The good news is that, at A Level, there is no new content on Quadratic Graphs. However, you will need to have a thorough understanding of the GCSE content.

To further revise GCSE content on linear and quadratic graphs, try this preparation for A Level resource or these multiple-choice prior knowledge questions. If you think you’ve grasped it, try this A Level task to test your understanding, or these exam-style questions on the whole of AS level coordinate geometry.

All quadratic graphs are parabolas: a symmetrical curve. The orientation of this parabola depends on whether the coefficient of *x*^{2} is positive or negative.

You can use your skills working with quadratic equations to find other key features of the graph:

- The
*y*-intercept. This is found by setting*x*= 0 and solving the given equation for*y*. - The
*x*-intercept(s). This is found by setting*y*= 0 and solving the given equation for*x*. There may be 0, 1, or 2*x*-intercepts. - The turning point. For a quadratic function, this will be the minimum (lowest point – if the coefficient of
*x*^{2}is positive) or the maximum (highest point – if the coefficient of*x*^{2}is negative). This can either be found by working out the average of the*x*-intercept values (where they exist) or by completing the square. A graph with an equation rearranged into the form*y*=*a*(*x*+*b*)^{2}+*c*has a turning point at (-*b*,*c*).

**Example Question**

Sketch the graph of *y* = *x*^{2} – 7*x* + 10, clearly indicating any points of intersection with the axes and the location of the turning point of the curve.

The coefficient of *x*^{2} is 1. This is positive, so our graph will be u-shaped.

The *y*-intercept is found by substituting *x* = 0 into the equation:

The x-intercepts are found by substituting *y* = 0 then solving the resulting equation:

To find the turning point, either find the average of the two *x*-intercepts or complete the square.

Finding the average gives us an* x*-coordinate of:

And a *y*-coordinate of:

Giving a turning point of .

Finding the turning point using completing the square will give the same value:

As a graph with an equation in the form *y* = *a*(*x* + *b*)^{2} + *c* has a turning point at (-*b*, *c*), the turning point in this example is .

The question also asks you to sketch the graph. When sketching a graph, it does not need to be to scale but should be the right shape and roughly in proportion, as shown:

**Practice Questions**

For each graph, find the *x*-intercept, *y*-intercept and turning point:

1. *y* = *x*^{2} + 4*x* – 5.

2. *y* = *x*^{2} + 8*x* – 9

3. *y* = –*x*^{2} + 7*x*

4. *y* = 2*x*^{2} + 17*x* + 8

5. *y* = 5*x*^{2} – 20*x* + 15

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