A parabola representing a quadratic equation has x-intercepts at (1,0)

Question

A parabola representing a quadratic equation has x-intercepts at  (1,0)
  and  (−3,0)
  and a third point with coordinates at  (−1,4)
 . Which quadratic equation in intercept form accurately represents this information?(1 point)
Responses

y=1(x−1)(x+3)
y is equal to 1 open paren x minus 1 close paren open paren x plus 3 close paren

y=−1(x−1)(x+3)
y is equal to negative 1 open paren x minus 1 close paren open paren x plus 3 close paren

y=−1(x−1)(x−3)
y is equal to negative 1 open paren x minus 1 close paren open paren x minus 3 close paren

y=−1(x+1)(x+3)

GPT-4o mini · Answer

To find the correct quadratic equation in intercept form that represents the given information, we can start from the x-intercepts.

We know that the x-intercepts are at $(1, 0)$ and $(- 3, 0)$ . The quadratic equation in intercept form can be expressed as:

$y = a (x - r_{1}) (x - r_{2})$

where $r_{1}$ and $r_{2}$ are the x-intercepts and $a$ is a constant that determines the orientation and steepness of the parabola. In this case, since the intercepts are $r_{1} = 1$ and $r_{2} = - 3$ , this gives us:

$y = a (x - 1) (x + 3)$

Now, to determine the value of $a$ , we can use the third point given, which is $(- 1, 4)$ . Plugging the coordinates of this point into the equation gives us:

$4 = a (- 1 - 1) (- 1 + 3)$

Calculating the terms inside the parentheses:

$4 = a (- 2) (2)$ $4 = - 4 a$

Solving for $a$ :

$a = - 1$

Now we substitute $a$ back into the equation:

$y = - 1 (x - 1) (x + 3)$

Thus, the quadratic equation that accurately represents this information is:

$y = - 1 (x - 1) (x + 3)$

So, the correct response is:

y = −1(x−1)(x+3).