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QuestionTransform y=x4y=x^{4} to y=4[3(x+2)]4−6y=4[3(x+2)]^{4}-6. Identify transformations, complete the table, and sketch the graph.

Studdy Solution

STEP 1

Assumptions1. The original function is y=x4y=x^{4}. . The transformed function is y=4[3(x+)]4−6y=4[3(x+)]^{4}-6.
3. The transformations are represented by the parameters in the transformed function.

STEP 2

The general form of the transformed function is y=a[b(x−h)]n+ky=a[b(x-h)]^{n}+k. Here, aa is the vertical stretch or compression factor, bb is the horizontal stretch or compression factor, hh is the horizontal shift, kk is the vertical shift, and nn is the power of xx.

STEP 3

By comparing the transformed function y=[3(x+2)]−6y=[3(x+2)]^{}-6 with the general form, we can identify the parameters.a=a=, b=3b=3, h=−2h=-2, k=−6k=-6, and n=n=.

STEP 4

The transformations corresponding to the parameters are as follows1. a=4a=4: Vertical stretch by a factor of4.
2. b=3b=3: Horizontal compression by a factor of1/3.
3. h=−2h=-2: Horizontal shift2 units to the left.
4. k=−6k=-6: Vertical shift6 units down. . n=4n=4: The graph remains unchanged as the power of xx is the same in both the original and transformed functions.

STEP 5

To complete the table, we need to apply the transformations to the xx-coordinates of the original function and then calculate the corresponding yy-coordinates.

STEP 6

For the point (−2,16)(-2,16) in the original function, after applying the transformations, the new xx-coordinate is −2∗1/3−2=−2.67-2*1/3-2=-2.67 and the new yy-coordinate is 4[3(−2.67+2)]4−64[3(-2.67+2)]^{4}-6.

STEP 7

Similarly, for the other points in the original function, calculate the new xx and yy coordinates.

STEP 8

To sketch the graph of y=4[3(x+2)]4−6y=4[3(x+2)]^{4}-6, plot the transformed points and connect them with a smooth curve.

STEP 9

The domain of the function y=4[3(x+2)]4−6y=4[3(x+2)]^{4}-6 is all real numbers because the function is defined for all xx.

STEP 10

The range of the function y=4[3(x+2)]4−6y=4[3(x+2)]^{4}-6 is y≥−6y \geq -6 because the function is a quartic function with a minimum value of −6-6.

STEP 11

The vertex of the function y=4[3(x+)]4−6y=4[3(x+)]^{4}-6 is the point (−,−6)(-,-6) because this is the minimum point of the function.

STEP 12

The equation of the axis of symmetry is x=−2x=-2 because the function is symmetric about the line x=−2x=-2.

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