Non-linear graphs and real-life graphs

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Notes de leçon

Plotting Quadratic Graphs

A quadratic graph has equation y = ax² + bx + c, where a, b and c are numbers and a ≠ 0.
To plot one, make a table of values: choose x-values, work out y for each, then plot the (x, y) points.
The points join to form a smooth curve called a parabola — never join them with straight lines.
If a is positive the parabola opens upwards (a U shape); if a is negative it opens downwards (an n shape).
The graph is symmetric about a vertical line through its turning point (the vertex).
Example: for y = x², values are (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4).

Roots from a Graph

The roots of an equation are the x-values where the graph crosses the x-axis (where y = 0).
Read them off by finding the x-coordinates of the crossing points.
A quadratic can cross the x-axis twice (two roots), touch it once (one repeated root), or miss it (no real roots).
Example: y = x² − 4 crosses at x = −2 and x = 2, so its roots are −2 and 2.

Cubic and Reciprocal Graphs

A cubic graph has an x³ term, e.g. y = x³; it has a characteristic S-shaped curve.
A cubic can cross the x-axis up to three times, so it can have up to three roots.
A reciprocal graph has equation y = k/x (k a number, x ≠ 0).
Reciprocal graphs have two separate branches and never touch either axis.
As x gets large, y = k/x gets close to 0, so the x-axis is an asymptote.

Distance–Time Graphs

On a distance–time graph, time is on the horizontal axis and distance from the start is on the vertical axis.
The gradient (steepness) of the line gives the speed.
A horizontal line means the object is stationary (not moving).
A steeper line means a faster speed; a straight line means constant speed.
Speed = distance ÷ time, so gradient = change in distance ÷ change in time.

Conversion Graphs

A conversion graph changes one unit into another, e.g. pounds to dollars or kilometres to miles.
It is usually a straight line through the origin, because the conversion rate is constant.
To convert a value, start on one axis, go up/across to the line, then across/down to the other axis.
The gradient equals the conversion rate (how much of one unit equals one of the other).

Real-Life Graphs: Filling Containers

A graph of depth against time shows how quickly a container fills.
A straight sloping line means the container fills at a steady rate (a uniform shape, like a cylinder).
A curve getting steeper means the container is widening as it fills.
A curve getting shallower means the container is narrowing as it fills.
A horizontal section means filling has paused.

Real-Life Graphs: Phone Tariffs

A phone tariff graph often plots cost against usage (minutes or data).
A fixed monthly fee appears as a starting value on the cost axis (the y-intercept).
A charge per unit appears as the gradient of the line.
A flat rate after an allowance shows as a horizontal line once the included amount is used up.
Comparing two tariffs means seeing where their lines cross — the point where costs are equal.

Reading and Interpreting Any Graph

Always check the axes: what each one measures and the scale used.
The gradient tells you the rate of change (how fast y changes as x increases).
A positive gradient means y increases as x increases; a negative gradient means y decreases.
Where a graph crosses the x-axis gives the roots; where it crosses the y-axis gives the intercept.
Use the shape of the curve to describe the real situation it represents.

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1.The graph of a function is the set of ordered pairs (x, y). Which equation shows the relationship between x and y in these pairs?
Easy
- Ax = f(y)
- By = f(x)
- Cy = x
- Df(x) = x
2.The graph of y = x² is a straight line.
Easy
True or false?
3.A table of values for y = x² + 2x + 1 is shown. What is the value of y when x = 2? | x | -2 | -1 | 0 | 1 | 2 | | y | 1 | 0 | 1 | 4 | ? |
Easy
- A5
- B7
- C9
- D11
4.Which of these graphs are non-linear? (select all that apply)
Medium
- Ay = x²
- By = 3x - 1
- Cy = x³
- Dy = 2/x
- Ey = 5
5.Match each equation to the shape of its graph.
Medium
- y = x²
- y = x³
- y = 1/x
- y = x
- Parabola (U-shaped curve)
- Curve with an S-shape through the origin
- Two curved branches with asymptotes
- Straight line through the origin
6.On a distance-time graph, a horizontal line means the object is:
Medium
- Amoving at constant speed
- Baccelerating
- Cstationary
- Dmoving backwards
7.Put these steps in order to plot the graph of y = x² - 3x + 2.
Medium
- Draw the axes with a suitable scale
- Complete a table of values for chosen x-values
- Plot the coordinate pairs
- Join the points with a smooth curve
8.A conversion graph for pounds (£) and euros (€) passes through (0, 0) and (10, 12). How many euros is £25?
Medium
- A24
- B27
- C30
- D32

