EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Quadratic graphs
Graphs · Lesson 6 of 8
Last Lesson and Before
Answer each one, then check.
1. Work out (−3)².
9
2. Work out −3².
−9
3. Substitute x = −1 into x² − 4x + 3.
8
4. Factorise x² − 4x + 3.
(x − 1)(x − 3)
Learning Objectives
1. Complete a table of values for a quadratic and plot its graph.
2. Recognise the shape of a quadratic graph.
3. Find the roots, turning point, line of symmetry and y-intercept.
4. Use a quadratic graph to solve equations.
Plotting a Quadratic
A quadratic has an x² term and no higher power. Its graph is a smooth curve called a parabola.
▸ Brackets for negatives. (−2)² = 4, so put negative values in brackets on your calculator.
▸ Use the symmetry. The y-values repeat either side of the turning point - a quick check on your table.
▸ Smooth curve. Join the points with a smooth curve, not straight lines, and don't flatten the bottom.
▸ Shape. If the x² term is positive, the curve is U-shaped; if it is negative, it is upside down (∩).
A Table of Values for y = x² − 4x + 3
|
x |
x² |
−4x |
+3 |
y |
|---|---|---|---|---|
|
−1 |
1 |
4 |
3 |
8 |
|
0 |
0 |
0 |
3 |
3 |
|
1 |
1 |
−4 |
3 |
0 |
|
2 |
4 |
−8 |
3 |
−1 |
|
3 |
9 |
−12 |
3 |
0 |
|
4 |
16 |
−16 |
3 |
3 |
|
5 |
25 |
−20 |
3 |
8 |
The Features of a Parabola
|
The roots are where the curve crosses the x-axis: they solve x² − 4x + 3 = 0. The turning point is the lowest point, halfway between the roots, on the line of symmetry x = 2. The y-intercept is the number on its own: 3. |
Roots, y-intercept, turning point and line of symmetry. |
Solving from the Graph
A graph gives the solutions of an equation as the x-coordinates where two things meet.
▸ Equal to 0. The solutions of x² − 4x + 3 = 0 are where the curve crosses the x-axis: x = 1 and x = 3.
▸ Equal to a number. For x² − 4x + 3 = 3, draw the line y = 3: it meets the curve at x = 0 and x = 4.
▸ Estimates. When the curve does not cross exactly on a grid line, give answers to 1 decimal place.
▸ No solution. If the line misses the curve, there is no solution: x² − 4x + 3 = −2 has none.
Reading Solutions
|
Use the graph of y = x² − 4x + 3 to solve (a) x² − 4x + 3 = 0 (b) x² − 4x + 3 = 8. |
1. (a) Where the curve crosses the x-axis
x = 1 and x = 3
2. (b) Draw the line y = 8
It meets the curve twice
3. Read the x-coordinates
x = −1 and x = 5
4. Check (b): (−1)² − 4(−1) + 3 = 8
Correct
Answer: (a) x = 1, x = 3 (b) x = −1, x = 5
Key Terms
|
Quadratic An expression whose highest power of x is x². |
Parabola The U-shaped (or ∩-shaped) curve of a quadratic graph. |
|
Roots The x-values where a graph crosses the x-axis. |
Turning point The point where the curve changes direction: a minimum or maximum. |
|
Line of symmetry The vertical line through the turning point. |
|
Your Task: Spot the Mistakes
10 minutes
|
A student made this table for y = x² + 2x − 1: x = −3 gives 2, x = −2 gives −9, x = −1 gives −2, x = 0 gives −1, x = 1 gives 2. They plotted it and joined the points with straight lines. Find every mistake and describe the correct graph. 1. Check each value. 2. Look for symmetry. 3. Check how the points are joined. |
A good answer shows: x = −2 should give 4 − 4 − 1 = −1 (they worked out −2² as −4). The rest are right. Correct values: 2, −1, −2, −1, 2 - symmetrical about x = −1, with minimum (−1, −2). The points should be joined with a smooth curve.
Can I...?
☐ Complete a table of values, including negative x.
☐ Plot a smooth quadratic curve.
☐ Recognise U-shaped and ∩-shaped quadratics.
☐ Find the roots from a graph.
☐ Find the turning point and line of symmetry.
☐ Find the y-intercept.
☐ Solve ax² + bx + c = k using a graph.
☐ Give estimates to 1 decimal place.
Summary
✓ A quadratic graph is a parabola: U-shaped for +x², ∩-shaped for −x².
✓ Roots are where y = 0; the y-intercept is the constant term.
✓ The turning point lies on the line of symmetry, halfway between the roots.
✓ To solve f(x) = k, draw y = k and read the x-values.
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EXAM FOCUS The graph of y = x² − 2x − 3 is drawn. Use it to find estimates for the solutions of x² − 2x − 3 = 2. (2 marks) When a question says "use the graph", draw the horizontal line on the graph and read off where it meets the curve. Calculated answers without the line drawn may not get the marks. |