Math / Algebra

QuestionHomework: Final Exam Review Question 28, Setup \& Solve-5.3.41 Part 1 of 7 HW Score: 41.54\%, 394.62 of 950 points Points: 0 of 20 Save estion list
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For the function $\mathrm{G}(\mathrm{x})=-3+\frac{4}{(x-2)^{2}}$ , (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. (a) Which of the following transformations is required to graph the given function? A. Vertically stretch the graph of $y=\frac{1}{x^{2}}$ by a factor of 4 , shift it 2 units to the left, and 3 units down. B. Vertically stretch the graph of $y=\frac{1}{x^{2}}$ by a factor of 4 , shift it 2 units to the right, and 3 units down. C. Vertically stretch the graph of $y=\frac{1}{x^{2}}$ by a factor of 4 , shift it 3 units to the right, and 2 units down. D. Vertically stretch the graph of $y=\frac{1}{x^{2}}$ by a factor of 4 , shift it 2 units to the left, and 3 units up.

Studdy Solution

STEP 1

What is this asking? We need to figure out how to transform the simple function $y = \frac{1}{x^2}$ into our more complicated function $G(x) = -3 + \frac{4}{(x-2)^2}$ using stretches and shifts! Watch out! Don't mix up the order of the transformations!
And be super careful about whether shifts are up/down or left/right.

STEP 2

1. Rewrite the function
2. Analyze the transformations

STEP 3

Let's rewrite our function $G(x)$ to look a bit more like the basic function $y = \frac{1}{x^2}$ .
We can rewrite $G(x) = -3 + \frac{4}{(x-2)^2}$ as $G(x) = \frac{4}{(x-2)^2} - 3$ .
This helps us see the transformations more clearly.

STEP 4

The 4 multiplied in the numerator means we vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4.
This makes the graph taller!
We now have $y = \frac{4}{x^2}$ .

STEP 5

Replacing $x$ with $(x-2)$ means we shift the graph 2 units to the right.
Think about it: we need a bigger $x$ in $(x-2)$ to get the same result as a smaller $x$ in the original.
We now have $y = \frac{4}{(x-2)^2}$ .

STEP 6

Finally, subtracting 3 means we shift the graph 3 units down.
This moves the entire graph downwards.
We now have $y = \frac{4}{(x-2)^2} - 3$ , which is our function $G(x)$ !

STEP 7

The correct transformation is B.
We vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4, shift it 2 units to the right, and 3 units down.

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Studdy Solution

STEP 1

STEP 2

1. Rewrite the function
2. Analyze the transformations

STEP 3

Let's rewrite our function $G(x)$ to look a bit more like the basic function $y = \frac{1}{x^2}$ .
We can rewrite $G(x) = -3 + \frac{4}{(x-2)^2}$ as $G(x) = \frac{4}{(x-2)^2} - 3$ .
This helps us see the transformations more clearly.

STEP 4

The 4 multiplied in the numerator means we vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4.
This makes the graph taller!
We now have $y = \frac{4}{x^2}$ .

STEP 5

Replacing $x$ with $(x-2)$ means we shift the graph 2 units to the right.
Think about it: we need a bigger $x$ in $(x-2)$ to get the same result as a smaller $x$ in the original.
We now have $y = \frac{4}{(x-2)^2}$ .

STEP 6

Finally, subtracting 3 means we shift the graph 3 units down.
This moves the entire graph downwards.
We now have $y = \frac{4}{(x-2)^2} - 3$ , which is our function $G(x)$ !

STEP 7

The correct transformation is B.
We vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4, shift it 2 units to the right, and 3 units down.

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Studdy Solution

STEP 1

STEP 2

1. Rewrite the function2. Analyze the transformations

STEP 3

STEP 4

The **4** multiplied in the numerator means we **vertically stretch** the graph of y=1x2y = \frac{1}{x^2}y=x21​ by a factor of **4**.This makes the graph taller!We now have y=4x2y = \frac{4}{x^2}y=x24​.

STEP 5

Replacing xxx with (x−2)(x-2)(x−2) means we **shift** the graph **2 units to the right**.Think about it: we need a bigger xxx in (x−2)(x-2)(x−2) to get the same result as a smaller xxx in the original.We now have y=4(x−2)2y = \frac{4}{(x-2)^2}y=(x−2)24​.

STEP 6

Finally, subtracting **3** means we **shift** the graph **3 units down**.This moves the entire graph downwards.We now have y=4(x−2)2−3y = \frac{4}{(x-2)^2} - 3y=(x−2)24​−3, which is our function G(x)G(x)G(x)!

STEP 7

The correct transformation is **B**.We **vertically stretch** the graph of y=1x2y = \frac{1}{x^2}y=x21​ by a factor of **4**, **shift it 2 units to the right**, and **3 units down**.

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1. Rewrite the function
2. Analyze the transformations

The 4 multiplied in the numerator means we vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4.
This makes the graph taller!
We now have $y = \frac{4}{x^2}$ .

Replacing $x$ with $(x-2)$ means we shift the graph 2 units to the right.
Think about it: we need a bigger $x$ in $(x-2)$ to get the same result as a smaller $x$ in the original.
We now have $y = \frac{4}{(x-2)^2}$ .

Finally, subtracting 3 means we shift the graph 3 units down.
This moves the entire graph downwards.
We now have $y = \frac{4}{(x-2)^2} - 3$ , which is our function $G(x)$ !

The correct transformation is B.
We vertically stretch the graph of $y = \frac{1}{x^2}$ by a factor of 4, shift it 2 units to the right, and 3 units down.