Question: Question 6, 1.5.23 Graph the function by starting with a function from the library of…

Question 6, 1.5.23

Graph the function by starting with a function from the library of functions and then using the techniques of shifting, compressing, stretching, and/or reflecting.

\[ g(x) = (x - 1)^2 + 4 \]

Use the graphing tool to graph the function.

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Solution

To graph the function \( g(x) = (x - 1)^2 + 4 \), we can use transformations of the basic quadratic function \( f(x) = x^2 \). Start with the basic function: \[ f(x) = x^2 \] Apply horizontal shift: The expression \( (x - 1) \) indicates a shift to the right by 1 unit. \[ f(x) = (x - 1)^2 \] Apply vertical shift: The \( +4 \) indicates a shift upward by 4 units. \[ g(x) = (x - 1)^2 + 4 \] The transformations applied are: 1. Shift the basic parabola \( f(x) = x^2 \) to the right by 1 unit. 2. Shift it upward by 4 units. The vertex of the transformed parabola is at \( (1, 4) \). Use these transformations to graph \( g(x) \).