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Question Find the vertex of the quadratic function y=−2x2+4x−3y=-2x^2+4x-3 and determine if it is a maximum or minimum.
The given function has a Minimum\textbf{Minimum} at y=−1\textbf{y}=-\textbf{1}.

Studdy Solution

STEP 1

Assumptions
1. The given function is a quadratic function in the form y=ax2+bx+cy = ax^2 + bx + c
2. The vertex of a quadratic function is given by the point (h,k)(h, k), where h=−b2ah = -\frac{b}{2a} and kk is the value of the function at hh
3. The function has a maximum if a<0a < 0 and a minimum if a>0a > 0

STEP 2

First, we need to find the xx-coordinate of the vertex, hh, which is given by h=−b2ah = -\frac{b}{2a}. In the given function, a=−2a = -2 and b=4b = 4.
h=−b2ah = -\frac{b}{2a}

STEP 3

Now, plug in the given values for aa and bb to calculate hh.
h=−42×−2h = -\frac{4}{2 \times -2}

STEP 4

Calculate the value of hh.
h=−4−4=1h = -\frac{4}{-4} = 1

STEP 5

Now that we have the xx-coordinate of the vertex, we can find the yy-coordinate, kk, by substituting hh into the function.
k=−2(1)2+4(1)−3k = -2(1)^2 + 4(1) - 3

STEP 6

Calculate the value of kk.
k=−2+4−3=−1k = -2 + 4 - 3 = -1

STEP 7

Now that we have the vertex of the function, we can determine whether it is a maximum or a minimum by looking at the sign of aa. If a<0a < 0, the function has a maximum. If a>0a > 0, the function has a minimum. In the given function, a=−2a = -2.
Since a=−2<0a = -2 < 0, the function has a maximum.
The correct statements are: - Maximum - C. -1

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