Plus de questions

9.A phone tariff charges a fixed monthly fee plus a cost per minute. Which graph shape best models the total monthly cost against minutes used?
Medium
- AA straight line starting at the origin
- BA straight line starting above the origin
- CA parabola
- DA curve that flattens off
10.Which of the following equations produce a curved (non-linear) graph? (select all that apply)
Medium
- Ay = x² + 3x - 1
- By = 2x + 5
- Cy = x³ - 2x
- Dy = 1/x
- Ey = 7 - 4x
11.Which of these statements about distance-time graphs are true? (select all that apply)
Medium
- AA horizontal line means the object is stationary.
- BA steeper line means a greater speed.
- CA straight line sloping downwards means the object is returning to the start.
- DThe gradient of the line gives the speed.
- EA curved line means the object is travelling at a constant speed.
12.Put these steps in the correct order to plot the graph of y = x² - 4x + 3 for x from 0 to 5.
Medium
- Draw a table of values for x = 0, 1, 2, 3, 4, 5.
- Substitute each x value into y = x² - 4x + 3 to find y.
- Plot the points (x, y) on a coordinate grid.
- Join the points with a smooth curve.
- Label the graph and axes.
13.The graph of y = x² - 2x - 3 crosses the x-axis at two points. What are the x-coordinates of these points?
Medium
- Ax = -1 and x = 3
- Bx = 1 and x = -3
- Cx = -1 and x = -3
- Dx = 1 and x = 3
14.A conversion graph for dollars ($) and pounds (£) is a straight line through (0, 0) and (5, 4). How many dollars is £20?
Medium
- A$16
- B$25
- C$24
- D$20
15.On a speed-time graph, a horizontal line means the object is accelerating.
Easy
True or false?
16.A table of values for y = x² + 3x is shown. What is the value of y when x = 2? | x | -2 | -1 | 0 | 1 | 2 | | y | -2 | -2 | 0 | 4 | ? |
Easy
- A10
- B7
- C4
- D12
17.The graph of y = x² - 4x + 3 is drawn. What are the x-coordinates of the points where the graph crosses the x-axis?
Medium
- Ax = 1 and x = 3
- Bx = -1 and x = -3
- Cx = 1 and x = -3
- Dx = -1 and x = 3
18.A distance-time graph shows a car travelling 120 km in 1.5 hours. What is the car's average speed in km/h?
Medium
- A60
- B80
- C180
- D90
19.The graph of y = x² + 2x - 3 is drawn. What is the y-intercept of the graph?
Easy
- A(0, -3)
- B(0, 3)
- C(0, 2)
- D(0, -2)
20.Which of the following graphs are non-linear? (select all that apply)
Medium
- Ay = x²
- By = 2x + 1
- Cy = x³
- Dy = 3
- Ey = 1/x
21.On a distance-time graph, a horizontal line means the object is stationary.
Easy
True or false?
22.Put these steps in the correct order to plot the graph of y = x² - 2x + 1 for x from 0 to 4.
Medium
- Draw a table of values for x = 0, 1, 2, 3, 4.
- Substitute each x value into y = x² - 2x + 1 to find y.
- Plot the points (x, y) on a coordinate grid.
- Join the points with a smooth curve.
23.A quadratic graph has equation y = x² + bx + c and passes through (1, 0) and (3, 0). What is the value of b + c?
Medium
- A-1
- B1
- C-5
- D5
24.A straight conversion graph passes through (0, 0) and (8, 10). What is the value of x when y = 25?
Medium
- A20
- B22.5
- C18
- D31.25
25.The graph of y = x² - 4x + 3 is drawn. Which of the following statements are true? (select all that apply)
Hard
- AThe graph crosses the x-axis at x = 1 and x = 3.
- BThe graph has a minimum point.
- CThe y-intercept is at y = 3.
- DThe graph is a straight line.
- EThe graph crosses the x-axis at x = -1 and x = -3.
26.Put these steps in the correct order to plot the graph of y = x² - 5x + 6 for x from 0 to 5.
Medium
- Choose x-values from 0 to 5.
- Substitute each x-value into y = x² - 5x + 6 to find y.
- Plot the points (x, y) on a coordinate grid.
- Join the points with a smooth curve.
27.On a speed-time graph, a horizontal line above the time axis means the object is:
Medium
- Amoving at constant speed
- Bstationary
- Caccelerating
- Ddecelerating
28.The graph of y = x² - 2x - 3 is drawn. What are the x-coordinates of the points where the graph crosses the x-axis?
Medium
- A-1 and 3
- B1 and -3
- C3 and -1
- D-3 and -1
29.Which of the following statements about the graph of y = x² - 6x + 5 are true? (select all that apply)
Hard
- AThe graph crosses the x-axis at x = 1 and x = 5.
- BThe minimum point occurs at x = 3.
- CThe y-intercept is (0, 5).
- DThe graph is a straight line.
- EThe graph has a maximum point.
30.A container is filled with water at a steady rate. The graph of depth against time is a straight line that becomes less steep partway through. What does this mean?
Medium
- AThe container is wider at the top.
- BThe container is narrower at the top.
- CThe water is flowing in faster.
- DThe container is leaking.
31.For a function f, the graph is the set of ordered pairs (x, y) such that:
Easy
- Af(x) = y
- Bx = y
- Cf(y) = x
- Df(x) + y = 0
32.The graph of y = x² + 3x + 2 is a straight line.
Easy
True or false?
