C. Algebra · Linear and Non-Linear Relations

Linear and Non-Linear Relations: Unit Review

A1 A2 C3.1 C3.2 C3.3 C4.1 C4.2 C4.3 C4.4About 40 minutes

Learning goals

I can tell linear from non-linear relations using graphs, tables and equations, and describe their rates of change.
I can represent linear relations in several ways and find slopes, y-intercepts and equations.
I can graph x = k, y = k, x + y = k, x − y = k, ax + by = k, xy = k and their inequalities, and transform y = ax.
I can compare two linear relations, interpret their point of intersection and use models to make predictions.

Words to know

rate of change
Change in y ÷ change in x. For a line it is the slope.

initial value
The value of y when x = 0. For a line it is the y-intercept.

slope
Rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁).

y = mx + b
Slope–y-intercept form: m is the slope, b is the y-intercept (also written y = ax + b).

first differences
Differences between consecutive y-values when x goes up by equal steps. Constant means linear.

point of intersection
The point that satisfies both equations of two relations.

hyperbola
The graph of xy = k: two branches that never touch the axes.

Let's review!

In this unit you learned to tell linear from non-linear relations by their graphs, tables and equations; represented linear relations with models, tables, graphs and equations; connected rate of change to slope and initial value to y-intercept; found equations of lines; graphed x = k, y = k, x + y = k, x − y = k, ax + by = k, xy = k and their inequalities; transformed the line y = ax; compared two linear relations using their point of intersection; and built models from real data to make predictions.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

One linear relation, four representations. The rate of change 2 is the +2 in the table, the slope of the graph and the m in the equation. The initial value 1 is the first row of the table, the y-intercept and the b.

The graphs from Lessons 7 to 9: a vertical line, a line with slope −1, a line drawn from its intercepts, a two-branch hyperbola, and a shaded region with a solid boundary.

How to use this review

Do one set of questions at a time, and write your full working on paper.
Check each answer right away. If you miss more than one question in a set, go back to that lesson's examples and its "More practice" questions.
For every equation you write, check it by substituting a point. For every prediction, ask "Does this make sense?"
Finish with the 10-question quiz at the end.

Relations in Real Life: Linear or Non-Linear?

x: 0, 1, 2, 3 and y: 6, 4.5, 3, 1.5. Linear or non-linear? Growing or shrinking? Rate of change?

A savings account earns interest on its interest, so the amount added grows each year. Is the relation between years and balance linear?

Kai has $20 and saves $10 a week. When will he have $200? Is that interpolation or extrapolation if you only have data for weeks 0 to 6?

Tables, Graphs and Equations: Spotting Linear and Non-Linear Relations

Which are linear? (a) 4x − y = 7 (b) y = 5/x (c) y = 2x³

Make a table for y = x² + x for x = 0, 1, 2, 3 and find the first differences.

The first-differences program runs with ys = [5*x + 1 for x in [0, 1, 2, 3]]. What does it print?

One Relation, Four Representations: Rate of Change and Initial Value

Figures 1, 2, 3 of a toothpick pattern use 5, 9 and 13 toothpicks. Write an equation and find how many toothpicks figure 25 uses.

An electrician charges C = 45h + 60. Direct or partial variation? What do 45 and 60 mean?

Should the points for "number of concert tickets and total cost" be joined?

Slope: Measuring Steepness

Find the slope of the line through (−3, 2) and (1, −6).

Find the slope of the line through (2, 5) and (2, −1).

A road rises 12 m over a horizontal distance of 300 m. What is its slope?

The Equation y = mx + b: Slope and y-Intercept

State the slope and y-intercept of y = −(1/4)x + 9.

Write 6x + 2y = 10 in the form y = mx + b.

Find the x-intercept of y = 4x − 10.

Finding the Equation of a Line and Using It to Solve Problems

Find the equation of the line through (−2, −1) and (2, 7).

x: 3, 6, 9 and y: 10, 16, 22. Find the equation.

3 stacked cups are 14 cm tall and 7 stacked cups are 20 cm tall. Find an equation and the number of cups for a 32 cm stack.

Graphing x = k, y = k, x + y = k and x − y = k

Find the intercepts of x − y = −6.

Find the point of intersection of x + y = 9 and x − y = 1.

Write the equation of the line through (−2, 3) and (−2, 8).

Graphing ax + by = k and xy = k

Find the intercepts and slope of 3x + 5y = 30.

In which quadrants is xy = −12? Is (4, −3) on it?

Where do x + y = 8 and xy = 15 meet?

Inequalities: Graphing Regions

Is (1, 1) in the region 2x + 3y > 6?

Describe the graph of y ≤ −x + 2.

Describe the region xy > 0.

Transforming the Line y = ax: Translations, Reflections and Rotations

Translate y = −3x up 7.

Reflect y = 0.5x in the y-axis.

Rotate y = −2x by 90° about the origin.

Comparing Two Linear Relations: The Point of Intersection

Find the point of intersection of y = 4x − 3 and y = −x + 7.

Gym A costs $50 per month. Gym B costs $200 to join plus $30 per month. When are the costs equal? Which is cheaper for a year?

Do y = 3x + 2 and 6x − 2y = 8 intersect?

Modelling Real Data: Choosing a Linear or Non-Linear Model to Make Predictions

Measured first differences are 4.1, 3.9, 4.0, 4.2. What kind of model fits?

A bike worth $800 loses 25% of its value each year. What is it worth after 2 years? Is the relation linear?

Data were collected for x = 0 to 10. Why is a prediction at x = 100 risky?

Just for fun

You have now met lines that are vertical, horizontal, parallel, perpendicular, translated, reflected and rotated. That is a lot of lines. Luckily, none of them were lines at the grocery store.

Check your understanding

Quiz

Which relation is non-linear?

For x = 0, 1, 2, 3 the y-values are 7, 10, 13, 16. What is the equation?

What is the slope of the line through (−1, 4) and (3, −4)?

What is the y-intercept of 2x + 5y = 20?

Which equation is the line through (0, −3) with slope 4?

What does the graph of y = −5 look like?

Which point is in the region x − y < 2?

y = 3x is translated down 5. What is the image?

Where do y = 2x + 1 and y = −x + 10 intersect?

10.

Bounce heights 90, 63, 44, 31 cm. Which model fits?

For parents and teachers

Which lesson to revisit:

Mixes up linear and non-linear, or misreads graph shapes: Lessons 1 and 2 (Relations in Real Life; Tables, Graphs and Equations).
Swaps the rate of change and the initial value: Lesson 3 (One Relation, Four Representations).
Calculates slope as run over rise, or mixes up the subtraction order: Lesson 4 (Slope).
Reads the slope or y-intercept wrongly from an equation, or rearranges incorrectly: Lesson 5 (The Equation y = mx + b).
Cannot find an equation from two points or a table: Lesson 6 (Finding the Equation of a Line).
Mixes up x = k and y = k, or makes sign errors with x − y = k: Lesson 7.
Has trouble with intercepts of ax + by = k or with the shape of xy = k: Lesson 8.
Shades the wrong side or uses the wrong kind of boundary: Lesson 9 (Inequalities).
Gets the sign wrong in a horizontal translation, or forgets to flip the fraction in a 90° rotation: Lesson 10 (Transforming the Line y = ax).
Finds the intersection but cannot say what it means: Lesson 11 (Comparing Two Linear Relations).
Uses a linear model for percent changes, or trusts predictions far beyond the data: Lesson 12 (Modelling Real Data).

This unit leads straight into Grade 10 mathematics, where students solve linear systems and study quadratic relations. Solid understanding of slope, y-intercept and the point of intersection is the best preparation.