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Quadratic graphs

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Quadratic graphs

​​In a nutshell

Quadratic graphs have equations of the form y=ax2+bx+cy=ax^2+bx+c and, when plotted, are curved like a \cup or like an \cap. Solving a quadratic equation using the corresponding quadratic graph is a case of identifying the xx-intercepts.



This summary assumes you are comfortable with at least one of the following methods: factorising quadratics; completing the square of a quadratic;  using the quadratic formula.

What do quadratic graphs look like?

Quadratic graphs have either a \cup-shape (a positive quadratic, on the left below) or a \cap-shape (a negative quadratic, on the right below):

Maths; Other graphs; KS3 Year 7; Quadratic graphs

What do their equations look like?

Quadratic equations have the form ​


where aa, bb and cc are constant numbers and aa is not 00​. 

If aa is positive, then the graph will be \cup-shaped. If aa is negative, then it will be \cap-shaped. This is a quick, first indication as to what your quadratic curve will look like based on its equation.

Note: Unlike with linear equations, there is not a constant value for the gradient. This is because the gradient is not constant on a quadratic curve.

For example:


This is the equation of a quadratic graph, where a=1a=1, b=5b=5 and c=6c=6

Identifying the xx-intercepts

When looking for xx-intercepts on any graph, set yy​ in the equation to 00 and work out the corresponding xx-value(s). Hence, to find the xx-intercepts of the graph of y=ax2+bx+cy=ax^2+bx+c, you solve: 


This can be achieved using one of the three methods outlined in the prerequisites at the top.

Example 1

Find the xx-intercepts of the curve with equation 


First set y=0y=0, then factorise: 



x=3x=-3 and x=2x=-2

These give the two xx-intercepts:

(3,0)\underline{(-3,0)} and (2,0)\underline{(-2,0)}​​

​​Drawing a quadratic graph

Without using graphing software, only a rough sketch of a quadratic curve is enough. There are important features of the graph that you can get right and use in your sketch:

  • the shape: either \cup-shaped or \cap-shaped. This is based on the aa-term in y=ax2+bx+cy=ax^2+bx+c;​
  • the yy-intercept: this is the cc-term in y=ax2+bx+cy=ax^2+bx+c;
  • the xx-intercept(s): sometimes there are two, sometimes one and sometimes none. Solving 0=ax2+bx+c0=ax^2+bx+c find these.​

Once you have established these three characteristics of the quadratic curve, you can give a fairly accurate sketch of it.


Ensure the equation is in the form y=ax2+bx+cy=ax^2+bx+c. Rearrange to this if necessary.​
Identify the shape of the quadratic curve. If a>0a>0 then it is \cup-shaped, if a<0a<0, then it is \cap-shaped.​
Identify the yy-intercept. This is the cc-term in the equation of the curve. Mark this point on: (0,c)(0,c). Your curve will pass through it.​
By solving ax2+bx+c=0ax^2+bx+c=0 (either by factorising, completing the square or using the quadratic formula), identify the xx-intercept(s). Mark however many there are (zero, one or two) on the xx-axis. If there are no solutions and hence no xx-intercepts, then you may have to use graph transformations to draw the curve.
Join the points in the correct curve.

Example 2

Draw the curve that has equation


For this equation, a=1a=1 which is positive, so this curve is \cup-shaped. Since c=6c=6, the yy-intercept is at (0,6)(0,6). Finally, as given in the example above, the xx-intercepts are at (2,0)(-2,0) and (3,0)(-3,0). Combining these gives the following sketch:​

Maths; Other graphs; KS3 Year 7; Quadratic graphs

Solving quadratic equations using a graph

To find the xx-intercept(s) of a quadratic curve, you have to solve a quadratic equation. This works the other way too: if you have the graph of y=ax2+bx+cy=ax^2+bx+c​, then the xx-intercepts give the solutions to the quadratic equation ax2+bx+c=0ax^2+bx+c=0.