Question

The function
\[g\] is given in three equivalent forms.
Which form most quickly reveals the vertex?
Choose 1 answer:
Choose 1 answer:
(Choice A)

\[g(x)=\dfrac{1}{2}(x-12)(x-4)\]
A

\[g(x)=\dfrac{1}{2}(x-12)(x-4)\]
(Choice B)

\[g(x)=\dfrac{1}{2}(x-8)^2-8\]
B

\[g(x)=\dfrac{1}{2}(x-8)^2-8\]
(Choice C)

\[g(x)=\dfrac{1}{2}x^2-8x+24\]
C

\[g(x)=\dfrac{1}{2}x^2-8x+24\]
What is the vertex?
Vertex =
\[(\]

\[,\]

\[)\]

Answer 1

To determine which form of the function \(g\) most quickly reveals the vertex, we should analyze each option:

**Choice A**:
\[ g(x) = \frac{1}{2}(x - 12)(x - 4) \]

This is in factored form, and while you can find the vertex from this by first finding the x-coordinate of the vertex as the midpoint of the roots \(x = 12\) and \(x = 4\):
\[
x = \frac{12 + 4}{2} = 8.
\]
You would then need to substitute \(x = 8\) back into the function to find the y-coordinate.

**Choice B**:
\[ g(x) = \frac{1}{2}(x - 8)^2 - 8 \]

This is in vertex form \(g(x) = a(x - h)^2 + k\). Here, the vertex is \((h, k) = (8, -8)\). This form directly gives us the vertex without further calculations.

**Choice C**:
\[ g(x) = \frac{1}{2}x^2 - 8x + 24 \]

This is in standard form, and you can find the vertex by completing the square or using the vertex formula \(x = -\frac{b}{2a}\):
\[
x = -\frac{-8}{2 \cdot \frac{1}{2}} = \frac{8}{1} = 8.
\]
Then you would plug \(x = 8\) back into the function to find the y-coordinate.

From these forms, **Choice B** most quickly reveals the vertex because it gives it directly as \((8, -8)\) without any need of additional calculations.

So, the vertex is:
\[
\boxed{(8, -8)}.
\]