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:

$$\boxed{{\displaystyle y=2(x-2)(x-4)}}$$