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

Question

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=−1/3(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=1/3(x+1)(x−4)
y is equal to 1 third open paren x plus 1 close paren open paren x minus 4 close paren

y=1/2(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=−1/2(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

Answer 1

To find the quadratic equation in intercept form given the x-intercepts at \((-1,0)\) and \((4,0)\) and the point \((5,2)\), we start with the intercept form of a quadratic equation:

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

Next, we need to find the value of \(a\). We'll use the point \((5, 2)\) to do this. Plugging \(x = 5\) and \(y = 2\) into the equation gives us:

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

This simplifies to:

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

So:

\[ 2 = 6a \]

Solving for \(a\) gives:

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

Now we substitute \(a\) back into the quadratic equation:

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

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

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

Thus, the correct response is:

y = 1/3(x+1)(x−4)