Quadratic Function Graphs: Theory and Exercises

From predicting the trajectory of a kicked football to designing satellite dishes and calculating maximum profit in business economics, quadratic functions are among the most essential mathematical models in algebra and calculus.

A quadratic function is a polynomial function of degree 2. When plotted on a Cartesian coordinate grid, its graph creates a distinctive curved U-shaped line known as a parabola.

Understanding how to analyse, factorise, and sketch quadratic functions is a foundational topic across all GCSE and A-Level Mathematics specifications.

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What Defines a Quadratic Function?

The word quadratic originates from the Latin word quadratum, meaning "square." An algebraic function is classified as quadratic if the highest exponent (power) of the independent variable x is exactly 2.

The standard form of a quadratic function is written as:

Where:

• a, b, and c are real constant numbers.
• $a \neq 0$ (if $a = 0$, the $x^2$ term disappears, leaving a linear function $y = bx + c$).
• a is the leading coefficient, b is the linear coefficient, and c is the constant term.

Identifying Quadratic Equations

To determine whether an equation represents a true quadratic function, expand all brackets, collect like terms, and check the highest power of x₂:

Equation	Is it Quadratic?	Reason
	Yes	Highest power of is 2.
	No	Degree is 4 due to the term.
	No	Linear function (degree 1).
	Yes	Expands to degree 2 polynomial.
	No	The terms cancel out, leaving a constant term (degree 0).

Sketching Parabola Graphs

Every quadratic function produces a symmetric curve called a parabola. To sketch a parabola accurately, you need to identify four key geometric features:

Direction of Opening

The sign of the leading coefficient a determines the orientation of the parabola:

• If a > 0 (positive): The parabola opens upwards ($\cup)$, forming a "u-shape." The vertex represents the absolute minimum point of the curve.
• If a < 0 (negative): The parabola opens downwards ($\cap$), forming an "n-shape." The vertex represents the absolute maximum point of the curve.

Vertex (Turning Point)

The vertex is the central peak or trough where the parabola changes direction.

• The x-coordinate of the vertex is given by the formula: $x = - \dfrac{b}{2a}$
• The y-coordinate is found by evaluating f(x) at $x = -\dfrac{b}{2a}$:

Axis of Symmetry

A parabola is perfectly symmetrical. The axis of symmetry is a vertical line that passes directly through the vertex, dividing the curve into two identical mirror halves:

Intercepts with the Axes

• y-intercept: The point where the curve crosses the vertical y-axis (where x = 0). For $f(x) = ax^2 + bx + c$, the y-intercept is always (0, c).
• x-intercepts (Roots or Zeros): The points where the curve crosses the horizontal x-axis (where y = 0). Found by solving $ax^2 + bx + c = 0$.

The number of x-intercepts is governed by the discriminant ($\Delta = b^2 - 4ac$):

• $\Delta > 0$: Two distinct real roots (the curve crosses the x-axis twice).
• $\Delta = 0$: Exactly one real root (the vertex sits directly on the x-axis).
• $\Delta < 0$: No real roots (the curve floats entirely above or below the x-axis).

Step-by-Step Graphing Method

To graph any quadratic function $f(x) = ax^2 + bx + c$, follow these five steps:

1. Determine direction: Check the sign of a to see if the parabola opens upwards or downwards.
2. Calculate the vertex: Find $x = -\dfrac{b}{2a}$ and evaluate $y = f(x)$.
3. Find the y-intercept: Set $x = 0 \implies (0, c)$.
4. Find the x-intercepts: Set $y = 0$ and solve $ax^2 + bx + c = 0$ using factorisation, completing the square, or the quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
5. Plot and sketch: Mark all calculated points on a grid and draw a smooth, continuous U-shaped curve.

Worked Example

Problem: Sketch the graph of the quadratic function $f(x) = x^2 + 6x + 8$

Step-by-Step Solution:

• Step 1: Determine orientation: Here a = 1, b = 6, and c = 8. Since a = 1 > 0, the parabola opens upwards ($\cup$).
• Step 2: Find the vertex: Calculate the x-coordinate:

Calculate the y-coordinate by substituting x = -3 into f(x):

The vertex is at (-3, -1) and the axis of symmetry is the line x = -3.

• Step 3: Find the y-intercept: Set x = 0:

• Step 4: Find the x-intercepts: Set f(x) = 0:

Factorise the quadratic expression:

Solve for x:

The x-intercepts are at (-2, 0) and (-4, 0).

• Step 5: Plot the graph:

Practice Questions & Answers