176e. Gradient and y-Intercept Form

Learning Intentions

• Determine the gradient and -intercept for the Draw of a rule in the form
• Determine the rule for a graph where the -intercept is visible and the gradient can be determined

Pre-requisite Summary

• A straight line can be described by a linear rule
• The gradient measures the slope of a line
• The -intercept is the point where the graph crosses the -axis
• In the rule , represents the gradient
• In the rule , represents the -intercept
• The -intercept has coordinates
• The gradient can be found from two points Use change in over change in
• A line can be sketched or identified from its gradient and -intercept

Worked Examples

Worked Example 1

For the rule , State the gradient and the -intercept.

Worked Example 2

For the rule , state the gradient and the -intercept.

Worked Example 3

A graph crosses the -axis at and rises units for every unit moved to the right. Determine the rule.

Worked Example 4

A graph crosses the -axis at and falls units for every unit moved to the right. Determine the rule.

Worked Example 5

A line passes through the points and . Determine the gradient, the -intercept and the rule.

Worked Example 6

A student says the rule of a graph with gradient and -intercept is . Test the claim and determine the correct rule.

Problems

Problem 1

For the rule , state the gradient and the -intercept.

Problem 2

For the rule , state the gradient and the -intercept.

Problem 3

A graph crosses the -axis at and rises units for every unit moved to the right. Determine the rule.

Problem 4

A graph crosses the -axis at and falls unit for every unit moved to the right. Determine the rule.

Problem 5

A line passes through the points and . Determine the gradient, the -intercept and the rule.

Problem 6

A student says the rule of a graph with gradient and -intercept is . Test the claim and determine the correct rule.

Exercises

Understanding and Fluency

Exercise 1

For each rule, state the gradient and the -intercept.

a)

b)

c)

Exercise 2

For each rule, state the gradient and the -intercept.

a)

b)

c)

Exercise 3

Write the rule in the form for each graph description.

a) gradient , -intercept

b) gradient , -intercept

c) gradient , -intercept

Exercise 4

Write the rule in the form for each graph description.

a) gradient , -intercept

b) gradient , -intercept

c) gradient , -intercept

Exercise 5

Determine the gradient and -intercept, then write the rule.

a) The graph crosses the -axis at and rises unit for every units to the right

b) The graph crosses the -axis at and falls units for every unit to the right

c) The graph crosses the -axis at and is horizontal

Exercise 6

Determine the gradient and -intercept, then write the rule.

a) The line passes through and

b) The line passes through and

c) The line passes through and

Exercise 7

For each graph description, determine the rule.

a) -intercept , gradient

b) -intercept , gradient

c) -intercept , gradient

Exercise 8

For each set of points, determine the rule.

a) and

b) and

c) and

Reasoning

Exercise 9

Explain why the number in is the -intercept.

Exercise 10

Explain why the number in is the gradient.

Exercise 11

A student says the rule has gradient and -intercept . Explain the error.

Exercise 12

Explain why the point where a graph crosses the -axis always has .

Exercise 13

A student sees that a graph crosses the -axis at and rises for every across, then writes . Explain the mistake.

Problem-solving

Exercise 14

A taxi fare graph crosses the -axis at and has gradient . The rule is in the form , where is kilometres and is cost in dollars. Determine the rule.

Exercise 15

A plant starts at height cm and grows cm each week. Let be the number of weeks and be the height in cm. Determine the rule.

Exercise 16

A water tank starts with L and loses L each hour. Let be the number of hours and be the amount of water left. Determine the rule.

Exercise 17

A student reads a graph and sees that it crosses the -axis at and passes through . Determine the gradient and write the rule.

Exercise 18

A line has gradient and crosses the -axis at . Determine the rule and state the coordinates of another point on the line when .

Potential Misunderstandings

• Thinking the gradient and the -intercept are in the wrong order in
• Confusing the coefficient of with the constant term
• Thinking the -intercept is the point where the graph crosses the -axis
• Forgetting that a negative gradient means the line falls from left to right
• Believing that the rule must be guessed rather than built from gradient and intercept information
• Reading the -intercept as a single number but not writing it as the point when needed
• Making sign errors when the gradient or intercept is negative
• Thinking a horizontal line has no -intercept
• Confusing rise over run with run over rise when determining the gradient

Next: 177. Linear Graphs in Real-World Situations