A parabola representing a quadratic equation has x-intercepts at (2,0)

Question

(
2
,
0
)
and (4,0)
(
4
,
0
)
and a third point with coordinates at (5,6)
(
5
,
6
)
. Which quadratic equation in intercept form accurately represents this information?(1 point)
Responses

y=2(x−2)(x+4)
𝑦
=
2
(
𝑥
−
2
)
(
𝑥
+
4
)
y is equal to 2 open paren x minus 2 close paren open paren x plus 4 close paren

y=2(x−2)(x−4)
𝑦
=
2
(
𝑥
−
2
)
(
𝑥
−
4
)
y is equal to 2 open paren x minus 2 close paren open paren x minus 4 close paren

y=2(x+2)(x−4)
𝑦
=
2
(
𝑥
+
2
)
(
𝑥
−
4
)
y is equal to 2 open paren x plus 2 close paren open paren x minus 4 close paren

y=−2(x−2)(x−4)

Answer 1

To find the quadratic equation in intercept form that represents the given information, we need to consider the x-intercepts (roots) and also use the point (5, 6) to determine the correct vertical stretch or compression.

Given Information:

• X-intercepts: (2, 0) and (4, 0)
• Point: (5, 6)

The intercept form of a quadratic equation is given by:

\[ y = a(x - p)(x - q) \]

where \(p\) and \(q\) are the x-intercepts. Here, the x-intercepts are 2 and 4, so we can write:

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

Next, we need to determine the value of \(a\) using the point (5, 6). We substitute \(x = 5\) and \(y = 6\) into the equation:

\[ 6 = a(5 - 2)(5 - 4) \]

Calculating the factors:

\[ 6 = a(3)(1) \]

So:

\[ 6 = 3a \]

Now, solve for \(a\):

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

Substituting \(a\) back into the equation gives:

\[ y = 2(x - 2)(x - 4) \]

Conclusion:

Thus, the correct quadratic equation that represents the given information is:

\[ \boxed{y = 2(x - 2)(x - 4)} \]