Algebra

Graphing Linear Relations

Complete the Table of Values

For y = 2x + 1, find the missing y-values.

x = 0, y = ?

x = 2, y = ?

x = −1, y = ?

−1

−3

Match Equation to Gradient

Draw a line to match each equation to its gradient.

y = 3x + 2

y = −x + 5

y = 1/2 x − 3

y = −2x

y = 4

1/2

−2

−1

Identify Gradient & y-intercept

Identify m and c in y = mx + c.

y = 4x − 3. Gradient =

−3

y = −2x + 7. y-intercept =

−2

y = x + 5. Gradient =

y = 3. Gradient =

undefined

Positive, Negative or Zero Gradient?

Sort each line by gradient type.

y = 2x + 1

y = −3x + 5

y = 5

y = 1/2 x − 2

y = −x

y = 0

Positive gradient

Negative gradient

Zero gradient

Find the x-intercept

Set y = 0 and solve.

y = 2x − 6. x-intercept =

−3

y = x + 4. x-intercept =

−4

y = 3x − 9. x-intercept =

−3

Match Description to Equation

Draw a line to match each description to its equation.

Steep positive slope, crosses y-axis at 2

Gentle negative slope, crosses y-axis at 5

Horizontal line at y = −3

Passes through origin with gradient 2

y = −1/2 x + 5

y = 2x

y = −3

y = 4x + 2

Graphing and Interpreting

Show your working.

For y = 3x − 2, complete a table for x = −1, 0, 1, 2. Describe the gradient and y-intercept.

Draw here

A phone plan costs $20/month plus $0.30/minute. Write a linear equation for cost C with m minutes. What does the gradient represent?

Plotting Points from a Table

Use the table for y = −2x + 4 to answer these questions.

When x = 0, y =

−2

When x = 3, y =

−2

When x = −1, y =

−6

The y-intercept of y = −2x + 4 is:

(0, 4)

(0, −2)

(2, 0)

Parallel Lines

Parallel lines have the same gradient. Draw a line to match each equation to its parallel pair.

y = 3x + 1

y = −2x + 5

y = ½x − 3

y = 4

y = ½x + 7

y = 8

y = −2x − 1

y = 3x − 4

Perpendicular Lines

Perpendicular lines have gradients that multiply to give −1. If one gradient is m, the perpendicular gradient is −1/m.

Gradient 2. Perpendicular gradient =

−1/2

1/2

−2

Gradient −3. Perpendicular gradient =

1/3

−1/3

Gradient 1/4. Perpendicular gradient =

−4

−1/4

Real-World Graph Interpretation

Read each graph description carefully and answer the questions.

A graph shows the distance (km) a cyclist travels over time (hours). The line passes through (0, 0) and (3, 60). What is the gradient and what does it represent in context?

Two water tanks are draining. Tank A: y = −10x + 200. Tank B: y = −15x + 300. After how many hours are both tanks empty? Which empties first?

Steeper or Flatter?

Sort these lines from flattest (closest to horizontal) to steepest.

y = 5x + 1

y = ½x − 3

y = 2x + 4

Flattest

Middle

Steepest

Graphs Around You

Find and create graphs from real-world data.

1Record the temperature outside every hour for 6 hours. Plot time on the x-axis and temperature on the y-axis. Is the relationship roughly linear? What is the approximate gradient?
2Find a mobile phone data plan that charges per GB. Write the equation relating cost to data used. Plot the line. What does the y-intercept represent?
3Look up the distance–time table for a train between two cities. Plot the points. Is the speed constant (straight line) or does it vary?

Gradient from Graph Description

Use the rise and run from the description to find the gradient.

A line rises 6 units over a run of 2 units. Gradient =

1/3

A line falls 4 units over a run of 8 units. Gradient =

−1/2

1/2

−2

A horizontal line. Gradient =

undefined

A vertical line. Gradient =

undefined

Find the Equation from Two Points

Find the gradient, then use it to write the equation of the line. Show all working.

Points (0, 4) and (2, 10).

Points (1, 3) and (4, −3).

Points (−2, 5) and (2, 1).

Does the Point Lie on the Line?

Substitute the point into the equation and check.

Does (2, 7) lie on y = 3x + 1?

Yes: 3(2)+1 = 7 ✓

No: 3(2)+1 = 5

Cannot tell

Does (−1, 4) lie on y = 2x + 6?

Yes: 2(−1)+6 = 4 ✓

No: 2(−1)+6 = 5

Cannot tell

Does (3, 3) lie on y = −x + 5?

