Practice Linear functions
A linear function has the form f(x) = mx + b, with constant slope m and vertical intercept b.
Its graph is a straight line.
Exercises with solutions
A linear function has the form y = mx + b. b is the y-intercept; m is the change in y for each unit increase in x. Calculate two points and draw the straight line. When b = 0, the relationship is proportional.
Exercise 1
For y = 2x + 1, find y when x = 0, 1 and 2.
Show solution and working
1, 3, 5
Substitute: 2 × 0 + 1 = 1, 2 × 1 + 1 = 3 and 2 × 2 + 1 = 5. These points lie on the plotted line.
Exercise 2
Find the slope and y-intercept of y = −3x + 4.
Show solution and working
m = −3; b = 4
The coefficient of x is the slope. The constant 4 is the y-intercept. Each step right reduces y by 3.
Exercise 3
A line passes through (0, 2) and (3, 8). Find its equation.
Show solution and working
y = 2x + 2
The slope is (8 − 2) ÷ (3 − 0) = 2. At x = 0, y = 2, so b = 2.
Exercise 4
A taxi charges €4 plus €2 per kilometre. Write its cost function. Is it proportional?
Show solution and working
K(x) = 2x + 4; not proportional
x is the distance in kilometres and K(x) is the fare in euros. The fixed fee gives b = 4, so the line does not pass through the origin.
Common mistake
Do not swap m and b. Slope is the change in y divided by the change in x. A straight-line relationship is proportional only when b = 0.
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