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

Question

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

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

y=1(x−1)(x+3)
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)

Answer 1

To find the quadratic equation in intercept form that represents the given information, we can start with the x-intercepts and the vertex point provided.

The x-intercepts are at \( (1, 0) \) and \( (-3, 0) \). In intercept form, a quadratic function can be expressed as:

\[ y = a(x - x_1)(x - x_2) \]

where \( x_1 \) and \( x_2 \) are the x-intercepts.

Substituting the x-intercepts into the equation:

\[ y = a(x - 1)(x + 3) \]

Next, we need to determine the value of \( a \). We know the point \( (-1, 4) \) lies on the parabola, meaning when \( x = -1 \), \( y = 4 \). We can substitute these values into the equation to find \( a \).

\[ 4 = a(-1 - 1)(-1 + 3) \]

Calculating the factors:

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

Solving for \( a \):

\[ a = \frac{4}{-4} = -1 \]

Thus, substituting \( a \) back into the equation gives:

\[ y = -1(x - 1)(x + 3) \]

Therefore, the quadratic equation in intercept form that accurately represents the information is:

\[ y = -1(x - 1)(x + 3) \]

So the correct response is:

y = −1(x−1)(x+3)