Sec 3 E-Math Functions & Graphs topical questions

Find the gradient of the line through (0, 2) and (4, 10).

A line has gradient 3 and passes through (2, 5). Find its equation.

Write 2y = 6x − 4 in the form y = ax + b and state the gradient and y-intercept.

The table shows values of y = x² − 2x − 3 for x = −2 to 4. (a) Copy and complete the table. (b) Use the table to state the x-intercepts of the graph. (c) State the coordinates of the turning point.

1. (a)Complete the table.[2]
2. (b)State the x-intercepts.[1]
3. (c)State the turning point.[1]

For the graph of y = (x + 1)(x − 3), state (a) the x-intercepts, (b) the y-intercept, and (c) whether the turning point is a maximum or a minimum, with a reason.

Find the equation of the line through (1, 2) and (3, 10).

For y = x² − 2x − 3, find the values of y at x = −2, 0 and 2, and state whether the turning point is a maximum or a minimum.

A line passes through the points (−2, 1) and (6, 5). (a) Find its gradient. (b) Find its equation. (c) Write down the coordinates of its x-intercept.

1. (a)Find the gradient.[2]
2. (b)Find the equation.[1]
3. (c)State the x-intercept.[1]

A quadratic curve has equation y = 2x² − 8x + 3. (a) Express 2x² − 8x + 3 in the form a(x − p)² + q. (b) State the coordinates and nature of the turning point. (c) Write down the y-intercept of the curve.

1. (a)Complete the square.[3]
2. (b)State the turning point.[1]
3. (c)State the y-intercept.[1]

A curve has equation y = (x − 4)² − 3. (a) State the coordinates of its turning point. (b) Find the x-intercepts, giving your answers correct to 3 significant figures. (c) Describe the sketch (shape and key points).

1. (a)Turning point.[1]
2. (b)x-intercepts to 3 s.f.[2]
3. (c)Describe the sketch.[1]

A curve has equation y = 3 × 2ˣ. (a) Copy and complete the table for x = 0, 1, 2, 3. (b) Describe how the y-values change as x increases. (c) Find the value of x for which y = 96.

1. (a)Complete the table.[2]
2. (b)Describe the growth.[1]
3. (c)Find x when y = 96.[2]

The curve y = 6/x is drawn for positive x. (a) Copy and complete the table for x = 1, 2, 3, 6. (b) Describe what happens to y as x increases. (c) Does the curve touch the x-axis? Give a reason.

1. (a)Complete the table.[2]
2. (b)Describe the behaviour.[1]
3. (c)Axis contact?[1]

A line y = x + 1 and a curve y = x² − 1 are drawn on the same axes. (a) Show that the x-coordinates of their intersection points satisfy x² − x − 2 = 0. (b) Find the coordinates of both intersection points.

1. (a)Show the equation.[2]
2. (b)Find both points.[3]

A bacteria culture numbers P = 500 × 2ᵗ after t hours. (a) Copy and complete the table of P for t = 0, 1, 2, 3. (b) Use the pattern to find when the culture reaches 4 000.

1. (a)Complete the table.[2]
2. (b)Find when P = 4 000.[2]

Tangents are drawn to a curve at two points P and Q. The tangent at P(2, 6) passes through (6, 16). The tangent at Q passes through (1, 2) and (3, 2). (a) Estimate the gradient of the curve at P. (b) Find the gradient of the tangent at Q and explain what it tells you about the curve at Q.

1. (a)Gradient at P.[2]
2. (b)Gradient at Q and interpretation.[3]

A curve has equation y = −x² + 2x + 8. (a) Factorise −x² + 2x + 8 and state the x-intercepts of the curve. (b) Write down the y-intercept. (c) Find the coordinates and nature of the turning point. (d) Describe the shape of the curve.

1. (a)Factorise and give x-intercepts.[2]
2. (b)y-intercept.[1]
3. (c)Turning point.[2]
4. (d)Shape.[1]

The line y = x + 1 and the curve y = x² − 1 are drawn on the same axes. (a) Form an equation in x for their intersection points. (b) Solve it to find the x-coordinates. (c) Write down the coordinates of the intersection points.

1. (a)Form the equation.[1]
2. (b)Solve for x.[2]
3. (c)Coordinates of the points.[2]

A ball is thrown upwards and its height after t seconds is h = 20t − 5t² metres. (a) Copy and complete the table for t = 0, 1, 2, 3, 4. (b) State the maximum height and the time it is reached. (c) At what other time is the ball at 15 m? (d) What feature of the parabola explains your answer to (c)?

1. (a)Complete the table.[2]
2. (b)Maximum height.[2]
3. (c)Other time at 15 m.[1]
4. (d)Reason.[1]

A stone is thrown upwards from 1 m above the ground and its height is h = 1 + 4t − t² metres after t seconds. (a) Write h in the form a − (t − b)². (b) State the greatest height and the time it occurs. (c) Find when the stone lands, correct to 3 significant figures.

1. (a)Complete the square.[3]
2. (b)Greatest height and time.[1]
3. (c)Time of landing (3 s.f.).[2]

Two phone plans charge per month: Plan A costs 15 + 0.5g dollars for g GB of data; Plan B costs a flat $25. (a) Write both cost equations. (b) Find the data amount where the plans cost the same. (c) Which plan is cheaper for 30 GB, and by how much? (d) Interpret the meaning of the constant 15 in the graph of Plan A.

1. (a)Cost equations.[1]
2. (b)Break-even data amount.[2]
3. (c)Cheaper plan at 30 GB.[2]
4. (d)Meaning of the constant.[1]