Using quadratic graphs Revision Notes for AQA GCSE Maths

Using quadratic graphs

What are quadratic graphs?

Quadratic graphs are U-shaped curves (called parabolas) that represent equations containing an $x^2$ term. These graphs can open upwards or downwards and are essential for solving quadratic equations visually.

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You can read values from quadratic graphs in two main ways:

Looking at points where the graph crosses the x-axis
Drawing horizontal lines on your graph to find specific values

Creating a table of values

To draw a quadratic graph, you first need to create a table of values. This systematic approach ensures accuracy when plotting your quadratic curve.

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Worked Example: Creating a Table of Values

For the equation $y = x^2 - x - 3$ :

Table

Key steps:

Choose x-values (usually including negative and positive integers)
Substitute each x-value into the equation
Calculate the corresponding y-values
Plot these coordinate pairs on a graph

The table of values forms the foundation for accurate graph plotting and helps you identify the key features of the quadratic curve.

Drawing the quadratic graph

Once you have your table of values, you can plot the graph using proper mathematical techniques:

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Critical Points for Drawing Quadratic Graphs:

Always join points with a smooth curve, never straight lines
Label your graph with its equation
Use graph paper for accuracy
The graph should be symmetrical

Remember that quadratic graphs have a distinctive smooth, curved shape that must be drawn carefully to maintain mathematical accuracy.

Finding solutions from graphs

The solutions of a quadratic equation like $x^2 - x - 3 = 0$ are the x-coordinates where the graph crosses the x-axis. These crossing points are also called the roots of the equation.

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Worked Example: Finding Solutions Graphically

Method:

Look where the curve crosses the x-axis (where $y = 0$ )
Read the x-values at these crossing points
Read values accurate to the nearest small square on your graph

For the graph of $y = x^2 - x - 3$ , if it crosses the x-axis at $x = -1.3$ and $x = 2.3$ , then these are the solutions to $x^2 - x - 3 = 0$ .

This graphical method provides a visual understanding of how many solutions exist and their approximate values.

Using horizontal lines to solve equations

You can solve more complex equations like $x^2 - 4x - 1 = 8$ using your quadratic graph by employing the horizontal line method.

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Worked Example: Solving Using Horizontal Lines

Method:

Draw a horizontal line on the graph at the required y-value (in this case, $y = 8$ )
Find where this line crosses the curve
Read down to the x-axis at these crossing points
These x-values are your solutions

This method works because you're finding where the quadratic expression equals the constant value.

This technique extends the usefulness of your quadratic graph beyond finding simple roots.

Exam tips

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Essential Exam Strategies:

Always read graphs carefully - accuracy is important
Use a ruler when drawing horizontal lines
Check your answers make sense by substituting back into the original equation
Show your working by indicating where you read values from the graph
Round answers to 1 decimal place unless told otherwise

Remember!

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Key Points to Remember:

Quadratic graphs are always smooth U-shaped curves - never join points with straight lines
Solutions to quadratic equations are found where the graph crosses the x-axis
Use horizontal lines to solve equations where the quadratic expression equals a specific value
Always label your graphs with their equations
Read values accurately from the graph using the grid squares to help you

Using quadratic graphs (AQA GCSE Maths): Revision Notes