Graphs Of Functions
Quadratic and cubic graphs
In this note
- 2.11Find the roots, y-intercept, line of symmetry and turning point of a parabola.
- 2.11Find the equation of a parabola from its graph.
- 2.11Sketch cubics from factorised form, including a repeated root.
Before you start: Graph shapes and completing the square. Next: Reciprocal and exponential graphs.
Formula bank
The features of a parabola
A parabola is symmetrical about a vertical line through its turning point. So the turning point is exactly halfway between the two roots.
General methodHow to find the features of a parabola
- y-intercept: put x = 0.
- Roots: put y = 0 and solve (factorise).
- Line of symmetry: x = halfway between the roots.
- Turning point: put that x into the equation for y (or complete the square: (−p, q)).
Example Find the features of \(y=x^{2}-4x-5\).
Intercepts, then the turning point halfway between the roots.
- \(x=0\): \(y=-5\)
- \((x-5)(x+1)=0\Rightarrow x=-1,\ 5\)
- symmetry \(x=\tfrac{-1+5}2=2\)
- \(y=4-8-5=-9\)
Answer roots −1, 5; y-intercept −5; minimum (2, −9)
Try it yourself
Show answer
Answer 2, 4
Show answer
Answer (−3, −5)
Finding the equation from a graph
General methodHow to find a parabola's equation from its graph
- Roots p and q shown: write \(y=a(x-p)(x-q)\).
- Put in one more known point (often the y-intercept or the turning point) to find a.
- Turning point \((h,\ k)\) shown and the \(x^{2}\) coefficient is 1: write \(y=(x-h)^{2}+k\).
- Expand if they want it as ax² + bx + c.
Example A parabola crosses the x-axis at 1 and 4 and the y-axis at 8. Find its equation.
Start with \(y=a(x-1)(x-4)\); use (0, 8).
- \(8=a(-1)(-4)=4a\)
- \(a=2\)
Answer \(y=2(x-1)(x-4)=2x^{2}-10x+8\)
Cubic graphs from factors
Each factor \((x-k)\) gives a root at \(x=k\). A squared factor \((x-k)^{2}\) makes the curve touch the axis and turn back, because a square never changes sign.
- Roots from each bracket: −1 and 2 (squared → touch).
- y-intercept: \((0-2)^{2}(0+1)=4\).
- Positive x³: comes up from bottom-left, ends top-right.
Try it yourself
Show answer
Answer 4
Ways they test it
Type 1 Complete the square, then sketch
"(a) Write \(x^{2}-8x+7\) as \((x-a)^{2}-b\). (b) Sketch the graph, showing the turning point and where it crosses the axes." (4 marks)
Spot it a sketch after a completed square.
- Turning point from (a).
- Roots by factorising; y-intercept from x = 0.
For example \((x-4)^{2}-9\)
- roots 1 and 7, y-intercept 7
Answer U-shape, minimum (4, −9)
Type 2 Line of symmetry
"The minimum point is (−3, −5). Write down the equation of the line of symmetry." (1 mark)
Spot it 'line of symmetry'.
- It is vertical: x = the turning point's x.
Type 3 Equation from the graph
"A ∩-shaped parabola crosses the x-axis at 1 and 5, maximum (3, 4). Find its equation." (3 marks)
Spot it roots marked on a graph.
- \(y=a(x-1)(x-5)\).
- Put in the maximum to find a.
For example \(4=a(2)(-2)\)
- \(a=-1\)
Answer \(y=-x^{2}+6x-5\)
Type 4 Cubic with a repeated root
"Sketch \(y=(x+1)^{2}(x-3)\)."
Spot it a squared bracket.
- Touch at the squared root, cross at the other.
- y-intercept; ends from the sign of x³.
Mistakes to avoid
- ✗ turning point of \((x-4)^{2}-9\) at \((-4,\ -9)\)✓ \((4,\ -9)\)The x-coordinate has the opposite sign.
- ✗ line of symmetry y = −3✓ x = −3It is a vertical line.
- ✗ a squared factor drawn crossing✓ touching the axisA square doesn't change sign.
Summary
- Roots (y = 0), y-intercept (x = 0), symmetry halfway between the roots, turning point on it.
- \(y=(x+p)^{2}+q\): turning point \((-p,\ q)\).
- Equation from roots: \(y=a(x-p)(x-q)\), then find a.
- A squared factor touches the axis; a single factor crosses.
If you see …
- "line of symmetry"x = (the turning point's x-coordinate).
- roots and one more point on a graphy = a(x − p)(x − q) and solve for a.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer −1, 5
Show answer
Answer 2
Show answer
Answer −3
Medium
Guide me
What is it asking? a point
What do we know? \((x-4)^{2}-9\)
Plan (−p, q)
Show working
- \((x-4)^{2}-16+7\)
- \((4,\ -9)\)
Answer (4, −9)
Guide me
What is it asking? a point
What do we know? \(2(x+3)(x-1)\): roots −3, 1
Plan halfway: x = −1
Show working
- \(x=-1\)
- \(y=2-4-6=-8\)
Answer (−1, −8)
Guide me
What is it asking? a
What do we know? use (0, 8)
Plan \(8=a(-1)(-4)\)
Show working
- \(8=4a\)
- \(a=2\)
Answer 2
Hard
Guide me
What is it asking? a
What do we know? use (3, 4)
Plan \(4=a(2)(-2)\)
Show working
- \(4=-4a\)
- \(a=-1\)
Answer −1
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(4=a(3-1)(3-5)\) | M1 | ||
| a = −1 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? b and c
What do we know? \(y=(x+3)^{2}-5\)
Plan expand
Show working
- \(x^{2}+6x+9-5\)
- \(x^{2}+6x+4\)
Answer b = 6, c = 4
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \((x+3)^{2}-5\) | M1 | ||
| b = 6, c = 4 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a count
What do we know? roots −1 (squared) and 3
Plan touch + cross
Show working
- touches at x = −1
- crosses at x = 3
Answer 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 2 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can find the roots, y-intercept, line of symmetry and turning point of a parabola.
- I can find the equation of a parabola from its graph.
- I can sketch cubics from factors, including repeated roots.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.
Your progress on this note is saved when JavaScript is on.