What you'll learn
Quadratic graphs are the U-shaped (or ∩-shaped) curves of quadratic functions, and they have important features such as roots, the turning point and the line of symmetry. In this guide you will learn how to plot a quadratic graph, recognise its features, find the roots and turning point, and interpret quadratic graphs in context. This builds on non-linear graphs and quadratic equations.
Key terms and definitions
Quadratic graph — the graph of y = ax² + bx + c, a parabola.
Parabola — the symmetrical U- or ∩-shaped curve of a quadratic.
Roots — the x-values where the curve crosses the x-axis (where y = 0).
Turning point — the maximum or minimum point of the parabola.
Line of symmetry — the vertical line through the turning point.
Core concepts
The shape of a quadratic graph
A quadratic y = ax² + bx + c gives a parabola. If a is positive, the parabola is U-shaped with a minimum turning point; if a is negative, it is ∩-shaped with a maximum. The parabola is symmetrical about a vertical line through its turning point.
Plotting the graph
To plot, complete a table of values, substituting x-values into the equation (taking care with negatives and squaring), then plot the points and join them with a smooth curve. Choose enough x-values to show the turning point and where the curve crosses the axes.
Roots (where it crosses the x-axis)
The roots are where the curve crosses the x-axis, i.e. where y = 0. These are the solutions of the quadratic equation ax² + bx + c = 0. A parabola may cross the x-axis twice (two roots), once (a repeated root, just touching), or not at all (no real roots).
The turning point and line of symmetry
The turning point is the lowest point (minimum) or highest point (maximum) of the parabola. The line of symmetry is the vertical line through the turning point; it lies halfway between the two roots. You can find the line of symmetry by averaging the roots, and read the turning point's y-value from the graph or by substituting.
Interpreting in context
Quadratic graphs model real situations such as the height of a projectile over time. The maximum point gives the greatest height, the roots give when the height is zero (e.g. launch and landing), and the y-intercept (where x = 0) gives the starting value.
Worked examples
Example 1: Identifying shape and turning point
Describe the graph of y = x² − 4x + 3.
The x² term is positive, so it is a U-shaped parabola with a minimum. The roots (from x² − 4x + 3 = 0) are x = 1 and x = 3, so the line of symmetry is x = 2.
Example 2: Reading roots
A parabola crosses the x-axis at x = −2 and x = 4. What is the line of symmetry?
Halfway between the roots: (−2 + 4)/2 = x = 1.
Example 3: y-intercept
Where does y = x² + 2x − 5 cross the y-axis?
Set x = 0: y = −5. So it crosses at (0, −5).
Common mistakes and how to avoid them
Wrong orientation. Positive x² → U-shape (minimum); negative x² → ∩-shape (maximum).
Joining points with straight lines. Always draw a smooth curve.
Confusing roots and turning point. Roots are on the x-axis; the turning point is the max/min.
Forgetting symmetry. The line of symmetry is halfway between the roots.
Errors squaring negatives in the table of values.
Exam technique for Quadratic Graphs
Complete the table accurately and draw a smooth parabola.
Identify the shape from the sign of the x² term.
Find the roots where y = 0 and the turning point as the max/min.
Use symmetry to locate the line of symmetry between the roots.
Interpret features in context — maximum height, start value, and so on.
Quick revision summary
A quadratic graph is a parabola: U-shaped with a minimum if the x² coefficient is positive, ∩-shaped with a maximum if negative, and symmetrical about a vertical line through its turning point. Plot it by completing a table of values (careful with negatives and squaring) and drawing a smooth curve. The roots are where the curve crosses the x-axis (y = 0), i.e. the solutions of ax² + bx + c = 0; there may be two, one (touching), or none. The turning point is the maximum or minimum, and the line of symmetry runs vertically through it, halfway between the roots. The y-intercept (x = 0) gives the constant term. In context, the maximum gives the greatest value (e.g. height), the roots give where it's zero, and the y-intercept gives the start. Identify the shape from the x² sign, plot a smooth curve, find roots and turning point, use symmetry, and interpret the features in real situations.