Question

What is the vertex form of the graphed equation?
(4, 8)
(1 point)
Responses

y=2(x−4)2−8
𝑦
=
2
(
𝑥
−
4
)
2
−
8
y equals 2 left parenthesis x minus 4 right parenthesis squared minus 8

y=2(x−4)2+8
𝑦
=
2
(
𝑥
−
4
)
2
+
8
y equals 2 left parenthesis x minus 4 right parenthesis squared plus 8

y=−2(x−4)2+8
𝑦
=
−
2
(
𝑥
−
4
)
2
+
8
y equals negative 2 left parenthesis x minus 4 right parenthesis squared plus 8

y=2(x+4)2+8

Answer 1

To find the vertex form of the graphed equation, we start with the vertex coordinates provided, which are (4, 8). The vertex form of a quadratic equation is given by:

\[
y = a(x - h)^2 + k
\]

where (h, k) is the vertex of the parabola, and \(a\) is a constant that determines the width and direction of the parabola.

From the vertex (4, 8):
- \(h = 4\)
- \(k = 8\)

Now we have:

\[
y = a(x - 4)^2 + 8
\]

The sign and value of \(a\) determine whether the parabola opens upwards or downwards. In the options provided, we see different possible values for \(a\). However, the only way to determine the correct value of \(a\) would be if we know another point that lies on the graph.

If we assume there is no specific value provided for \(a\) and that the parabola opens upwards, the general form of the vertex equation when \(a\) is positive is:

\[
y = 2(x - 4)^2 + 8
\]

This indicates that \(a\) is 2 and the parabola opens upwards.

If the parabola were to open downwards (which seems less likely given the vertex at (4, 8) being above the axis), the equation would be:

\[
y = -2(x - 4)^2 + 8
\]

Based on typical conventions and without further information about the direction of the parabola, it is more common to use a positive value for \(a\).

Here are the options you listed:

1. **y = 2(x - 4)^2 - 8** (Incorrect; the vertex should have a +8)
2. **y = 2(x - 4)^2 + 8** (Correct; assuming a is positive)
3. **y = -2(x - 4)^2 + 8** (Could be correct if we knew it opens downwards)
4. **y = 2(x + 4)^2 + 8** (Incorrect; h should be -4, not +4)

The most likely correct answer, with standard assumptions, is:

**y = 2(x - 4)^2 + 8**