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Math  /  Algebra

QuestionWhich function transforms the graph of v=x2v=x^{2} so that it is first shifted down 2 units and is then reflected across the 1λ\frac{1}{\lambda}-axis? A. y=−x2+2y=-x^{2}+2 B. y=−x2−2y=-x^{2}-2 c. y=(−x)2+2y=(-x)^{2}+2 D. y=−(x+2)2y=-(x+2)^{2}

Studdy Solution

STEP 1

1. The original function is v=x2 v = x^2 .
2. The transformation involves a vertical shift and a reflection.
3. The graph is shifted down 2 units.
4. The graph is reflected across the x-axis.

STEP 2

1. Determine the effect of shifting the graph down 2 units.
2. Determine the effect of reflecting the graph across the x-axis.
3. Combine both transformations to find the resulting function.

STEP 3

Shifting the graph of v=x2 v = x^2 down 2 units results in the function v=x2−2 v = x^2 - 2 .

STEP 4

Reflecting the graph across the x-axis changes the sign of the function. Thus, v=x2−2 v = x^2 - 2 becomes v=−(x2−2) v = -(x^2 - 2) .

STEP 5

Simplify the expression v=−(x2−2) v = -(x^2 - 2) to get v=−x2+2 v = -x^2 + 2 .
The function that transforms the graph of v=x2 v = x^2 by shifting it down 2 units and reflecting it across the x-axis is:
y=−x2+2 \boxed{y = -x^2 + 2}

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