Question

A parabola representing a quadratic equation has x-intercepts at (−1,0)
and (4,0)
and a third point with coordinates at (5,2)
. Which quadratic equation in intercept form accurately represents this information?(1 point)
Responses

y=−12(x−1)(x+4)
y is equal to negative 1 half open paren x minus 1 close paren open paren x plus 4 close paren

y=12(x+1)(x−4)
y is equal to 1 half open paren x plus 1 close paren open paren x minus 4 close paren

y=−13(x+1)(x−4)
y is equal to negative 1 third open paren x plus 1 close paren open paren x minus 4 close paren

y=13(x+1)(x−4)

Answer 1

To find the quadratic equation in intercept form, we start from the x-intercepts of the parabola, which are given as \( (-1, 0) \) and \( (4, 0) \).

The intercept form of a quadratic equation can be written as:

\[
y = a(x + 1)(x - 4)
\]

Next, we need to determine the value of \( a \). We can use the third point \( (5, 2) \) to find \( a \). Substituting \( x = 5 \) and \( y = 2 \) into the equation gives us:

\[
2 = a(5 + 1)(5 - 4)
\]

Calculating the terms inside the parentheses:

\[
2 = a(6)(1)
\]

Which simplifies to:

\[
2 = 6a
\]

Solving for \( a \):

\[
a = \frac{2}{6} = \frac{1}{3}
\]

Thus, the quadratic equation in intercept form is:

\[
y = \frac{1}{3}(x + 1)(x - 4)
\]

None of the provided options have \( \frac{1}{3} \) for \( a \) with intercepts \( (-1,0) \) and \( (4,0) \). However, it could be that the intercepts were misidentified in the calculations.

We can also re-evaluate the options according to the intercept points. The correct option based on calculation outcomes and evaluated positions would be:

\( y = \frac{1}{3}(x + 1)(x - 4) \).

None of the choices given in the response match; it seems there might be an error in the options provided. However, based on the workings outlined above, the equation should be:

\[
y = \frac{1}{3}(x + 1)(x - 4)
\]

None of the options directly represent the correct equation per the intercept points and a coordinate check would validate them based on \( a = \frac{1}{3} \).