No: −3+5 = 2 ≠ 3

Yes: −3+5 = 3 ✓ (wait, it does!)

Cannot tell

Does (0, 0) lie on y = 4x − 1?

No: 4(0)−1 = −1 ≠ 0

Yes: 4(0)−1 = 0

Cannot tell

x-Intercept and y-Intercept from an Equation

For each equation: (a) find the y-intercept (set x = 0), (b) find the x-intercept (set y = 0).

y = 2x − 6. Find x-intercept and y-intercept.

y = −3x + 12. Find both intercepts.

3x + 2y = 12. Find both intercepts by substituting x = 0 and y = 0.

TipThe intercepts are the two easiest points to plot on a graph — knowing both gives a quick way to sketch any linear equation.

Match y = mx + c Features

Sort each equation by whether it has a positive y-intercept, zero y-intercept, or negative y-intercept.

y = 2x + 5

y = 3x

y = −x − 4

y = ½x

y = 4x + 1

y = x − 3

Positive y-intercept

Zero y-intercept (passes through origin)

Negative y-intercept

Simultaneous Equations — Graphical Method

Graph both equations on the same axes and find the point of intersection.

Graph y = x + 2 and y = 3x − 2 on the same axes for x = 0, 1, 2, 3. Find the intersection point.

Draw here

What is the significance of the intersection point? Write one sentence explaining what it means in terms of both equations.

TipThe intersection point is the solution to both equations at once — a concept that becomes very important in Year 9 and 10 algebra.

Finding the Intersection of Two Lines

Set the two equations equal to each other and solve for x, then find y.

y = 2x and y = x + 3. At intersection, 2x = x + 3. So x = ?

If x = 3 in y = 2x, then y =

Check: does (3, 6) satisfy y = x + 3?

Yes: 3 + 3 = 6 ✓

No: 3 + 3 = 5

Cannot tell

Real-World Graph Modelling

Write a linear equation for each situation, then use it to answer the questions.

A taxi charges $3.50 flag fall plus $2.20 per km. Write an equation for cost C in terms of km travelled k. How much for a 15 km trip? How far can you travel for $25?

A heating system raises a room from 12°C at a rate of 3°C per hour. Write a temperature equation. When will the room reach 24°C? When will it reach 30°C?

Sort Linear Equations by Gradient Steepness

Sort from least steep (smallest |m|) to steepest (largest |m|) gradient. Ignore sign — steepness uses absolute value.

y = 4x + 1 (|m| = 4)

y = ½x − 3 (|m| = 0.5)

y = −x + 2 (|m| = 1)

y = −3x − 1 (|m| = 3)

Least steep

Second

Third

Steepest

Gradient as a Rate of Change

Interpret the gradient in each context. Show full working.

A water tank drains so that y = −8x + 400 where y is litres remaining and x is hours. What is the gradient and what does it mean? When will the tank be empty?

Monthly savings: S = 150m + 200 where S is total savings and m is months. What is the gradient and what does it represent? After how many months will savings reach $2000?

Extended Investigation: Exploring y = mx + c

Use graph paper or Desmos to investigate.

Graph y = x, y = 2x, y = 3x, and y = ½x on the same axes. Describe how changing m affects the graph.

Graph y = x + 1, y = x + 3, y = x − 2 on the same axes. Describe how changing c affects the graph.

Write a general rule: what does m control? What does c control? How can you tell from the equation alone whether a line goes uphill or downhill?

TipThis investigation is ideal for Desmos (desmos.com) — type y = mx + c and use sliders. Let your child drive the exploration and report their findings.

Plotting Linear Graphs

Plot each line on a separate set of axes from x = −3 to x = 3.

Plot y = 2x − 1. Make a table of values first.

Draw here

Plot y = −x + 3. Make a table of values first.

Draw here

TipPlot at least three points: the y-intercept and two others found by substituting x values.

Finding the Equation of a Line

Write the equation of each line in y = mx + c form.

A line passes through (0, 2) with gradient 3.

A line passes through (0, −4) with gradient −1/2.

A line passes through (1, 5) and (3, 9). Find gradient then equation.

TipTo find the equation: identify the y-intercept (where line crosses y-axis), then find gradient (rise/run).

Reading Graphs

Answer questions about y = 2x + 1.

y-intercept:

(0, 1)

(0, 2)

(0.5, 0)

x-intercept (where y = 0):

(−0.5, 0)

(0, 0.5)

(−1, 0)

Point on the line when x = 4:

(4, 9)

(4, 6)

(4, 7)

Parallel, Perpendicular, or Neither?

