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Chapter 1: Problem 10

How do you obtain the graph of \(y=4(x+3)^{2}+6\) from the graph of \(y=x^{2} ?\)

Short Answer

Expert verified
Answer: The transformations applied to the base function are: 1. A horizontal shift of 3 units to the left. 2. A vertical stretch by a factor of 4. 3. A vertical shift of 6 units upwards.

Step by step solution

01

Identify the base function

The base function is \(y=x^2\). Our goal is to obtain the graph of the given function \(y=4(x+3)^2+6\) through transformations.
02

Identify horizontal shift

The given function is \(y=4(x+3)^2+6\). Comparing it with the base function, we can see that the term \((x+3)\) implies that there is a horizontal shift (translation) of \(-3\) units to the left. So, the first transformation is to shift the graph of the base function \(3\) units to the left.
03

Identify vertical stretch/compression

Now, we can see that the term \(4(x+3)^2\) in the given function indicates a vertical stretch. The coefficient \(4\) suggests that the original graph will be stretched vertically by a factor of \(4\). This means that the y-values of the base function will be multiplied by \(4\).
04

Identify vertical shift

Lastly, we can see that there is a term \(+6\) at the end of the given function. This indicates a vertical shift (translation) of \(6\) units upwards. The third and final transformation involves shifting the graph of the base function \(y=4(x+3)^2\) up by \(6\) units to obtain the graph of the desired function \(y=4(x+3)^2+6\).
05

Combine the transformations

Now, we will combine all the identified transformations to obtain the graph of the desired function. Start with the graph of the base function \(y=x^2\). 1. Shift the graph \(3\) units to the left. 2. Stretch the graph vertically by a factor of \(4\). 3. Shift the graph upwards by \(6\) units. After performing these transformations, you obtain the graph of the given function \(y=4(x+3)^2+6\).

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