QuestionIdentify the parent function for $f(x)=\frac{2}{3}(x-8)^{3}$ and its transformations: reflections, stretches, vertical shifts, horizontal shifts.

Studdy Solution

STEP 1

Assumptions1. The given function is $f(x)=\frac{}{3}(x-8)^{3}$ . . We need to identify the parent function from a given list.
3. We need to identify the transformations applied to the parent function to obtain the given function.

STEP 2

First, we need to identify the parent function. The parent function is the simplest form of the given function, without any transformations. Looking at the given function, we can see that the base function is x^.So, the parent function is $y=x^{}$ .

STEP 3

Next, we need to identify the transformations applied to the parent function. The general form of a cubic function with transformations is $y=a(x-h)^{3}+k$ , where $a$ is the vertical stretch/compression factor, $h$ is the horizontal shift, and $k$ is the vertical shift.
Comparing this with the given function, we can see that $a=\frac{2}{3}$ , $h=8$ , and $k=0$ .

STEP 4

The value of $a=\frac{2}{3}$ is less than1 and greater than0, which indicates a vertical compression by a factor of $\frac{2}{3}$ . So, the correct transformation for Stretches/Compressions is "Vertical compression by a factor of $\frac{2}{3}$ ".

STEP 5

The value of $h=8$ indicates a horizontal shift to the right by8 units. So, the correct transformation for Horizontal Shifts is "Shift right by8 units".

STEP 6

The value of $k=0$ indicates that there is no vertical shift. So, the correct transformation for Vertical Shifts is "No vertical shift".

STEP 7

Since the value of $a$ is positive, there is no reflection over the x-axis. And since the function is not written in the form $y=a(-x)^{3}$ , there is no reflection over the y-axis. So, the correct transformation for Reflections is "No reflection".
In conclusion, the parent function is $y=x^{3}$ , and the transformations applied are a vertical compression by a factor of $\frac{2}{3}$ , a shift to the right by units, and no reflection or vertical shift.

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QuestionIdentify the parent function for $f(x)=\frac{2}{3}(x-8)^{3}$ and its transformations: reflections, stretches, vertical shifts, horizontal shifts.

Studdy Solution

STEP 1

Assumptions1. The given function is $f(x)=\frac{}{3}(x-8)^{3}$ . . We need to identify the parent function from a given list.
3. We need to identify the transformations applied to the parent function to obtain the given function.

STEP 2

First, we need to identify the parent function. The parent function is the simplest form of the given function, without any transformations. Looking at the given function, we can see that the base function is x^.So, the parent function is $y=x^{}$ .

STEP 3

STEP 4

The value of $a=\frac{2}{3}$ is less than1 and greater than0, which indicates a vertical compression by a factor of $\frac{2}{3}$ . So, the correct transformation for Stretches/Compressions is "Vertical compression by a factor of $\frac{2}{3}$ ".

STEP 5

The value of $h=8$ indicates a horizontal shift to the right by8 units. So, the correct transformation for Horizontal Shifts is "Shift right by8 units".

STEP 6

The value of $k=0$ indicates that there is no vertical shift. So, the correct transformation for Vertical Shifts is "No vertical shift".

STEP 7

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QuestionIdentify the parent function for f(x)=23(x−8)3f(x)=\frac{2}{3}(x-8)^{3}f(x)=32​(x−8)3 and its transformations: reflections, stretches, vertical shifts, horizontal shifts.

Studdy Solution

STEP 1

Assumptions1. The given function is f(x)=3(x−8)3f(x)=\frac{}{3}(x-8)^{3}f(x)=3​(x−8)3. . We need to identify the parent function from a given list.3. We need to identify the transformations applied to the parent function to obtain the given function.

STEP 2

First, we need to identify the parent function. The parent function is the simplest form of the given function, without any transformations. Looking at the given function, we can see that the base function is x^.So, the parent function is y=xy=x^{}y=x.

STEP 3

STEP 4

The value of a=23a=\frac{2}{3}a=32​ is less than1 and greater than0, which indicates a vertical compression by a factor of 23\frac{2}{3}32​. So, the correct transformation for Stretches/Compressions is "Vertical compression by a factor of 23\frac{2}{3}32​".

STEP 5

The value of h=8h=8h=8 indicates a horizontal shift to the right by8 units. So, the correct transformation for Horizontal Shifts is "Shift right by8 units".

STEP 6

The value of k=0k=0k=0 indicates that there is no vertical shift. So, the correct transformation for Vertical Shifts is "No vertical shift".

STEP 7

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QuestionIdentify the parent function for $f(x)=\frac{2}{3}(x-8)^{3}$ and its transformations: reflections, stretches, vertical shifts, horizontal shifts.

Assumptions1. The given function is $f(x)=\frac{}{3}(x-8)^{3}$ . . We need to identify the parent function from a given list.
3. We need to identify the transformations applied to the parent function to obtain the given function.

First, we need to identify the parent function. The parent function is the simplest form of the given function, without any transformations. Looking at the given function, we can see that the base function is x^.So, the parent function is $y=x^{}$ .

The value of $a=\frac{2}{3}$ is less than1 and greater than0, which indicates a vertical compression by a factor of $\frac{2}{3}$ . So, the correct transformation for Stretches/Compressions is "Vertical compression by a factor of $\frac{2}{3}$ ".

The value of $h=8$ indicates a horizontal shift to the right by8 units. So, the correct transformation for Horizontal Shifts is "Shift right by8 units".

The value of $k=0$ indicates that there is no vertical shift. So, the correct transformation for Vertical Shifts is "No vertical shift".