Mathematics
Graphs and functions — Topic 2
- 1.
A straight line passes through the points (1, 5) and (4, 17). Find the gradient of the line and write its equation in the form y = mx + c.
[4 marks] - 2.
The curve y = x² − 2x − 8 crosses the x-axis at two points. Find the coordinates of both points by factorising.
[3 marks] · no calculator - 3.
Find the gradient of the line perpendicular to y = 2x + 3.
[2 marks] · no calculator - 4.
A ball is thrown and its height h metres after t seconds is given by h = 20t − 5t². Calculate the time at which the ball reaches its maximum height, and state the maximum height.
[4 marks] - 5.
A function is given by f(x) = 3x − 2. Find f⁻¹(x), the inverse function.
[3 marks] · no calculator - 6.
Given f(x) = x² + 1 and g(x) = 2x, find the composite function fg(x).
[3 marks] · no calculator - 7.
A straight line has equation y = −3x + 6. State its gradient and y-intercept, and find its x-intercept.
[3 marks] · no calculator - 8.
Sketch, in words, how the graph of y = x² is transformed to give the graph of y = (x − 3)² + 2.
[2 marks] · no calculator - 9.
A car's distance-time graph shows it travelling at a constant 15 m/s for 20 seconds, then stationary for 10 seconds. Calculate the total distance travelled and the average speed over the whole 30 seconds.
[4 marks] - 10.
A quadratic graph y = x² − 4x + k touches the x-axis at exactly one point. Calculate the value of k.
[3 marks] · no calculator - 11.
Points A(2, 3) and B(8, 11) are given. Find the midpoint of AB and the length of AB.
[4 marks] - 12.
Find the equation of the line parallel to y = 2x − 5 that passes through the point (3, 4).
[3 marks] · no calculator - 13.
Express y = x² − 6x + 11 in the form (x − a)² + b, and hence state the coordinates of the turning point of the curve.
[3 marks] · no calculator - 14.
Find the coordinates of the points where the line y = 2x − 1 intersects the curve y = x² − 2x − 1.
[4 marks] · no calculator - 15.
The value of a car, $V, after t years is modelled by V = 18000 × (0.85)ᵗ. Calculate the value of the car after 5 years, to the nearest dollar.
[3 marks] - 16.
State the equations of the asymptotes of the graph y = 3/x + 2.
[2 marks] · no calculator - 17.
A speed-time graph shows a car accelerating uniformly from 0 to 20 m/s over 8 seconds, then travelling at a constant 20 m/s for a further 12 seconds. Calculate the total distance travelled.
[3 marks] - 18.
A speed-time graph is a straight line showing speed increasing from 4 m/s to 22 m/s over 6 seconds. Calculate the acceleration.
[2 marks] - 19.
The curve y = x² − 5x + 6 crosses the x-axis at two points. Find the x-coordinates of these points, and state the equation of the curve's line of symmetry.
[3 marks] · no calculator - 20.
A conversion graph is a straight line through the origin, converting miles (x) to kilometres (y). The line passes through the point (5, 8). Find the equation connecting y and x, and use it to convert 20 miles to kilometres.
[3 marks] - 21.
Solve the inequality x² − x − 6 > 0.
[3 marks] · no calculator - 22.
A straight-line graph has x-intercept (6, 0) and y-intercept (0, −4). Find the equation of the line.
[3 marks] · no calculator - 23.
The graph of y = (x − 1)(x + 2)(x − 3) crosses the x-axis at three points. State the coordinates of these three points.
[3 marks] · no calculator - 24.
A distance-time graph for a runner is curved and becomes steeper as time passes. Describe what this shape indicates about the runner's speed.
[2 marks] · no calculator - 25.
A table shows a plant's height (cm) over time: at week 0 it is 5 cm, and at week 4 it is 17 cm. Assuming the growth is linear, find the weekly growth rate and predict the plant's height at week 10.
[3 marks]