The Linear function

1. Investigate the effect of m on the graph of y = ax2. Work with gradient and parallel and perpendicular

lines.3. Investigate the effect of c on the graph of y = ax + q.4. Use linear equations to solve real life problems

Drawing graphs 1

Consider the function y=2x+1

1. TABLE METHOD

Choose values for x and substitute to find the corresponding y-values. Plot the (x;y) coordinate pairs.

x -1 0 1 2

y -1 1 3 5

Drawing graphs 2

Consider the function y=2x+1

2. DUAL INTERCEPT METHOD

Find the value of the x-intercept (let y=0) and plot this point.Find the value of the y-intercept (let x=0) and plot this point

x-int: 0 = 2x+1 y-int: y=2(0)+1 -1=2x = 0+1 -½ =x y = 0

Now connect the two intercepts to form a straight line.

Drawing graphs 3

Consider the function y=2x+1

3. GRADIENT INTERCEPT METHOD

From the equation, determine the y-intercept (c-value)Plot the y-intercept and use the “rise over run” method to use the gradient of the graph to find one other point. Join these points to form a straight line.

y-int = 1… then rise 2 and run 1

Investigate the effect of m on the graph of y = mx

The equation of the straight line graph can be written as:Standard equation: y=ax+qGeneral equation:

The gradient of a line (a):

0ax by c

2 1

2 1

change in vertical distance

change in horizontal distance

y =

xy

=

m

y

x x

Investigate the effect of m on the graph of y = mx

A decreasing function: ( m is negative)

An increasing function: ( m is positive)

A greater m value will have a “steeper slope”

Worked example: find the gradient of the line which passes through (-2;3) and (1;9).

0m

0m

9 3 62

1 ( 2) 3m

Work with gradient and parallel and perpendicular lines.

Use subscript to indicate the gradient of different lines:represents the gradient of line AB and line CD

Parallel and Perpendicular lines:

and AB CDm m

If: AB CD,

then:

1AB CDm m

B

A

D

C

y

x

A

BC

D

y

x

Worked Example:Determine k if the line joining P(5;7) and R(-3;-1) is perpendicular to the line joining A(7;-11) and B(k;-9).

7-(-1) =

5-(-3)

8 =

8 =1

P R

PR

P R

y ym

x x

AB1 m 1

11 91

7

-2 1

7-k 2 7

5

AB PR

A B

AB

A B

m m

y ym

x x k

k

k

Test your knowledgeQuestion 1

Determine k if the line joining A(2; 1) and P(5; 7) is parallel to the Line through

R( k; 6) and T(-3; -2)

Answer

A) k = -3 B) k = 2 C) k = 1 D) k = 4

3. Investigate the effect of q on the graph of y = ax + q.

A line parallel with the y – axis is: x = c i.e. x = 4 and its gradient =0A line parallel with the x – axis is: y = c i.e. y = 2 and its gradient is

undefinedy = c, is a line parallel with the x axis and cut the y – axis at y = cm = gradient and c = the y – intercept.

c<0 c=0 c>0

3. Investigate the effect of a on the graph of y = ax + q.

To determine the y – intercept, put x = 0

To determine the x –intercept, put y = 0

c = 0c

y=3c

x = 4

y y

y

x x

x

To determine the equation of a linear function

Determine the gradient:

if: and

If q (y-int) is given, substitute into your equationIf a co – ordinate pair of one point is given, substitute into the

given equation and solve for q.

2 1

2 1

y ym

x x

If: AB CD,

then:

1AB CDm m

Worked Example:

Determine the equation of a line that passes through (-2;-3) and (-7;-13)

13 ( 3)2

7 ( 2)

2 and uses (-2;-3):

-3=2(-2)+c

c=1

2 1

m

y x c

y x

Test your knowledge

Question 2

Determine the equation of a line that passes through (1; 6) and (-2; 3)

Answer

A) y = x -3B) y = x +5 C) y = - x +3 D) y = -2x +4

4. Use linear equations to solve real – life problems

Example:Mr. Naidoo uses wooden boards as shelves for plant holders. Each

board rests on supports fixed at equal distances along the plank. Mr Flowers finds that if the supports are 50 cm apart, he can load 110 kg on a plank. If the supports are 100cm apart, he can load only 10kg.on the plank.

a) Write down two pairs of coordinates (distance; Load) b) If the relationship between distance in centimeters and load in

kilograms is a linear function, find the equation of the function.c) Make a graphical representation of the function.

4. Use linear equations to solve real – life problems

Solution:a) (50;110) and (100;10)

110 102

50 1002 and use the point: (50; 110)

110=-2(50)+c

c = 210

m

y x c

y = -2x + 210

105

210y

x

Test your knowledge

Question 3

Determine the equation of a line through (-1; 2) and (-3; -2)

Answer

A) y = 3x +4 B) y = - 2x + 5 C) y = 2x – 3 D) y = 2x +4

Test your knowledge

Question 4 Determine the line perpendicular to the line 3y – 2x =4 and passing through (2; -3)

a. y = -3x +4

b. y = x23

c. y = 12

3 x

d. y = 12

3 x

Bibliography

Oxford Mathematics Plus Grade 10. Maths Workshop by Support and Tuition in Mathematics