Graphs Of Functions
Graph shapes and sketching
In this note
- 2.11Recognise and sketch linear, quadratic, cubic, reciprocal and exponential graphs.
- 2.11Show the important features: intercepts, turning points, asymptotes.
- 2.11Match an equation to its graph and give a reason.
Before you start: straight lines and solving quadratics. Asymptotes in detail: Asymptotes.
Formula bank
The shape families
The highest power of \(x\) decides the overall shape, and its sign decides which way up it is. Squaring makes numbers positive (a parabola), cubing keeps the sign (an S-shape), dividing by \(x\) breaks the graph at \(x=0\), and powers of a positive number are always positive.
Linear: y = mx + c
A straight line. Gradient \(m\), y-intercept \(c\).
Quadratic, \(a>0\)
U-shape (parabola), one minimum turning point, symmetric.
Quadratic, \(a<0\)
∩-shape, one maximum turning point.
Cubic, \(a>0\)
Rises left to right overall; up to two turning points; S-shape.
Cubic, \(a<0\)
Falls left to right overall.
Reciprocal: \(y=\dfrac ax\)
Two branches; asymptotes on the axes. \(a<0\): the other two quadrants.
Exponential growth: \(y=a^x\), \(a>1\)
Through (0, 1), always positive, steeper and steeper.
Exponential decay: \(y=a^{-x}\)
Through (0, 1), falls towards 0.
Reciprocal \(y=\dfrac ax\): if \(a>0\) the branches are in the top-right and bottom-left quadrants; if \(a<0\), top-left and bottom-right. (Full steps for shifted curves on Asymptotes.)
Try it yourself
Show answer
Answer cubic
Show answer
Answer 1
Sketching
General methodHow to sketch any graph
- Decide the shape from the highest power and its sign.
- y-intercept: put x = 0.
- x-intercepts: put y = 0 and solve (read them off the brackets if factorised).
- Add turning points or asymptotes if you know them.
- Draw a smooth curve through these points; label the values. A sketch does not need to be to scale.
Example Sketch \(y=12+4x-x^{2}\).
\(-x^{2}\): ∩-shape. Find the intercepts and the top.
- \(x=0\): \(y=12\)
- \(y=0\): \(x^{2}-4x-12=0\Rightarrow x=6\) or \(-2\)
- maximum halfway between the roots: \(x=2\), \(y=16\)
Answer ∩ through (−2, 0), (0, 12), (6, 0), maximum (2, 16)
Try it yourself
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Answer −2, 3
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Answer −3, 0, 2
Ways they test it
Type 1 Name the type
"Put a ring around the type of function shown." (1 mark, 0580 Paper 2)
Spot it a graph and a list of words.
- Count the turning points and look at the ends.
- Branches + asymptotes = reciprocal; through (0, 1), never negative = exponential.
Type 2 Sketch with intercepts
"Sketch \(y=(x-1)(x+2)(x-3)\), showing where it crosses the axes." (3 marks)
Spot it 'sketch' and 'show where it crosses'.
- Roots from the brackets.
- y-intercept with x = 0.
- Shape from the sign of x³.
For example \(y=(x-1)(x+2)(x-3)\)
- roots 1, −2, 3
- y-intercept (−1)(2)(−3) = 6
Answer S-shape through (−2, 0), (0, 6), (1, 0), (3, 0)
Type 3 Exponential sketch
"Sketch \(y=3^{x}\), showing the y-intercept." (2 marks)
Spot it a number to the power x.
- Through (0, 1).
- Above the x-axis everywhere; steeper to the right.
Type 4 Match equations to graphs
"Match each equation to its graph."
Spot it several equations and pictures.
- Test one feature each: the sign of the highest power, the y-intercept, asymptotes.
Mistakes to avoid
- ✗ A parabola with a pointed vertex✓ a smooth curveQuadratics are smooth.
- ✗ An exponential crossing the x-axis✓ it stays above the x-axis\(a^{x}>0\) always.
- ✗ A sketch with no values✓ label the interceptsThose are the marks.
Summary
- Shape from the highest power; a negative coefficient flips it.
- Sketch = shape + y-intercept + x-intercepts + turning points/asymptotes, labelled.
- Every \(y=a^{x}\) passes through (0, 1) and stays above the x-axis.
- \(y=\dfrac ax\): \(a>0\) top-right & bottom-left; \(a<0\) top-left & bottom-right.
If you see …
- "sketch"shape and labelled intercepts, not plotted points.
- brackets in the equationthe x-intercepts can be read straight off.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer −3
Show answer
Answer 4
Show answer
Answer 1
Medium
Guide me
What is it asking? a y-value
What do we know? put x = 0
Plan (−1)(2)(−3)
Show working
- \((-1)(2)(-3)=6\)
Answer 6
Guide me
What is it asking? x-values
What do we know? y = 0
Plan \(x(3-x)=0\)
Show working
- \(x(3-x)=0\)
- x = 0 or 3
Answer 0, 3
Guide me
What is it asking? the top of the ∩
What do we know? roots −2 and 6
Plan x = 2 halfway
Show working
- \(x=2\)
- \(y=12+8-4=16\)
Answer 16
Hard
Guide me
What is it asking? a point
What do we know? roots 1 and 5
Plan halfway: x = 3
Show working
- \((x-1)(x-5)=0\)
- \(x=3\)
- \(y=9-18+5=-4\)
Answer (3, −4)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| x = 3 | B1 | ||
| y = −4 | B1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? two quadrants
What do we know? a < 0
Plan try x = 1: y = −2
Show working
- x = 1 gives y = −2 (quadrant 4)
- x = −1 gives y = 2 (quadrant 2)
Answer 2, 4
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 2nd and 4th | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? a y-value
What do we know? put x = 0
Plan 0 − 3
Show working
- \(0-3=-3\)
Answer −3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| −3 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can recognise each graph family from its equation or shape.
- I can sketch a graph showing intercepts and turning points.
- I can match equations to graphs with reasons.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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