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Summary
Quadratic graphs have equations of the form and, when plotted, are curved like a or like an . Solving a quadratic equation using the corresponding quadratic graph is a case of identifying the -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.
Quadratic graphs have either a -shape (a positive quadratic, on the left below) or a -shape (a negative quadratic, on the right below):
Quadratic equations have the form
where , and are constant numbers and is not .
If is positive, then the graph will be -shaped. If is negative, then it will be -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 , and .
When looking for -intercepts on any graph, set in the equation to and work out the corresponding -value(s). Hence, to find the -intercepts of the graph of , you solve:
This can be achieved using one of the three methods outlined in the prerequisites at the top.
Find the -intercepts of the curve with equation
.
First set , then factorise:
Hence
and
These give the two -intercepts:
and
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:
Once you have established these three characteristics of the quadratic curve, you can give a fairly accurate sketch of it.
1. | Ensure the equation is in the form . Rearrange to this if necessary. |
| 2. | Identify the shape of the quadratic curve. If then it is -shaped, if , then it is -shaped. |
| 3. | Identify the -intercept. This is the -term in the equation of the curve. Mark this point on: . Your curve will pass through it. |
| 4. | By solving (either by factorising, completing the square or using the quadratic formula), identify the -intercept(s). Mark however many there are (zero, one or two) on the -axis. If there are no solutions and hence no -intercepts, then you may have to use graph transformations to draw the curve. |
5. | Join the points in the correct curve. |
Draw the curve that has equation
For this equation, which is positive, so this curve is -shaped. Since , the -intercept is at . Finally, as given in the example above, the -intercepts are at and . Combining these gives the following sketch:
To find the -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 , then the -intercepts give the solutions to the quadratic equation .