Grade 8

Linear functions: practice problems

A linear function $y = ax + b$ has a constant rate of change. Its graph is a straight line with slope $a$ and $y$ -intercept $b$ .

Basic, Standard, Advanced: Grade 8 · Term 2

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Rate of change and slope

Rate of change

\text{rate of change} = \frac{\text{increase in } y}{\text{increase in } x} = a

For a linear function the rate of change is always $a$ , so the increase in $y$ is $a$ times the increase in $x$ .

Finding the equation of a line

Write $y = ax + b$ and use the conditions to find $a$ and $b$ . Through two points, first find the slope $\dfrac{y_2 - y_1}{x_2 - x_1}$ , then substitute one point. Parallel lines have equal slopes.

Example: through $(1,\ 3)$ and $(3,\ 7)$ the slope is $2$ , and $3 = 2 + b$ gives $y = 2x + 1$ .

Intersections, ranges and areas

The intersection of two lines is the solution of their simultaneous equations. For the range of $y$ , evaluate at both ends of the domain; a negative slope reverses the order.

Areas

For a triangle bounded by two lines and the $y$ -axis, use the side on the $y$ -axis as the base and $|x|$ of the intersection as the height. A line through a vertex and the midpoint of the opposite side halves the area.

Q1BasicGr 8 · Term 2Linear functions

For the linear function $y = 4x + 3$ , as $x$ increases from $0$ to $2$ , find the following.

(1)the rate of change
(2)the increase in $y$

Answer

(1) $4$
(2) $8$

Solution

The rate of change equals the coefficient of $x$ .
$4$
The increase in $x$ is
$2 - 0 = 2$
So the increase in $y$ is
$4 \times 2 = 8$

Q2StandardGr 8 · Term 2Linear functions

Find the slope and the $y$ -intercept of the graph of $5x - 2y = 8$ .

Answer

slope $\frac{5}{2}$
intercept $-4$

Solution

Solve for $y$ .
$-2y = -5x + 8 \quad\Rightarrow\quad y = \frac{5}{2}x - 4$

Q3AdvancedGr 8 · Term 2Linear functions

Lines $\ell: y = x + 3$ and $m: y = -2x - 3$ meet at P, and cross the $y$ -axis at A and B respectively. Find the area of $\triangle \mathrm{PAB}$ (1 unit = 1 cm).

Answer

6\text{ cm}^2

Solution

Find P.
$x + 3 = -2x - 3 \quad\Rightarrow\quad \mathrm{P}(-2,\ 1)$
Base AB and height
$\mathrm{AB} = 3 - (-3) = 6,\quad \text{height} = 2$
Area:
$\frac{1}{2} \times 6 \times 2 = 6$

Q1BasicGr 8 · Term 2Linear functions

Find the equation of the line through $(-1,\ 2)$ and $(3,\ -14)$ .

Q2StandardGr 8 · Term 2Linear functions

For $y = -\frac{1}{3}x - 2$ with $-6 < x \leqq 0$ , find the range of $y$ .

Q3AdvancedGr 8 · Term 2Linear functions

When a mass of $x$ g hangs from a spring, its length is $y$ cm, and $y$ is a linear function of $x$ . With 25 g the length is 24 cm, and with 50 g it is 29 cm.

(1)Express $y$ in terms of $x$ .
(2)Find the length when a 75 g mass hangs from the spring.

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Linear functions: practice problems

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Simultaneous equations

The function y = ax²