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How Can You Find a Quadratic’s Equation from Its Graph?

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.

y=g(x). This parabola has x-intercepts at x=-1 and x=3, y-intercept at y=-3, and vertex at (0, -4).

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 \inline \fn_phv g\left ( 0 \right )=-3, 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.

Related: “Finding a Second Degree Equation from Its Solutions.”

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.