Quadratic Graphs and Graphical Solutions
Foundation & Higher — sections 4–5 are Higher extensions.
A quadratic graph turns an equation into a picture you can read. Learn to plot it accurately, identify its key features and use crossings to solve equations.
Before you start: Linear Graphs, Factorising and Quadratic Equations. Use Rounding and Estimation when reporting estimates.
1. Recognise and plot a quadratic
A quadratic has an x² term as its highest power: y = ax² + bx + c, where a ≠ 0. Its graph is a smooth parabola. When a is positive it opens upwards; when a is negative it opens downwards. It has a vertical line of symmetry.
An accurate plot uses calculated points and a numbered grid. A sketch shows the correct overall shape and labelled features, without needing every point to scale. For either, do not replace the curve with a chain of straight lines.
Worked example — y = x² − 2x − 3
Choose values on both sides of the bend. At x = −2, use brackets: (−2)² − 2(−2) − 3 = 4 + 4 − 3 = 5. At x = 1, y = 1 − 2 − 3 = −4.
| x | y = x² − 2x − 3 |
|---|---|
| -2 | 5 |
| -1 | 0 |
| 0 | -3 |
| 1 | -4 |
| 2 | -3 |
| 3 | 0 |
| 4 | 5 |
Label both axes, use evenly spaced scales and plot each ordered pair. Draw one smooth curve through them. Add extra points near the bend if needed, and recheck a point that breaks the smooth pattern.
2. Read roots, intercepts and the turning point
Roots are the x-values where y = 0: here x = −1 and x = 3. The corresponding x-intercepts are the points (−1, 0) and (3, 0). The y-intercept occurs at x = 0: (0, −3).
The turning point (or vertex) is (1, −4), the lowest point. The minimum value of y is −4, reached when x = 1. The symmetry line is x = 1: points equally far left and right of it have the same height. A y-intercept need not be the turning point.
Worked example — read the solutions of x² − 2x − 3 = 2
Keep the same quadratic. Draw the horizontal line y = 2. From each crossing, read down to the x-scale. The readings are approximately x ≈ −1.45 and x ≈ 3.45 (about −1.4 and 3.4 on a coarse hand-drawn grid). They are estimates, not exact roots.
To solve f(x) = 0, read the x-axis crossings. To solve f(x) = k, read crossings with the horizontal line y = k. The equation decides which level to use.
3. Downward curves and the number of roots
Worked example — y = −2x² + 2
The coefficient −2 makes the curve open downwards. For x = −2, −1, 0, 1, 2, the y-values are −6, 0, 2, 0, −6. In particular, −2(−2)² + 2 = −8 + 2 = −6: square the input before multiplying by −2.
The maximum is 2 at (0, 2). Reading a negative or non-unit leading coefficient is part of the shared Foundation/Higher route.
An upward quadratic may cross the x-axis twice, touch it once, or miss it. For example, y = x² − 1 has two distinct roots (−1 and 1); y = x² touches at (0, 0), giving one repeated root; y = x² + 1 stays above the axis, so has no real roots. A graph window can hide roots, so check the shape and displayed domain before concluding there are none.
Worked example — use a sensible domain
A model for height is h = 4 − t², for 0 ≤ t ≤ 2, where t is time in seconds and h is height in metres. The full parabola meets h = 0 at t = −2 and t = 2, but only t = 2 seconds belongs to the model. Its maximum on this domain is 4 m at t = 0. Negative time is excluded by the stated domain.
4. Higher — find a vertex from the equation
The form y = a(x − p)² + q puts the vertex at (p, q). The square is smallest when x = p. Positive a gives a minimum q; negative a gives a maximum q.
Worked example — y = −2(x − 2)² + 5
At x = 2, the square is zero and y = 5. Everywhere else, subtracting a positive amount makes y smaller. So the vertex is (2, 5), a maximum, and the symmetry line is x = 2.
For a form already expanded, use completing the square. For example, x² − 2x − 3 = (x − 1)² − 4, confirming vertex (1, −4). Also 2x² + 6x + 1 = 2(x + 1.5)² − 3.5 gives minimum (−1.5, −3.5). The sign inside the bracket must make the bracket zero.
5. Higher — line–curve intersections
At an intersection the graphs have the same x and the same y. The pair of coordinates solves both equations simultaneously. Plot both graphs on the same axes and read every crossing in the required domain.
Worked example — y = x² − 2x − 3 and y = x + 1
The simultaneous solutions are (−1, 0) and (4, 5). Check (−1, 0): 1 + 2 − 3 = 0 and −1 + 1 = 0. Check (4, 5): 16 − 8 − 3 = 5 and 4 + 1 = 5.
