Answered

Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

Use transformations of the graph of [tex]f(x) = x[/tex] to determine the graph of the given function.

[tex]g(x) = (x + 7)^2[/tex]

Sagot :

GinnyAnswer
To determine the graph of the function [tex]\( g(x) = (x+7)^2 \)[/tex] using transformations of the graph of [tex]\( f(x) = x \)[/tex], we can follow these steps:

1. Start with the original function [tex]\( f(x) = x \)[/tex].
- This is a straight line, and we will use it as our starting point.

2. Apply the horizontal transformation involving a shift:
- To transform [tex]\( f(x) = x \)[/tex] into [tex]\( g(x) = (x+7)^2 \)[/tex], we first observe that the function [tex]\( x \)[/tex] is shifted horizontally. The expression [tex]\( x + 7 \)[/tex] indicates a shift to the left by 7 units.
- So, [tex]\( f(x) = x \)[/tex] becomes [tex]\( f(x+7) \)[/tex] by shifting the graph 7 units to the left. The new function after shifting will be [tex]\( f_{\text{shift}}(x) = x + 7 \)[/tex].

3. Transform the resulting function by squaring:
- Now, we need to apply the squaring transformation to our shifted function [tex]\( f_{\text{shift}}(x) = x + 7 \)[/tex].
- Squaring the function [tex]\( x+7 \)[/tex] results in [tex]\( (x+7)^2 \)[/tex].
- So, the function [tex]\( f_{\text{shift}}(x) \)[/tex] which is [tex]\( x + 7 \)[/tex], when squared, transforms to the final function [tex]\( g(x) = (x+7)^2 \)[/tex].

4. Combine the transformations to get the final function:
- Start from [tex]\( f(x) = x \)[/tex].
- Shift the graph left 7 units to get [tex]\( f_{\text{shift}}(x) = x + 7 \)[/tex].
- Finally, square the resulting function to obtain [tex]\( g(x) = (x+7)^2 \)[/tex].

As a visual aid, when you graph [tex]\( g(x) \)[/tex], you can expect the following:
- The vertex of the parabola [tex]\( g(x) = (x + 7)^2 \)[/tex] will be at the point [tex]\((-7, 0)\)[/tex].
- The graph is symmetric about the vertical line [tex]\( x = -7 \)[/tex].
- The parabola opens upwards because the leading coefficient is positive.

In summary:
1. Start with [tex]\( f(x) = x \)[/tex].
2. Shift the graph of [tex]\( f(x) = x \)[/tex] to the left by 7 units to obtain [tex]\( f_{\text{shift}}(x) = x + 7 \)[/tex].
3. Square the shifted graph to arrive at the final function [tex]\( g(x) = (x + 7)^2 \)[/tex].

Thus, applying these transformations step-by-step, we obtain the graph of [tex]\( g(x) \)[/tex] from the graph of [tex]\( f(x) = x \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.