33.In the equation y = x² - 5x + 6, what is the value of y when x = 2?
Easy
- A0
- B2
- C4
- D6
34.On a distance–time graph, what does the gradient of the line represent?
Medium
- ASpeed
- BDistance
- CTime
- DAcceleration
35.Which of the following equations will produce a curved graph? (select all that apply)
Medium
- Ay = 3x + 1
- By = x² - 4
- Cy = 2x
- Dy = x³
- Ey = 1/x
36.On a speed–time graph, a horizontal line above the time axis means the object is moving at a constant speed.
Easy
True or false?
37.Put these steps in the correct order to plot the graph of y = x² - 3x + 2 for x from -1 to 3.
Medium
- Draw a table of values for x and y
- Substitute each x value into the equation to find y
- Plot the points on a coordinate grid
- Join the points with a smooth curve
38.Which of the following statements about the graph of y = x² + 4x + 3 are true? (select all that apply)
Medium
- AIt crosses the x-axis at x = -3 and x = -1
- BIt has a y-intercept at y = 3
- CIts shape is a parabola
- DIt is a straight line
39.Put these steps in the correct order to plot the graph of y = x² - 6x + 8 for x from 0 to 6.
Medium
- Draw and label a set of axes covering x from 0 to 6 and a suitable range of y
- Work out the y-values by substituting each value of x into y = x² - 6x + 8
- Plot the coordinate pairs (x, y) from the table
- Join the plotted points with a smooth curve
40.A graph shows the depth of water in a container against time as the container is filled at a steady rate. Which of the following statements are true? (select all that apply)
Medium
- AA steeper section means the depth is increasing faster
- BA horizontal section means the depth is not changing
- CA less steep section means the water is being added more slowly
- DThe graph must be a curve for a container that gets wider
41.A distance-time graph shows a cyclist travelling 30 km in 2 hours. Work out the cyclist's speed in km/h.
Medium
42.On a speed-time graph, what does the area under the line represent?
Medium
43.Complete the table of values for y = x² - 1.
Medium
When x = -3, y = ____; when x = 0, y = ____; when x = 3, y = ____.
44.A distance-time graph shows a runner covering 100 metres in 20 seconds. Work out the runner's average speed in metres per second.
Medium
45.Complete the sentence about non-linear graphs.
Easy
The graph of y = x² is a ____ shaped curve called a parabola.
46.What is the name of the shape of the graph of a quadratic function such as y = x²?
Easy
47.A conversion graph for pounds (£) and euros (€) is a straight line through (0, 0) and (5, 6). How many euros is £15?
Medium
48.Complete the table of values for y = x² + 1. | x | -2 | -1 | 0 | 1 | 2 | | y | 5 | 2 | 1 | 2 | ? |
Easy
When x = 2, y = ____
49.A table of values for y = x² - 2x is shown. Work out the value of y when x = 3. | x | -1 | 0 | 1 | 2 | 3 | | y | 3 | 0 | -1 | 0 | ? |
Medium
50.Complete the sentence about the shape of a quadratic graph.
Easy
The graph of y = x² is a curve called a ____.
51.What is the name of the point where a graph crosses the y-axis?
Medium
52.A conversion graph for pounds (£) and dollars ($) is a straight line through (0, 0) and (4, 5). How many dollars is £20?
Medium
53.A container is filled with water at a steady rate. The graph of depth against time is a straight line that becomes less steep partway through. Explain what this tells you about the container.
Hard
54.A container is filled with water at a steady rate. The graph of depth against time is a straight line that becomes less steep partway through. Explain what this means about the container.
Hard
55.A phone tariff charges a fixed monthly fee of £10 plus 5p per minute. Explain how the graph of total monthly cost against minutes used would look and what the gradient represents.
Hard
56.A car travels 120 km in 1.5 hours. Explain how to find its average speed and state the value.
Hard
57.A container is filled with water at a steady rate. The graph of depth against time is a straight line that becomes less steep partway through. Explain what this means.
Hard
58.A distance-time graph shows a cyclist travelling 30 km in 2 hours. Work out the cyclist's speed in km/h.
Medium
- A14 km/h
- B15 km/h
- C16 km/h
- D17 km/h
59.A distance-time graph shows a runner covering 100 metres in 20 seconds. Work out the runner's average speed in metres per second.
Medium
- A4 m/s
- B7 m/s
- C5 m/s
- D6 m/s
60.A conversion graph for pounds (£) and euros (€) is a straight line through (0, 0) and (5, 6). How many euros is £15?
Medium
- A18 €
- B20 €
- C19 €
- D17 €
61.A table of values for y = x² - 2x is shown. Work out the value of y when x = 3. | x | -1 | 0 | 1 | 2 | 3 | | y | 3 | 0 | -1 | 0 | ? |
Medium
- A2
- B5
- C3
- D4
62.A conversion graph for pounds (£) and dollars ($) is a straight line through (0, 0) and (4, 5). How many dollars is £20?
Medium
- A22 $
- B31 $
- C25 $
- D28 $