If asked only to solve x² − 2x − 3 = x + 1, give the x-values −1 and 4. If asked for the simultaneous solution, give both coordinate pairs. For algebraic methods, return to Simultaneous Equations.
Worked example — use a supplied graph in a new way
Suppose only y = x² − 2x − 3 is supplied and you must solve x² − 3x − 4 = 0. Rearrange it to x² − 2x − 3 = x + 1. Draw y = x + 1 on the supplied axes; the x-values of the crossings give −1 and 4. Choose the extra line by making one side exactly match the supplied curve.
Think Like a Mathematician
Two roots of a quadratic are −3 and 5. Where is its symmetry line? It is halfway between them: x = 1. But the roots alone do not fix its height: y = (x + 3)(x − 5) and y = 2(x + 3)(x − 5) have the same roots and different minimum values. Think about what the graph tells you and what remains unknown.
Common Mistakes
Root or point? A root is an x-value; an intercept is a point. Wrong level? For f(x) = 2 use y = 2, not the x-axis. Lost signs? (−2)² = 4. Wrong bend? A y-intercept is not automatically a vertex. Too many digits? A graph estimate is limited by the grid.
Practice — Try It Yourself
Work on squared paper where a graph is requested. Try each question before opening its answer. Graph readings are estimates: give only the precision your scale supports.
1. Plot y = x² − 4 for integer x from −3 to 3. State both roots, the y-intercept and the turning point.
| x | y |
|---|---|
| -3 | 5 |
| -2 | 0 |
| -1 | -3 |
| 0 | -4 |
| 1 | -3 |
| 2 | 0 |
| 3 | 5 |
Roots: −2 and 2. Y-intercept and minimum: (0, −4). Symmetry line: x = 0. Check at least one negative input and use a smooth curve.
2. Sketch y = 4 − x². Give its maximum and x-intercepts.
It opens downwards. Maximum: (0, 4); x-intercepts: (−2, 0), (2, 0). Mark these and join with a smooth symmetric downward curve. For an extra check, (±1, 3) lies on it.
3. Without solving algebraically, explain how many real roots x² + 3 = 0 has.
None: the graph y = x² + 3 has minimum y = 3, above the x-axis. It cannot meet y = 0.
4. Use the worked graph of y = x² − 2x − 3 to estimate where y = −2.
Draw a horizontal line at −2. Read x ≈ −0.4 and x ≈ 2.4. A sensible hand-drawn tolerance is about ±0.1. Both readings are equally far from x = 1; this symmetry checks them.
5. For y = 2x² − 2, calculate the points at x = −1, 0, 1 and identify the turning point.
The points are (−1, 0), (0, −2), (1, 0). The curve opens upwards and is symmetric about x = 0, so the turning point is a minimum at (0, −2). The coefficient 2 does not make this graph-reading question Higher-only.
6. A height model h = 9 − t² is valid for 0 ≤ t ≤ 3. Its full graph reaches h = 5 at t = −2 and t = 2. Which answer is relevant?
t = 2. The other mathematical solution is outside 0 ≤ t ≤ 3. A domain restriction can remove a solution even when it is correct for the unrestricted equation.
7. Higher: state the vertex and whether it is a maximum or minimum for y = 3(x + 2)² − 7.
The bracket is zero at x = −2. The vertex is (−2, −7), a minimum because 3 is positive. The least value of y is −7.
8. Higher: using the supplied curve y = x² − 2x − 3, which line solves x² − x − 5 = 0? Estimate both simultaneous intersection points.
Rearrange to x² − 2x − 3 = 2 − x, so draw y = 2 − x. The x-values are about −1.79 and 2.79; using y = 2 − x gives intersections about (−1.79, 3.79) and (2.79, −0.79). Read approximately (−1.8, 3.8), (2.8, −0.8) from a suitable grid.
Frequently Asked Questions
Can a quadratic have just one real root?
Yes. If its turning point lies on the x-axis, it touches the axis once. This is a repeated root.
Should I calculate exact solutions?
If the question asks for a graphical estimate, show and read the graph. Quadratic Equations teaches exact algebraic methods; they can check your reading.
Do all plotted points need equal spacing?
The axis scales must be even. Your chosen x-values need not all be equally spaced: extra points near a turning point can improve the curve.
Continue Learning
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Next: Cubic, Reciprocal and Exponential Graphs
Higher connection: Circle Equations and Tangent Lines