GM Lesson 066 Constructing Straight-Line Graphs

Learning Intentions

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

• Construct a straight-line graph from an equation in the form .
• Use the y-intercept and gradient to plot points.
• Draw and label a straight-line graph accurately.

Prerequisites

Students should already be able to:

• Identify the gradient and y-intercept in an equation of the form .
• Interpret gradient as a rate of change.
• Plot ordered pairs on a Cartesian plane.
• Read coordinates from a graph.
• Understand that the y-intercept occurs where .

Key Idea Summary

A straight-line graph can be constructed efficiently from the equation:

where:

• is the gradient.
• is the y-intercept.
• The y-intercept gives the first point: .
• The gradient tells us how to move from one point to another.

The gradient can be interpreted as:

For example:

• If , then , so move up and right .
• If , then , so move down and right .
• If , then move up and right .
• If , then move down and right .

A straight-line graph should include:

• Clearly drawn axes.
• A suitable scale.
• Correctly plotted points.
• A straight line drawn through the points.
• A label for the equation of the line.

Direct Instruction and Worked Examples

Timing Guide

• Introduction and review: minutes
• Direct instruction: minutes
• Worked examples: minutes
• Understanding checks: minutes
• Exercises: minutes

Direct Instruction

To graph a line in the form :

1. Identify the y-intercept .
2. Plot the point .
3. Write the gradient as a fraction.
4. Use the gradient to find at least one more point.
5. Draw a straight line through the points.
6. Label the line with its equation.

The y-intercept is useful because it gives an immediate point on the graph.

If the equation is:

then:

• The y-intercept is .
• Since , from move up and right to get another point.

Worked Example 1: Positive Integer Gradient

Construct the graph of:

Step 1: Identify the gradient and y-intercept

So the y-intercept is .

Step 2: Plot the y-intercept

Plot the point:

Step 3: Use the gradient

Since:

the line rises units for every unit moved to the right.

Starting at :

Step 4: Draw the graph

Plot the points:

, and

Draw a straight line through the points and label it:

Worked Example 2: Negative Gradient

Construct the graph of:

Step 1: Identify the gradient and y-intercept

So the y-intercept is .

Step 2: Plot the y-intercept

Plot the point:

Step 3: Use the gradient

Since:

the line falls units for every unit moved to the right.

Starting at :

Step 4: Draw the graph

Plot the points:

, and

Draw a straight line through the points and label it:

Worked Example 3: Fractional Gradient

Construct the graph of:

Step 1: Identify the gradient and y-intercept

So the y-intercept is .

Step 2: Plot the y-intercept

Plot the point:

Step 3: Use the gradient

The gradient is:

So from the y-intercept, move up unit and right units.

Starting at :

Step 4: Draw the graph

Plot the points:

, and

Draw a straight line through the points and label it:

Worked Example 4: Choosing Points Using a Table

Construct the graph of:

Step 1: Identify the gradient and y-intercept

So the y-intercept is .

Step 2: Use the gradient

Since:

move down and right from the y-intercept.

Starting at :

We can also move in the opposite direction: up and left .

Step 3: Plot the points

Plot:

, and

Step 4: Draw and label the line

Draw a straight line through the points and label it:

Understanding Checks

Check 1

For the equation:

identify:

• the gradient
• the y-intercept
• the first point that should be plotted

Expected responses:

The first point is:

Check 2

For the equation:

complete the sentence:

The line starts at the y-intercept and then moves __________ units and __________ unit to the right.

Expected response:

The line moves down units and unit to the right.

Check 3

For the equation:

explain how to use the gradient to find another point after plotting the y-intercept.

Expected response:

The y-intercept is . Since the gradient is , move up units and right units to get another point.

Check 4

A student says that the graph of should go upward from left to right because the y-intercept is positive.

Explain the mistake.

Expected response:

The y-intercept only tells us where the line crosses the y-axis. The gradient is negative, so the graph should decrease from left to right.

Exercises

Simple Familiar Exercises

Exercise 1

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

a.

b.

c.

d.

Exercise 2

For each equation, write the y-intercept as a coordinate.

a.

b.

c.

d.

Exercise 3

For each equation, state how to move from the y-intercept to find a second point.

a.

b.

c.

d.

Exercise 4

Construct each straight-line graph using the y-intercept and gradient.

a.

b.

c.

d.

Exercise 5

Construct each straight-line graph using the y-intercept and gradient.

a.

b.

c.

d.

Complex Familiar Exercises

Exercise 6

Construct the graph of:

Use at least three plotted points. Label the graph clearly.

Exercise 7

Construct the graph of:

Use the gradient to find points on both sides of the y-intercept.

Exercise 8

A straight line has equation:

a. Write the gradient as a fraction.

b. State the y-intercept as a coordinate.

c. Construct the graph.

d. Explain why the graph increases from left to right.

Exercise 9

The height of a plant after weeks is modelled by:

where is the height in centimetres.

a. Identify the vertical intercept.

b. Interpret the vertical intercept in context.

c. Identify the gradient.

d. Interpret the gradient in context.

e. Construct the graph for .

Exercise 10

A tank contains litres of water and drains at a constant rate of litres per minute.

The amount of water remaining after minutes is modelled by:

a. Identify the gradient.

b. Identify the vertical intercept.

c. Construct the graph for .

d. Explain what the graph shows about the water in the tank.

Homework Problems

Problem 1

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

a.

b.

c.

d.

Problem 2

For each equation, use the y-intercept and gradient to find two more points on the line.

a.

b.

c.

d.

Problem 3

Construct and label the graph of:

Use at least three plotted points.

Problem 4

Construct and label the graph of:

Use at least three plotted points.

Problem 5

A cyclist travels at a constant speed after already travelling kilometres. The distance travelled after hours is modelled by:

where is measured in kilometres.

a. Identify the gradient.

b. Interpret the gradient in context.

c. Identify the vertical intercept.

d. Interpret the vertical intercept in context.

e. Construct the graph for .

Problem 6

A candle has height centimetres and burns down at a constant rate of centimetres per hour. Its height after hours is modelled by:

a. Identify the gradient.

b. Identify the vertical intercept.

c. Construct the graph for .

d. Explain why the gradient is negative.

Problem 7

Two students graph the equation:

Student A starts at and moves down and right .

Student B starts at and moves up and right .

a. Which student is correct?

b. Explain the error made by the other student.

c. Construct the correct graph.

Problem 8

Construct the graphs of the following equations on the same set of axes.

Then answer:

a. What do the three lines have in common?

b. Which line is the steepest?

c. Which line decreases from left to right?

d. Explain how the equations show these features.

Next: GM Lesson 067 Types of Gradients