The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and...
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![Page 1: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/1.jpg)
The Linear function
![Page 2: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/2.jpg)
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
![Page 3: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/3.jpg)
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
![Page 4: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/4.jpg)
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.
![Page 5: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/5.jpg)
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
![Page 6: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/6.jpg)
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
![Page 7: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/7.jpg)
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
![Page 8: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/8.jpg)
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
![Page 9: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/9.jpg)
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
![Page 10: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/10.jpg)
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
![Page 11: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/11.jpg)
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
![Page 12: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/12.jpg)
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
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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
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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
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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
![Page 16: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/16.jpg)
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.
![Page 17: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/17.jpg)
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
![Page 18: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/18.jpg)
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
![Page 19: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/19.jpg)
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
![Page 20: The Linear function. 1. Investigate the effect of m on the graph of y = ax 2. Work with gradient and parallel and perpendicular lines. 3. Investigate.](https://reader030.fdocuments.in/reader030/viewer/2022032702/56649cac5503460f9496e198/html5/thumbnails/20.jpg)
Bibliography
Oxford Mathematics Plus Grade 10. Maths Workshop by Support and Tuition in Mathematics