## Question

This post about how to find a quadratic function’s equation from its graph is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test. This graph represents a polynomial function . Which of the following could be g‘s equation?

## Solution

What can you tell from the graph?

You can read the coordinates of the x-intercepts, the y-intercept, and the turning point.

 x-intercepts (-1, 0), (3, 0) y-intercept (0, -3) turning point (1, -4)

The x-intercepts are clues to g‘s factors: The x-intercept (-1, 0) tells you that is a zero, and the x-intercept (3, 0) tells you that is a factor.

Sidebar: The factor theorem says that a polynomial that has a polynomial has a factor has a root at c, and a polynomial that has a root at c has a factor . A root is the x-value at the x-intercept: for this example, -1 and 3. So the factors are and .

Multiply those factors together and see what you get:

The graph looks like a parabola, a second-degree curve. And is second degree. So you may be done. How can you check? Use one of the two other points you can read from the graph. The y-intercept is (0, -3). Plug in 0 for x and see if the equation gives you -3, the y-intercept.

If

then

And as we saw from the graph, the y-intercept is (0, -3). Check. So answer choice #1 is the correct one.

That is one way to find a quadratic function’s equation from its graph.

Alternatively, since this question is multiple choice, you could try each answer choice. The easiest way to do that, I think, is to substitute 0 for and see if you get that , the y-intercept:

1. .  Then . This is the answer we found. It works.
2. . Not this one.
3. .  Looks like this works.
4. . Does not work.

So we have two contenders, answer choice #1 and answer choice #3. That means we have to try at least one more point besides the y-intercept. We have already done that for answer choice #1; we know the x-intercepts work for it. Let’s see if  at  for answer choice #3, .

So answer choice #3 does not work after all, and answer choice #1 is the correct one.

For more about the end behavior of polynomial functions, see “End Behavior, Degree, and Leading Coefficient.”

The Accuplacer sample problem on which this one is based is #10 in the sample questions for the Accuplacer Advanced Algebra and Functions test.