GM Lesson 065 Meaning of Gradient and Intercepts

Learning Intentions

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

• Understand gradient as a rate of change.
• Understand the y-intercept as the value when .
• Interpret gradient and intercepts in simple contexts.

Prerequisites

Students should already be able to:

• Recognise the slope-intercept form .
• Identify as the gradient and as the y-intercept.
• Substitute values of into a linear equation to calculate values of .
• Read simple values from a table or graph.

Key Idea Summary

A linear equation in slope-intercept form is written as:

where:

• is the gradient.
• is the y-intercept.
• The gradient tells us the rate of change.
• The y-intercept tells us the value of when .

In a context, the gradient usually means:

The y-intercept usually means the starting value, fixed amount, or initial value.

For example, if the cost of hiring a bike is modelled by:

then:

• is the number of hours.
• is the total cost in dollars.
• The gradient means the cost increases by $ for each extra hour.
• The y-intercept means the starting cost is $ when .

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

Introduction: Interpreting and in — 5 minutes

Write the equation:

Ask students:

• What is the gradient?
• What is the y-intercept?
• What does the tell us?
• What does the tell us?

Explain that in pure graphing, tells us the line rises units for every unit moved to the right, while tells us the graph crosses the y-axis at .

In a practical situation, and need contextual meaning.

Worked Example 1: Taxi Fare — 8 minutes

A taxi fare is modelled by:

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

Identify the gradient and y-intercept.

The equation is in the form:

So:

and

Interpret the gradient.

The gradient is .

This means the taxi fare increases by $ for each extra kilometre travelled.

Interpret the y-intercept.

The y-intercept is .

This means the taxi fare is $ when .

In context, this is the fixed starting charge.

Calculate the cost for a km trip.

Substitute :

The cost of a km trip is $ .

Worked Example 2: Water in a Tank — 8 minutes

The amount of water in a tank is modelled by:

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

Identify the gradient and y-intercept.

The equation can be compared with:

So:

and

Interpret the gradient.

The gradient is .

This means the amount of water decreases by L each minute.

The negative gradient tells us the relationship is decreasing.

Interpret the y-intercept.

The y-intercept is .

This means there were L of water in the tank when .

In context, this is the starting amount of water.

Calculate the amount of water after minutes.

Substitute :

After minutes, there are L of water left.

Worked Example 3: Comparing Two Phone Plans — 7 minutes

Two phone plans are modelled by:

and

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

Interpret Plan A.

For Plan A:

The y-intercept is .

This means Plan A has a fixed monthly charge of $ .

The gradient is .

This means the cost increases by $ for each minute of calls.

Interpret Plan B.

For Plan B:

The y-intercept is .

This means Plan B has a fixed monthly charge of $ .

The gradient is .

This means the cost increases by $ for each minute of calls.

Compare the plans.

Plan A has the lower starting cost, but the higher call rate.

Plan B has the higher starting cost, but the lower call rate.

So Plan A may be cheaper for low call use, while Plan B may be cheaper for higher call use.

Understanding Checks

Check 1 — 3 minutes

A gym membership is modelled by:

where is the total cost in dollars and is the number of weeks.

Answer:

1. What is the gradient?
2. What does the gradient mean?
3. What is the y-intercept?
4. What does the y-intercept mean?

Expected responses:

1. The gradient is .
2. The cost increases by $ each week.
3. The y-intercept is .
4. There is an initial joining fee of $ when .

Check 2 — 3 minutes

The temperature of a drink is modelled by:

where is the temperature in degrees Celsius and is the time in minutes.

Answer:

1. Is the relationship increasing or decreasing?
2. What does the gradient mean?
3. What does the y-intercept mean?

Expected responses:

1. The relationship is decreasing.
2. The drink cools by C each minute.
3. The drink starts at C when .

Check 3 — 2 minutes

A student says:

“The y-intercept is always where the graph crosses the x-axis.”

Explain the error.

Expected response:

The y-intercept is where the graph crosses the y-axis. It is the value of when .

Exercises

Simple Familiar Exercises

Exercise 1

For each equation, identify the gradient and y-intercept.

a.

b.

c.

d.

Exercise 2

A parking fee is modelled by:

where is the total cost in dollars and is the number of hours parked.

a. Identify the gradient.

b. Interpret the gradient in context.

c. Identify the y-intercept.

d. Interpret the y-intercept in context.

e. Calculate the cost of parking for hours.

Exercise 3

A candle has height:

where is the height in centimetres and is the time in hours.

a. Identify the gradient.

b. Explain why the gradient is negative.

c. Identify the y-intercept.

d. Interpret the y-intercept in context.

e. Calculate the candle height after hours.

Exercise 4

A delivery company charges according to:

where is the total cost in dollars and is the delivery distance in kilometres.

a. What is the fixed charge?

b. What is the cost per kilometre?

c. Find the cost of a km delivery.

d. Explain the meaning of when .

Complex Familiar Exercises

Exercise 5

A swimming pool is being filled. The amount of water in the pool is modelled by:

where is the volume of water in litres and is the time in hours.

a. Interpret the gradient.

b. Interpret the y-intercept.

c. Calculate the volume of water after hours.

d. Explain why the y-intercept is not zero in this situation.

Exercise 6

A company compares two printing plans.

Plan A:

Plan B:

where is the cost in dollars and is the number of pages printed.

a. Interpret the gradient of Plan A.

b. Interpret the y-intercept of Plan A.

c. Interpret the gradient of Plan B.

d. Interpret the y-intercept of Plan B.

e. Which plan has the cheaper cost per page?

f. Which plan has the cheaper fixed charge?

g. Explain which plan might be better for a person who prints many pages.

Exercise 7

A car rental company uses the model:

where is the total rental cost in dollars and is the distance driven in kilometres.

a. Identify and interpret the y-intercept.

b. Identify and interpret the gradient.

c. Calculate the cost of driving km.

d. A customer paid $ . Write and solve an equation to find the distance driven.

Exercise 8

A phone battery percentage is modelled by:

where is the battery percentage and is the number of hours since full charge.

a. Interpret the y-intercept.

b. Interpret the gradient.

c. Calculate the battery percentage after hours.

d. Explain why the model cannot be used forever.

Homework Problems

Problem 1

For each equation, identify the gradient and y-intercept.

a.

b.

c.

d.

Problem 2

A gardener charges according to:

where is the total cost in dollars and is the number of hours worked.

a. Interpret the gradient.

b. Interpret the y-intercept.

c. Calculate the cost for hours of work.

d. Explain why the cost is not $ when .

Problem 3

The value of a machine is modelled by:

where is the value in dollars and is the number of years after purchase.

a. Interpret the y-intercept.

b. Interpret the gradient.

c. Calculate the value after years.

d. Explain what the negative gradient means in this situation.

Problem 4

A streaming service offers two plans.

Plan A:

Plan B:

where is the monthly cost in dollars and is the number of movie rentals.

a. Interpret the gradient of Plan A.

b. Interpret the y-intercept of Plan A.

c. Interpret the gradient of Plan B.

d. Interpret the y-intercept of Plan B.

e. Which plan has the lower fixed monthly cost?

f. Which plan has the lower cost per movie rental?

Problem 5

A tank is draining according to:

where is the number of litres left in the tank and is the time in minutes.

a. Identify the gradient.

b. Interpret the gradient in context.

c. Identify the y-intercept.

d. Interpret the y-intercept in context.

e. Find the amount of water left after minutes.

f. Explain why the model should not be used after the tank is empty.

Next: GM Lesson 066 Constructing Straight-Line Graphs