Notes de révision

## Non-linear graphs and real-life graphs **The graph of a function** is the set of ordered pairs (x, y) where f(x) = y. Plotted on Cartesian axes, these points usually form a curve. ### Quadratic graphs - Form: y = x^2 + bx + c - Plot from a table of values: choose x values, work out y, plot the points, join with a smooth curve. - Shape: a **parabola** — a symmetrical U-shape (or n-shape if the x^2 coefficient is negative). - **Roots**: where the curve crosses the x-axis (y = 0). Read them off the graph. ### Cubic and reciprocal graphs - **Cubic**: highest power x^3, e.g. y = x^3. An S-shaped curve. - **Reciprocal**: y = 1/x (a fraction with x on the bottom). Two separate branches that get close to the axes but never touch them. ### Real-life graphs - **Distance–time graph**: distance on the vertical axis, time on the horizontal. The **gradient** gives the speed. A horizontal line means stationary; a steeper line means faster. - **Conversion graphs**: straight lines used to change between units (e.g. pounds and kilograms). Read across and down to convert. - **Filling containers**: the shape of the graph shows how quickly the depth rises — wide containers fill slowly (shallow gradient), narrow parts fill quickly (steep gradient). - **Phone tariffs**: compare fixed charges plus a rate per unit by comparing the lines. ### Exam tips - Always label axes and use a sensible scale. - Join plotted points with a smooth curve for quadratics and cubics — never dot-to-dot straight lines. - For real-life graphs, check what each axis represents before reading values. - Gradient = change in y / change in x.

