Gradient and Intercept April 19, 2023

• Lesson Objective: • To Plot the graphs of

simple linear functions, and also find the equation of a straight line - L5

• Lesson Outcomes• Must: plot graphs

given equations of the line – D

• Should: find equation of line –D/C

• Could: Find the equation given the graph– C

Gradients of straight-line graphs

The gradient of a line is a measure of how steep the line is.

y

x

a horizontal line

Zero gradient

y

x

a downwards slope

Negative gradient

y

x

an upwards slope

Positive gradient

The gradient of a line can be positive, negative or zero if, moving from left to right, we have

If a line is vertical its gradient cannot by specified.

The equation of a straight line

y = mx + cm = gradient of the line. It tells us how steep the line is or the SLOPE of the line

c = intercept on the y axis tells us where the line crosses the y-axis.

y

x3

5

1

4-5 -4 -3 -2 -1

4

3

2

-2

-1

-4

-3

-6

-5

021

6

= Gradient is 1

1= 1

Intercept is 2

y = 1x + 2

= Gradient is 3

1= 3

Intercept is 3

y = 3x + 3

= Gradient is 0.5

1= 0.5

Intercept is 1

y = 0.5x + 1

The gradient and the y-intercept

equation gradient y-intercept

y = 3x + 4

y = – 5

y = 2 – 3x

1

–2

3 (0, 4)

(0, –5)

–3 (0, 2)

y = x

y = –2x – 7

x2

12

(0, 0)

(0, –7)

Finding the gradient from two given points

If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows,

the gradient =change in ychange in x

the gradient =y2 – y1

x2 – x1

x

y

x2 – x1

(x1, y1)

(x2, y2)

y2 – y1

Draw a right-angled triangle between the two points on the line as follows,

Draw a Graph Plot the points and write the equation

Level 7/8 Pack 1

Sketching Graphs

Y = x + 3Y = -x + 2

The sign in front of the x tells you if the gradient is positive or negative.

Positive gradient

Negative gradient

This tells you where the line cuts the vertical line

3

cuts the vertical line at 2

2

1. y = x + 5

2. y = x + 1

3. y = x + 3

4. y = x - 6

5. y = x - 5

6. y = x - 1

7. y = -x + 7

8. y = -x + 3

9. y = -x - 4

10. y = 2x + 5

Starter: sketch the graph of each of the equation.

51.

1

2. 3

3.

-6

4.

-5

5.

-1

6.

3

8.

-4

9.

77.

510.

Linear graphs with positive gradients

Investigating straight-line graphs

Rearranging equations into the form y = mx + c

Sometimes the equation of a straight line graph is not given in the form y = mx + c.

The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.

We can rearrange the equation by transforming both sides in the same way

2y + x = 4

2y = –x + 4

y =–x + 4

2

y = – x + 212

Rearranging equations into the form y = mx + c

Sometimes the equation of a straight line graph is not given in the form y = mx + c.

The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.

Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.

So the gradient of the line is 12

– and the y-intercept is 2.

y = – x + 212

What is the equation?

Look at this diagram:

C

A

B

E

G H

F

D

0 5

5

10-5

10

What is the equation of the line passing through the points

a) A and E

b) A and F

c) B and E

d) C and D

e) E and G

f) A and C?

x = 2

y = 10 – x

y = x – 2

y = 2

y = 2 – x

y = x + 6

x

y

Substituting values into equations

What is the value of m?

To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5.

This gives us: 11 = 3m + 5

6 = 3mSubtracting 5:

2 = mDividing by 3:

m = 2

The equation of the line is therefore y = 2x + 5.

A line with the equation y = mx + 5 passes through the point (3, 11).

Pairs

Matching statements

Exploring gradients