EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Using quadratic graphs
Equations and graphs · Lesson 4 of 6
Warm-up
Answer each one, then check.
1. Work out x² − 2x − 3 when x = 2.
−3
2. What shape is a graph of y = x²?
A U-shaped parabola
3. What is a root of a graph?
Where it crosses the x-axis
4. What is a line of symmetry?
A line the graph reflects onto itself in
5. What is y on the y-axis?
The vertical coordinate
Learning Objectives
1. Complete a table of values and draw a quadratic graph.
2. Find roots, the y-intercept, the turning point and the line of symmetry.
3. Solve ax² + bx + c = 0 using a graph.
4. Solve related equations such as x² − 2x − 3 = 2 by drawing a line.
Quadratic Graph
A quadratic graph is a smooth curve called a parabola. It is symmetrical about a vertical line through its turning point.
The roots are where y = 0. To solve f(x) = k, draw the line y = k and read the x-values where it meets the curve.
Key Features of a Parabola
|
Roots, minimum point, y-intercept and axis of symmetry. |
The graph of y equals x squared minus 2x minus 3 with its roots, turning point, y-intercept and line of symmetry marked.
Table of Values
y = x² − 2x − 3.
|
x |
−2 |
−1 |
1 |
2 |
|---|---|---|---|---|
|
y |
5 |
0 |
−3 |
−4 | −3 | 0 | 5 |
Roots and Turning Point
|
For y = x² − 2x − 3, find the roots, the turning point and the line of symmetry. |
1. Roots
y = 0: (x − 3)(x + 1) = 0, so x = 3 or −1
2. Line of symmetry
Halfway between the roots: x = 1
3. Turning point
y = 1 − 2 − 3 = −4, so (1, −4)
Answer: Roots −1 and 3; line of symmetry x = 1; minimum point (1, −4).
Solving a Related Equation
|
Use the graph of y = x² − 2x − 3 to solve x² − 2x − 3 = 2. |
1. Draw the line
y = 2 across the graph
2. Read the x-values where it meets the curve
x ≈ −1.4 and x ≈ 3.4
3. Check
1 ± √6 = −1.45, 3.45
Answer: x ≈ −1.4 or x ≈ 3.4
Completed Square Form
|
Write y = x² − 2x − 3 in the form (x − a)² + b and write down the turning point. |
1. Complete the square
(x − 1)² − 1 − 3
2. Simplify
y = (x − 1)² − 4
3. Turning point
(1, −4)
Answer: y = (x − 1)² − 4; the turning point is (1, −4).
Drawing Tips
Get a smooth curve.
▸ Plot every point. Then join with a smooth curve, not straight segments.
▸ No flat bottom. Near the turning point the curve rounds gently.
▸ Check symmetry. Values on each side of the turning point should match.
▸ Use a ruler for lines. Use a pencil for the curve.
Key Terms
|
Parabola The U-shaped graph of a quadratic. |
Root A value of x where y = 0. |
|
Turning point The lowest or highest point of the curve. |
Line of symmetry The vertical line through the turning point. |
|
y-intercept Where the graph crosses the y-axis. |
Minimum The lowest point of a U-shaped curve. |
Your Task: Sketch and Solve
12 minutes
|
Complete the table for y = x² − 4x for x = 0 to 4 and sketch the graph. Write down the roots and the turning point. 1. Substitute each x value. 2. Join with a smooth curve. |
A good answer shows: y = 0, −3, −4, −3, 0. Roots x = 0 and x = 4; turning point (2, −4).
Can I...?
☐ Complete a table.
☐ Plot the points.
☐ Draw a smooth curve.
☐ Find the roots.
☐ Find the turning point.
☐ Give the line of symmetry.
☐ Solve a related equation.
☐ Use the completed square form.
Summary
✓ Roots are where y = 0.
✓ The turning point lies on the line of symmetry.
✓ Solve f(x) = k with the line y = k.
✓ Completed square form gives the turning point directly.
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EXAM FOCUS The graph of y = x² − 2x − 3 is drawn. Use it to solve x² − 2x − 3 = 0 and to find the coordinates of the turning point. (3 marks) Draw your line across the graph and mark the intersections. |