how to find the quadratic equation from a graph

Question

This mail almost how to find a quadratic role's equation from its graph is function of a series of posts to assist you prepare for the Advanced Algebra and Functions function 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 $y=g(x)$ . Which of the following could be g's equation?

$g(x)=x^2-2x-3$
$g(x)=-x^2+2x+3$
$g(x)=(x-1)(x+3)$
$g(x)=-(x-1)^2-4$

Solution

What tin can you lot tell from the graph?

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

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

The x-intercepts are clues to m'southward factors: The x-intercept (-1, 0) tells you that $x+1$ is a nada, and the 10-intercept (3, 0) tells you lot that $x-3$ is a factor.

Sidebar: The gene theorem says that a polynomial that has a polynomial has a cistron $x-c$ has a root at c, and a polynomial that has a root atc has a factor $x-c$ . Aroot is the x-value at the 10-intercept: for this case, -1 and 3. And so the factors are $x+1$ and $x-3$ .

Multiply those factors together and meet what you go:

${\color{DarkBlue} (x+1)(x-3)=x^2-2x-3}$

The graph looks similar a parabola, a second-degree bend. And $x^2-2x-3$ is second degree. So you may be done. How can yous 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.

${\color{DarkBlue} g(x)=x^2-2x-3}$

then

${\color{DarkBlue} g(0)=0^2-2 \cdot 0-3}$

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 effort each answer option. The easiest way to practice that, I recollect, is to substitute 0 for xand run across if you get that $\inline \fn_phv g\left ( 0 \right )=-3$ , the y-intercept:

$g(x)=x^2-2x-3$ . Then $g(0)=-3$ . This is the answer we found. Information technology works.
$g(x)=-x^2+2x+3$ . $g(0)=3\neq -3$ . Not this one.
$g(x)=(x-1)(x+3)$ . $g(0)=-3$ . Looks like this works.
$g(x)=-(x-1)^2-4$ . $g(0)=-5$ . Does not work.

So we have two contenders, respond option #1 and answer pick #3. That means we have to try at least one more than point besides the y-intercept. We accept already done that for answer choice #1; nosotros know the ten-intercepts work for it. Let's see if $y=0$ at $x=-1$ for reply choice #three, $g(x)=(x-1)(x+3)$ .