Introduction to Functions: math practice, Grade 8

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Math practice Grade 8 : Introduction to Functions — Zyro the alien explorer of Planète Maths

22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

1 Function or not? ★★★

Decide whether each set of ordered pairs is a function. Explain.

\(\{(1, 4), (2, 7), (3, 4), (4, 9)\}\)
\(\{(2, 1), (2, 5), (3, 6)\}\)

2 Running a machine ★★★

A machine uses the rule \(y = 4x - 3\). Find the output for the inputs \(0, 2, 5\) and \(-1\).

3 Completing a table ★★★

Complete the table for \(y = -2x + 6\) with inputs \(-1, 0, 1, 3\).

4 Rate and initial value ★★★

Give the rate of change and the initial value of each function.

\(y = 5x + 12\)
\(y = -3x + 9\)
\(y = 0.5x\)

5 True or false? ★★★

True or false? Justify each answer.

An input of a function can have two different outputs.
Two different inputs can have the same output.
The graph of \(y = x^2\) is a straight line.

6 Is the change constant? ★★★

Look at this table. Is the function linear? If yes, find the rate of change.

\(x\)	0	1	2	3
\(y\)	7	10	13	16

7 A circle on the grid ★★★

The circle \(x^2 + y^2 = 25\) passes through the points \((3, 4)\) and \((3, -4)\). Is the circle the graph of a function? Name the test you use.

8 Taxi fare ★★★

A taxi charges a flat fee of 3 dollars plus 2 dollars for each mile. Let \(m\) be the number of miles and \(C\) the cost in dollars.

Write an equation for \(C\).
How much does a 6-mile ride cost?
A ride costs 27 dollars. How long is it?

9 Equation from two points ★★★

A linear function passes through \((2, 7)\) and \((6, 19)\). Find its rate of change, its initial value and its equation.

10 Linear or nonlinear? ★★★

Classify each function as linear or nonlinear.

\(y = 3x - 5\)
\(y = x^2 + 1\)
\(y = 7\)
\(y = x^3\)

11 Which function is greater? ★★★

Function \(f\) is given by the table below. Function \(g\) is given by \(g(x) = 4x - 1\).

\(x\)	0	1	2	3
\(f(x)\)	2	5	8	11

Find the rate of change and initial value of each function.
Which function has the greater value at \(x = 5\)?

12 Draining a tank ★★★

A water tank holds 80 gallons. It drains at a constant 5 gallons per minute. Let \(t\) be the time in minutes and \(V\) the volume in gallons.

Write an equation for \(V\).
What do \(-5\) and 80 mean in this situation?
When is the tank empty?

13 Phone battery ★★★

A phone battery shows 100% at hour 0, 84% at hour 2 and 68% at hour 4. Assume the drop is linear.

Find the rate of change and write the equation.
After how many hours does the battery reach 0%?

14 Temperature conversion ★★★

The rule \(F = 1.8C + 32\) converts degrees Celsius to degrees Fahrenheit.

Convert 0 °C, 25 °C and 100 °C.
Give the rate of change and the initial value. Is the function linear?

15 Cyclist’s trip ★★★

A cyclist rides at a constant speed of 12 miles per hour for half an hour. She then rests for half an hour. Finally she rides for one more hour at 8 miles per hour. Sketch the graph of her distance (in miles) against the time (in hours), and list the coordinates where the graph changes direction.

16 A table that fails ★★★

A student claims this table shows a function.

\(x\)	1	2	2	3
\(y\)	3	5	6	8

Is the student right? How could you change one value to make it a function?

17 Two membership plans ★★★

Plan A costs \(12 + 18x\) dollars after \(x\) months. Plan B costs \(40 + 10x\) dollars.

Which plan costs less after 3 months? After 4 months?
Solve \(12 + 18x = 40 + 10x\). What does the solution tell you?

18 Error analysis ★★★

A student says: “The outputs of \(y = x^2\) keep increasing, so it has a constant rate of change.” Use a table with \(x = 1, 2, 3, 4\) to explain the error.

19 Equation from a graph ★★★

The graph of a linear function passes through \((0, -2)\) and \((4, 6)\).

Find its equation.
At what value of \(x\) does the graph cross the \(x\)-axis?

20 A burning candle ★★★

A candle is 12 inches tall and burns down 1.5 inches per hour. Let \(t\) be the number of hours and \(h\) the height in inches.

Write \(h\) as a function of \(t\).
How tall is it after 5 hours?
For which values of \(t\) does the function make sense?

21 Area of a square ★★★

The area of a square of side \(s\) inches is \(A = s^2\).

Is \(A\) a function of \(s\)? Is it linear?
Complete: side 3 gives area ?, side 6 gives area ?. What happens to the area when the side doubles?

22 Missing coefficients ★★★

A linear function \(f(x) = ax + b\) satisfies \(f(0) = 5\) and \(f(4) = 17\). Find \(a\) and \(b\), then compute \(f(10)\) and find \(x\) when \(f(x) = 50\).