GCSE Maths Graphs of functions questions by topic

Matching graphs to equations.

1. (a)Which of y = x², y = 2x + 1 and y = 1⁄x is a straight line? Give its gradient and intercept.[2]
2. (b)Which of the three never crosses either axis?[1]
3. (c)State the coordinates of the y-intercept of y = 2x + 1.[1]

Drawing a quadratic.

1. (a)Complete a table for y = x² − 2x − 3 at x = −1, 0, 1, 2, 3.[2]
2. (b)Write down the roots of x² − 2x − 3 = 0 from your table.[2]

Distance–time reading.

1. (a)A journey graph is flat from t = 2 to t = 5 minutes. What is happening?[1]
2. (b)The graph rises 600 m between t = 5 and t = 10 minutes. Find the speed in m/min.[2]
3. (c)Was the speed in part (b) faster or slower than 150 m/min?[1]

Cubic and reciprocal shapes.

1. (a)Sketch y = x³, labelling the origin crossing.[2]
2. (b)Sketch y = 1⁄x, marking which two lines the curve never touches.[2]

A fuel-efficiency graph plots miles against litres.

1. (a)The line passes through (0, 0) and (6, 120). Work out the gradient and say what it means.[2]
2. (b)Work out the miles travelled on 8.5 litres.[2]
3. (c)Work out the litres needed for a 260-mile journey.[1]

Ava cycles to the library and back.

1. (a)She rides 3 km in the first 10 minutes, rests from minute 10 to minute 20, then rides the 3 km home from minute 20 to minute 32. Work out her speed on the way out, in km/min.[2]
2. (b)Work out her speed on the way home, in km/min.[1]
3. (c)Which leg was faster?[1]

Exponential graphs. (Higher)

1. (a)Complete a table for y = 3ˣ at x = 0, 1, 2, 3.[2]
2. (b)State the y-intercept of every curve of the form y = kˣ.[1]
3. (c)Describe how y = (½)ˣ differs from y = 3ˣ.[2]

Trig graphs. (Higher)

1. (a)Write down the values of sin x at x = 0°, 90°, 180°, 270°, 360°.[2]
2. (b)State the period of y = cos x.[1]
3. (c)What happens to tan x at x = 90°?[1]

Transformations of a known curve. (Higher)

1. (a)The point (3, 27) lies on y = x³. Write the coordinates of its image on y = x³ − 9.[2]
2. (b)Write the coordinates of the image of (3, 27) on y = (x + 2)³.[2]

Reflections. (Higher)

1. (a)Describe the difference between the graphs of y = f(x) and y = −f(x).[2]
2. (b)y = x² + 2x has a minimum at (−1, −1). State the coordinates of the maximum of y = −x² − 2x.[2]

Area under a velocity–time graph. (Higher)

1. (a)A car accelerates uniformly from rest to 20 m/s over 10 s, then travels at 20 m/s for 15 s. Sketch the velocity–time graph.[2]
2. (b)Find the distance travelled in the first 10 s.[2]
3. (c)Find the total distance for the whole 25 s journey.[2]

Estimating a gradient of a curve. (Higher)

1. (a)Explain how to estimate the gradient of a curved distance–time graph at a point.[2]
2. (b)A tangent drawn at t = 4 s passes through (0, 2) and (8, 18) in (s, m) coordinates. Estimate the speed at t = 4 s.[3]

Circle and tangent. (Higher)

1. (a)Show that the point (8, 6) lies on the circle x² + y² = 100.[1]
2. (b)Find the gradient of the radius to (8, 6) and hence the gradient of the tangent there.[3]
3. (c)Find the equation of the tangent at (8, 6).[2]

A full graph investigation. (Higher)

1. (a)Sketch y = x², labelling the turning point.[2]
2. (b)On the same axes, sketch y = (x − 3)² + 2, stating the new turning point.[2]
3. (c)Describe the single translation that maps the first curve onto the second.[2]
4. (d)Write down the equation of the axis of symmetry of y = (x − 3)² + 2.[1]

The curve y = 4^x is tabulated and used. (Higher)

1. (a)Complete a table of y = 4^x for x = 0, 1, 2 and 3.[2]
2. (b)Explain why the curve never meets the x-axis.[2]
3. (c)Estimate between which two whole values of x the curve reaches 30.[1]

A curve is shifted and reflected. (Higher)

1. (a)The curve y = f(x) passes through (1, 4), (3, 9) and (5, 2). Write down the corresponding points on y = f(x) + 2.[2]
2. (b)Write down the corresponding points on y = f(x − 3).[2]
3. (c)The point (1, 4) is reflected using y = −f(x). Write down its image.[1]

A tangent is built on a bigger circle. (Higher)

1. (a)Write down the radius of the circle x² + y² = 100.[1]
2. (b)Verify by substitution that C(6, 8) satisfies x² + y² = 100.[2]
3. (c)Work out the equation of the tangent line to the circle at C.[3]

A velocity–time graph forms a trapezium. (Higher)

1. (a)A vehicle accelerates uniformly from 0 to 20 m/s over the first 10 seconds, holds 20 m/s for the next 20 seconds, then decelerates to 0 over the last 20 seconds. Work out the acceleration in the first stage.[1]
2. (b)Work out the total distance travelled.[3]
3. (c)Work out the average speed for the whole 50 seconds.[2]