Graph of a function: The set of ordered pairs (x, y) where f(x) = y; plotted on axes it usually forms a curve.
Parabola: The symmetrical U-shaped curve produced by a quadratic graph such as y = x^2 + bx + c.
Roots: The x-values where a graph crosses the x-axis, i.e. where y = 0.
Cubic graph: A graph whose highest power of x is 3, e.g. y = x^3; it has an S-shaped curve.
Reciprocal graph: A graph of the form y = 1/x, with two separate branches that approach but never touch the axes.
Distance–time graph: A graph with distance on the vertical axis and time on the horizontal axis; its gradient gives the speed.
Conversion graph: A straight-line graph used to change between two units, read by going across and down.
Gradient: The steepness of a line, found by change in y divided by change in x.

Cartes mémo

What is the graph of a function f?

The set of ordered pairs (x, y) where f(x) = y. Plotted on axes, these points usually form a curve.

What shape is the graph of a quadratic function y = x² + bx + c?

A parabola — a smooth U-shaped (or upside-down U-shaped) curve.

How do you plot a quadratic graph such as y = x² + bx + c?

Make a table of values: choose x-values, work out each y-value, plot the points, then join them with a smooth curve.

What is a root of a graph?

An x-value where the graph crosses the x-axis, i.e. where y = 0.

How do you read the roots of a quadratic from its graph?

Find where the curve crosses the x-axis and read off the x-coordinates of those points.

What is the general shape of a cubic graph such as y = x³?

An S-shaped curve that can rise or fall steeply; it may cross the x-axis up to three times.

What is the general shape of a reciprocal graph such as y = 1/x?

Two separate curved branches that get closer to the axes but never touch them (asymptotes).

On a distance–time graph, what does the gradient represent?

Speed. A steeper line means a faster speed; a horizontal line means stationary (zero speed).

On a distance–time graph, what does a horizontal line mean?

The object is stationary — distance is not changing as time passes.

What is a conversion graph used for?

To convert between two units (e.g. miles and kilometres, or pounds and kilograms) by reading values from a straight-line graph.

How do you use a conversion graph to change 5 miles into kilometres?

Find 5 on the miles axis, go up to the line, then across to the kilometres axis and read the value.

What does a real-life graph of filling a container show?

How the depth (or volume) of water changes over time. The shape of the graph depends on the container's width at each height.

For a container that gets wider as it fills, what does the depth–time graph look like?

The graph gets less steep over time, because each extra unit of depth needs more water to raise the level.

What does a phone tariff graph usually compare?

The total monthly cost against the number of minutes (or texts/data) used, often as straight lines with different fixed charges and rates.

On a phone tariff graph, what does the y-intercept represent?

The fixed monthly charge — the cost when no minutes are used.

On a phone tariff graph, what does the gradient represent?

The cost per minute (or per unit) charged for usage.

How can you tell which of two phone tariffs is cheaper for a given number of minutes?

At that number of minutes, read the cost from each line; the lower line gives the cheaper tariff.

What is the difference between a linear and a non-linear graph?

A linear graph is a straight line (constant gradient); a non-linear graph is curved (gradient changes).

What does the gradient of a straight-line graph tell you?

The rate of change of y with respect to x — how much y increases for each unit increase in x.

How do you find the gradient of a straight line from a graph?

Choose two points on the line, then gradient = (change in y) ÷ (change in x).

What does the y-intercept of a graph represent in a real-life context?

The starting value of the quantity when the input (often time) is zero.

Why are graphs useful in real life?

They show how one quantity changes with another, making patterns, rates and comparisons easy to see.

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