Graphs of quadratic functions Revision Notes for AQA GCSE Further Maths

Graphs of quadratic functions

What are quadratic graphs?

Quadratic functions create graphs that form a distinctive curved shape called a parabola. When you plot any quadratic function of the form $y = ax^2 + bx + c$ , you'll always get this characteristic U-shaped curve, though it might be flipped upside down depending on the values in your equation.

Understanding these graphs is crucial for GCSE mathematics, as they appear frequently in exam questions and help you visualise how quadratic relationships work in real-world situations.

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The term "parabola" comes from Greek meaning "to place beside" - this refers to how the curve can be constructed geometrically. Every quadratic function produces this same basic shape, making pattern recognition a powerful tool in mathematics.

Shape and direction of parabolas

The most important thing to remember about parabola direction is that it's entirely determined by the coefficient of x² (the 'a' value in $y = ax^2 + bx + c$ ).

When the coefficient of x² is positive ( $a > 0$ ):

The parabola opens upward, creating a U-shape
The vertex represents the lowest point on the graph
This lowest point is called the minimum value

When the coefficient of x² is negative ( $a \< 0$ ):

The parabola opens downward, creating an upside-down U-shape
The vertex represents the highest point on the graph
This highest point is called the maximum value

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Key Rule: Always check the sign of your x² coefficient first to determine which way your parabola opens! This single check can prevent many common errors in exam questions.

Key features of quadratic graphs

The vertex

The vertex is the turning point of your parabola - it's either the highest or lowest point on the curve. Finding the vertex is essential because it tells you the maximum or minimum value of your quadratic function.

The vertex coordinates give you valuable information:

The x-coordinate tells you where the turning point occurs
The y-coordinate tells you the actual maximum or minimum value

Line of symmetry

Every parabola has a vertical line of symmetry that passes directly through the vertex. This means that if you were to fold the graph along this line, both halves of the parabola would match up perfectly.

The line of symmetry has a simple equation: $x = \text{(x-coordinate of vertex)}$

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This symmetry property is incredibly useful for checking your work and predicting values on either side of the vertex. If you know a point $(a, b)$ is on the parabola, then the point symmetric to it across the line of symmetry will also be on the parabola.

Y-intercept

The y-intercept is where your parabola crosses the y-axis. This occurs when $x = 0$ , so you can find it by substituting $x = 0$ into your quadratic equation.

For $y = ax^2 + bx + c$ , the y-intercept is simply the constant term 'c'.

Completing the square method

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a form that makes the vertex easy to identify. This method is particularly useful for finding key features of quadratic graphs.

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Worked Example: Finding Key Features Using Completing the Square

Find the vertex, line of symmetry, and y-intercept for $y = x^2 + 6x + 11$

Step 1: Complete the square Starting with: $x^2 + 6x + 11$

We want to express this as: $(x + a)^2 + b$

Expanding $(x + a)^2$ gives us: $x^2 + 2ax + a^2$

Step 2: Compare coefficients

Coefficient of x: $6 = 2a$ , so $a = 3$
For the constant terms: $11 = a^2 + b = 9 + b$ , so $b = 2$

Therefore: $x^2 + 6x + 11 = (x + 3)^2 + 2$

Step 3: Find the key features

From the completed square form $(x + 3)^2 + 2$ :

(i) The vertex: Since $(x + 3)^2$ is always positive or zero, its minimum value is 0, which occurs when $x = -3$ . At this point, $y = 0 + 2 = 2$ . So the vertex is at $(-3, 2)$ .

(ii) The line of symmetry: This passes through the vertex, so the equation is $x = -3$ .

(iii) The y-intercept: When $x = 0$ , $y = (0 + 3)^2 + 2 = 9 + 2 = 11$ . So the y-intercept is at $(0, 11)$ .

Plotting quadratic graphs using tables

When plotting quadratic graphs, creating a table of values helps ensure accuracy. Choose x-values that include points on both sides of the vertex to capture the parabola's complete shape.

For the function $y = x^2 - 5$ , you can see how the values decrease to a minimum at $x = 0$ (where $y = -5$ ) and then increase symmetrically on both sides.

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Pro tip: When creating your table of values, always include the vertex and at least two points on each side of the line of symmetry. This ensures you capture the parabola's true shape and can check your work using symmetry.

Common exam tips and problem-solving methods

The following strategies will help you tackle quadratic graph questions confidently and avoid common pitfalls.

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Exam Trap Alerts

Sign errors: Always double-check the sign of your x² coefficient - it determines everything about your parabola's direction
Vertex confusion: Remember that for $(x + a)^2 + b$ , the vertex x-coordinate is $-a$ , not $+a$
Symmetry mistakes: The line of symmetry always has the equation $x = \text{(vertex x-coordinate)}$

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Problem-Solving Approach

Identify the quadratic form: Look for $y = ax^2 + bx + c$
Determine direction: Check if $a$ is positive (opens up) or negative (opens down)
Complete the square: Transform to $(x + p)^2 + q$ form to find the vertex easily
Find key features: Use the completed square form to identify vertex, line of symmetry, and intercepts
Sketch carefully: Use symmetry to ensure both sides of your parabola match

Quick checks for accuracy:

The parabola should be symmetric about the line of symmetry
If $a > 0$ , the vertex should be the lowest point
If $a \< 0$ , the vertex should be the highest point
The y-intercept should match the constant term in your original equation

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Key Points to Remember:

Parabola direction is determined by the coefficient of x² - positive means opens upward, negative means opens downward
The vertex is the turning point - it's either the maximum or minimum value of your quadratic function
Every parabola has a line of symmetry that passes through the vertex and has equation $x = \text{(vertex x-coordinate)}$
Completing the square transforms $y = ax^2 + bx + c$ into $(x + p)^2 + q$ form, making it easy to identify the vertex at $(-p, q)$
Always check your work using symmetry - points equidistant from the line of symmetry should have the same y-value

Graphs of quadratic functions (AQA GCSE Further Maths): Revision Notes