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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]

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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].
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