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a. Sketch the graph obtained by shifting the graph of \( f(x) \) to the left 3 and up 1 unit(s) Label the new function \( g(x) \).

Ask by Goodwin Rogers. in the United States
Feb 27,2025

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To shift the graph of \( f(x) \) to the left by 3 units and up by 1 unit, replace \( x \) with \( x + 3 \) and add 1 to the function. The new function is \( g(x) = f(x + 3) + 1 \).

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The Deep Dive

To create the function \( g(x) \) from the original function \( f(x) \) by shifting it to the left by 3 units and up by 1 unit, you can apply the transformations to \( f(x) \). The left shift by 3 can be expressed as \( f(x + 3) \), and the upward shift by 1 unit can be added afterward. Therefore, the new function is: \[ g(x) = f(x + 3) + 1 \] When you sketch the graph of \( g(x) \), every point on the graph of \( f(x) \) will move left by 3 units and up by 1 unit. So, if a point on \( f(x) \) was at (a, b), it will now be at (a-3, b+1) on \( g(x) \). Now you’re ready to see how much fun it is to visualize data transformations!

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