GM Lesson 064 Slope-Intercept Form

Learning Intentions

By the end of this lesson, students will be able to:

• Understand the slope-intercept form .
• Identify the gradient and -intercept from an equation.
• Explain how and affect the graph of a straight line.

Prerequisites

Students should already be able to:

• Substitute values into linear equations.
• Solve simple linear equations.
• Recognise coordinates in the form .
• Understand that a straight-line graph represents a linear relationship.

Key Idea Summary

A straight-line equation can often be written in the form

where:

• is the gradient of the line.
• is the -intercept.
• The -intercept is the value of when .
• A positive gradient means the line rises from left to right.
• A negative gradient means the line falls from left to right.
• A larger absolute value of means a steeper line.

For example, in

the gradient is and the -intercept is .

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction: Recognising form

A linear equation is in slope-intercept form when is by itself on the left-hand side and the right-hand side has the form:

The number multiplying is the gradient.

The constant term is the -intercept.

Worked Example 1: Identifying and

Identify the gradient and -intercept of the line

The equation matches the form

Compare the two equations:

Therefore,

and

So the gradient is and the -intercept is .

This means the graph crosses the -axis at .

Worked Example 2: Negative gradient

Identify the gradient and -intercept of

Compare with

The coefficient of is , so

The constant term is , so

The graph has a negative gradient, so it falls from left to right.

The graph crosses the -axis at .

Worked Example 3: Fractional gradient

Identify and in

Compare with

The coefficient of is , so

The constant term is , so

The graph rises from left to right, but not as steeply as a line with gradient , or .

The graph crosses the -axis at .

Worked Example 4: Rewriting before identifying and

Identify the gradient and -intercept of

This is not written in the usual order, but it can be rearranged as

Now compare with

Therefore,

and

The line has a negative gradient and crosses the -axis at .

Worked Example 5: Understanding the effect of and

Compare the two equations:

and

Both equations have

so the lines have the same gradient.

The first equation has

and the second equation has

So the second line crosses the -axis higher than the first line.

Now compare

and

Both equations have

so both lines cross the -axis at .

The second equation has a larger gradient, so it is steeper.

Understanding Checks

Check 1

For each equation, identify and .

a.

b.

c.

d.

Check 2

State whether each line rises or falls from left to right.

a.

b.

c.

d.

Check 3

Two lines are given:

and

Explain one similarity and one difference between their graphs.

Check 4

Two lines are given:

and

Explain one similarity and one difference between their graphs.

Exercises

Simple Familiar Exercises

Exercise 1

Identify the gradient and -intercept for each equation.

a.

b.

c.

d.

e.

f.

Exercise 2

For each equation, state whether the line rises or falls from left to right.

a.

b.

c.

d.

e.

f.

Exercise 3

Rewrite each equation in the form , then identify and .

a.

b.

c.

d.

e.

f.

Exercise 4

Match each equation to the correct description.

Equations:

a.

b.

c.

d.

Descriptions:

i. Positive gradient and positive -intercept.

ii. Positive gradient and negative -intercept.

iii. Negative gradient and positive -intercept.

iv. Negative gradient and negative -intercept.

Complex Familiar Exercises

Exercise 5

A line has equation

a. State the gradient.

b. State the -intercept.

c. State the coordinates where the line crosses the -axis.

d. Explain whether the graph rises or falls from left to right.

Exercise 6

A line has equation

a. State the gradient.

b. State the -intercept.

c. Explain what the negative gradient tells you about the graph.

d. Explain why this line is steeper than .

Exercise 7

Compare the following two equations:

and

a. What is the gradient of each line?

b. What is the -intercept of each line?

c. Explain how the graphs are similar.

d. Explain how the graphs are different.

Exercise 8

Compare the following two equations:

and

a. What is the -intercept of each line?

b. Which line is steeper?

c. Explain how the value of affects the steepness of the graph.

Exercise 9

A taxi fare is modelled by

where is the cost in dollars and is the distance travelled in kilometres.

a. Identify the gradient.

b. Identify the vertical intercept.

c. Explain what the gradient means in this context.

d. Explain what the intercept means in this context.

Exercise 10

A water tank is being filled. The amount of water in the tank is modelled by

where is the amount of water in litres and is the time in minutes.

a. Identify the gradient.

b. Identify the vertical intercept.

c. Explain what the gradient means in context.

d. Explain what the intercept means in context.

e. Describe how the graph would change if the tank initially contained litres instead of litres.

Homework Problems

Homework 1

Identify and for each equation.

a.

b.

c.

d.

e.

Homework 2

For each equation, state whether the graph rises or falls from left to right.

a.

b.

c.

d.

Homework 3

Compare the two equations:

and

a. State the gradient of each line.

b. State the -intercept of each line.

c. Explain how the graphs are similar.

d. Explain how the graphs are different.

Homework 4

A phone plan is modelled by

where is the monthly cost in dollars and is the number of minutes of calls.

a. Identify the gradient.

b. Identify the vertical intercept.

c. Explain what the gradient means in context.

d. Explain what the intercept means in context.

Homework 5

Write a short explanation of the role of and in the equation

Your explanation should include:

• how affects the graph;
• how affects the graph;
• what happens when is positive;
• what happens when is negative.

Next: GM Lesson 065 Meaning of Gradient and Intercepts