Example 1

The zeros of a quadratic function are −2 and −4.
A point on the graph of the same quadratic function is (−5, −9).
Determine the equation of the quadratic function in factored form and in standard form.

Step 1: Organize the information that is given.

The zeros of the function are −2 and −4. So (−2, 0) and (−4, 0) are the x-intercepts of the graph of the function.

(−5, −9) is a point on the graph of the quadratic function, so when x = −5, y = −9.

Step 2: Select a form that works with the given information.

Because the zeros are known, the factored form of a quadratic function would be a good form to work with. Since the zeros are −2 and −4, two factors are (x + 2) and (x + 4). However, it's possible that there is also a GCF. As such, the function can be stated as , where a corresponds to the GCF.

Step 3: Determine the value of a.

Substitute the given point, (−5, −9), into the equation of the function and solve for a.

Step 4: Write the equation of the quadratic function in factored form.

Step 5: Write the equation of the quadratic function in standard form by expanding the factored form.

Step 6: Final statement.

Factored form:

Standard form: