will be
equal to 7/4.
0= -4x+7
-7= -4x = (7/4,0)
Question 15:
In question 11 there were two different types of functions used. The first function used
was a linear function, which has already been identified. The second function was a
quadratic function, a non-linear function. A quadratic function is a function that has
the power of 2. The highest power of a quadratic function is x2. A quadratic function is in
the form of f(x) = ax2 + bx + c. In that formula a, b and c are numbers with a not
equal to zero and the equation of ax^2 + bx + c = 0.
The graph of a quadratic function is a parabola. A parabola can be easily recognised as
“it is a curved line with a conic section which is produced by the intersection of a plane
parallel and a right circular cone to any element of the cone.” A parabola looks like this:
Next time you want to easily recognise a quadratic function you can see if the highest
power is 2 and if the equations forms a parabola.
1. Parabola | mathematics
Encyclopedia Britannica. (2018). Parabola | mathematics. [online] Available at:
https://www.britannica.com/science/parabola [Accessed 8 Jun. 2018].
2. Parabola
En.wikipedia.org. (2018). Parabola. [online] Available at: https://en.wikipedia.org/wiki/Parabola
[Accessed 8 Jun. 2018].
