174. Intercepts of Linear Graphs

Learning Intentions

• Know that the -intercept is the point on the Draw where the -coordinate is zero
• Understand that the -intercept is the point on the graph where the -coordinate is zero
• Solve the -intercept of a rule by solving an equation
• Find the -intercept of a rule by substituting
• Draw linear graphs by first finding the axes intercepts

Pre-requisite Summary

• A point on a graph is written as an ordered pair
• A point lies on a graph if its coordinates satisfy the rule
• The -axis is the horizontal axis, where
• The -axis is the vertical axis, where
• Substitution can be used to test whether a point lies on a graph
• Solving a linear equation can be used to find an unknown value
• A linear graph is a straight line
• Two points are enough to sketch a straight line

Worked Examples

Worked Example 1

Find the -intercept of the rule .

Worked Example 2

Find the -intercept of the rule .

Worked Example 3

Find both intercepts of the rule .

Worked Example 4

Find both intercepts of the rule .

Worked Example 5

Sketch the graph of by first finding the axes intercepts.

Worked Example 6

Sketch the graph of by first finding the axes intercepts.

Problems

Problem 1

Find the -intercept of the rule .

Problem 2

Find the -intercept of the rule .

Problem 3

Find both intercepts of the rule .

Problem 4

Find both intercepts of the rule .

Problem 5

Sketch the graph of by first finding the axes intercepts.

Problem 6

Sketch the graph of by first finding the axes intercepts.

Exercises

Understanding and Fluency

Exercise 1.

State the coordinate of each type of intercept in general form.

a) the -intercept

b) the -intercept

Exercise 2.

For each rule, find the -intercept by substituting .

a)

b)

c)

Exercise 3.

For each rule, find the -intercept by solving with .

a)

b)

c)

Exercise 4.

Find both intercepts of each rule.

a)

b)

c)

Exercise 5.

Find both intercepts of each rule.

a)

b)

c)

Exercise 6.

Find both intercepts of each rule.

a)

b)

c)

Exercise 7.

Sketch the graph of each rule by first finding the intercepts.

a)

b)

Exercise 8.

Sketch the graph of each rule by first finding the intercepts.

a)

b)

Exercise 9.

Match each rule to its intercepts.

a)

b)

c)

Intercepts:

i) and

ii) and

iii) and

Exercise 10.

A rule has -intercept and -intercept .

Sketch the line.

Reasoning

Exercise 11.

Explain why the -intercept must lie on the -axis.

Exercise 12.

Explain why substituting finds the -intercept.

Exercise 13.

A student says the -intercept of is found by substituting . Explain the error.

Exercise 14.

A student says the point is an -intercept because it lies on an axis. Explain why this is incorrect.

Exercise 15.

Explain why two intercepts are enough to sketch a straight-line graph when both exist.

Problem-solving

Exercise 16.

A taxi fare is modelled by , where is the distance in kilometres and is the total cost in dollars.

Find the -intercept and explain what it means in the context.

Exercise 17.

A line is given by the rule .

Find the intercepts and sketch the graph.

Exercise 18.

A pattern rule is , where is the step number.

Find the intercepts and explain whether the -intercept makes sense in the context of step number.

Exercise 19.

A student wants to sketch the graph of .

Find the -intercept and -intercept, then state the two points to plot.

Exercise 16.

A builder compares a rule for material length, .

Find both intercepts and Use them to sketch the graph.

Potential Misunderstandings

• Thinking the -intercept is where instead of where
• Thinking the -intercept is where instead of where
• Forgetting that an intercept is a point, not just a single number
• Substituting when trying to find the -intercept
• Substituting when trying to find the -intercept
• Solving the equation incorrectly when finding the -intercept
• Reversing the coordinates of an intercept, for example writing instead of
• Thinking every line must have two different intercepts in the first quadrant
• Plotting the intercepts correctly but not drawing a straight line through them
• Forgetting that the intercept method works because points on the axes have one coordinate equal to zero

Next: 175e. Gradient of a Straight Line