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The function

\[m\] is given in three equivalent forms.
Which form most quickly reveals the vertex?
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[m(x)=-2(x-6)^2+18\]
A
\[m(x)=-2(x-6)^2+18\]
(Choice B)
\[m(x)=-2x^2+24x-54\]
B
\[m(x)=-2x^2+24x-54\]
(Choice C)
\[m(x)=-2(x-3)(x-9)\]
C
\[m(x)=-2(x-3)(x-9)\]
What is the vertex?
Vertex =
\[(\]

\[,\]

\[)\]

1 answer

To determine which form quickly reveals the vertex of the function \( m(x) \), we analyze the three given forms:

  1. Choice A: \( m(x) = -2(x-6)^2 + 18 \)

    • This is in vertex form, \( m(x) = a(x-h)^2 + k \), where the vertex is at \( (h, k) \). Here, \( h = 6 \) and \( k = 18 \), so the vertex is \( (6, 18) \).

  2. Choice B: \( m(x) = -2x^2 + 24x - 54 \)

    • This is in standard form. To find the vertex, we would use the formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 24 \). \[ x = -\frac{24}{2 \cdot -2} = 6 \] Then we would substitute \( x = 6 \) back into the equation to find \( m(6) \): \[ m(6) = -2(6)^2 + 24(6) - 54 = -72 + 144 - 54 = 18 \] So the vertex is again \( (6, 18) \).

  3. Choice C: \( m(x) = -2(x-3)(x-9) \)

    • This form is factored, and to find the vertex, we need to find the midpoint of the roots. The roots are at \( x = 3 \) and \( x = 9 \): \[ x_{vertex} = \frac{3 + 9}{2} = 6 \] We find \( y \) by substituting back to find \( m(6) \): \[ m(6) = -2(6-3)(6-9) = -2(3)(-3) = 18 \] Again, we find the vertex is \( (6, 18) \).

Among the three choices, Choice A quickly reveals the vertex because it is already in vertex form. Thus, the correct answer is:

Vertex = (6, 18)