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

Question

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)

Answer 1

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:

\[ 4 = a(-2)(2) \] \[ 4 = -4a \]

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).