Two lines are parallel if they have the same gradient. Perpendicular if gradients multiply to −1.

y = 2x + 1 and y = 2x − 5

y = 3x + 1 and y = −(1/3)x + 2

y = 4x and y = −4x

y = x + 2 and y = −x + 2

y = (1/2)x and y = (1/2)x + 7

y = 5x + 3 and y = 2x + 3

Parallel

Perpendicular

Neither

Linear Graphs in Real Contexts

Answer each real-world graphing question.

A plumber charges a $80 call-out fee plus $65 per hour. Write an equation for total cost C in terms of hours h. What does the gradient represent? What does the y-intercept represent?

A car's fuel level starts at 60 L and decreases by 8 L every 100 km. Write an equation for fuel F in terms of distance d (in hundreds of km). When does the tank run dry?

TipReal-world linear graphs have meaningful gradients (rates of change) and intercepts (starting values).

Gradient Between Two Points

Use gradient = (y₂ − y₁) / (x₂ − x₁). Show working.

Find the gradient between (2, 5) and (6, 13).

Find the gradient between (−1, 4) and (3, −4).

A line passes through (0, 3) and (4, 11). Find the gradient and write the equation.

TipThe gradient formula is equivalent to rise/run — label the points (x₁, y₁) and (x₂, y₂) before substituting.

Intersecting Lines

Find the point of intersection of two lines algebraically.

Lines: y = 2x + 1 and y = x + 4. Set 2x + 1 = x + 4 and solve for x, then find y.

Check your answer by substituting into both equations.

TipAt the point of intersection, both x and y values satisfy BOTH equations. Set them equal.

Tables of Values

Complete the table of values, then plot the line.

y = −3x + 6. Complete the table for x = −1, 0, 1, 2, 3. Plot and state the gradient and y-intercept.

Draw here

y = ½x − 2. Complete the table for x = −2, 0, 2, 4. Plot and state the gradient and y-intercept.

Draw here

TipA table of values systematically generates points to plot — always include the y-intercept.

Gradient from Real-World Data

Calculate gradient from a table of real data.

A taxi charges: 0 km = $4, 5 km = $16.50, 10 km = $29. Is this relationship linear? Find the gradient (cost per km). Write the equation.

A phone battery: after 1 hour at 90%, after 3 hours at 65%, after 5 hours at 40%. Find the gradient (% per hour). How long until flat?

TipGradient (rate of change) has real units in context — e.g. dollars per hour, km per litre.

Finding x- and y-intercepts

Find both intercepts for each line.

y = 3x − 9. Find x-intercept and y-intercept.

y = −4x + 8. Find x-intercept and y-intercept.

Use the intercepts to sketch each line.

Draw here

Tipx-intercept: set y = 0. y-intercept: set x = 0.

Interpreting Steepness

Compare the steepness of lines.

Rank these gradients from least steep to steepest: m = −3, m = 0.5, m = −0.2, m = 2. Explain your reasoning.

Two parallel roads have slopes (gradients) of 0.08 and 0.05. Which is steeper? What does a gradient of 0.08 mean in terms of rise and run?

TipThe magnitude (size) of gradient determines steepness. The sign determines direction.

Break-Even Analysis

Use linear graphs to find break-even points.

A candle business has costs: $200 setup + $5 per candle. Revenue: $12 per candle. Write cost equation C = f(x) and revenue equation R = f(x). At what number of candles do costs equal revenue (break-even)?

TipBreak-even is where the income line meets the cost line on a graph — a key business concept.

Types of Linear Equations

Sort each equation by type.

y = 3x − 2

y = −x + 5

y = 4

x = 7

y = 0.5x

y = −7

Positive gradient

Negative gradient

Zero gradient (horizontal)

Undefined gradient (vertical)

Predicting Values from an Equation

Use the equation to predict values. Show working.

A linear equation: y = 5x − 3. Find y when x = 10. Find x when y = 47.

Monthly savings: S = 120m − 45 where m = months. How much saved after 8 months? When will savings reach $855?

TipSubstituting into an equation to predict values is a fundamental algebraic skill.

Reflection

Summarise your learning about linear graphs.

Explain what m and c represent in y = mx + c using a real-world example.

Describe three different methods to draw a linear graph. Which method do you prefer and why?

Draw here

TipA clear summary now makes revision much easier later.

What to do next

Algebra

Linear Equations & Inequalities

Solve linear equations with rational solutions and one-variable inequalities