Question: Graph the function by starting with a function from the library of functions and then using the…

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.

\[ g(x) = (x - 3)^2 + 1 \]

Use the graphing tool to graph the function.

[Graphing tool interface]

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Solution

To graph the function \( g(x) = (x - 3)^2 + 1 \), follow these steps: 1. Start with the parent function \( f(x) = x^2 \). 2. Shift the graph of \( f(x) \) horizontally to the right by 3 units. This is because the term \((x - 3)\) indicates a shift 3 units to the right. \[ h(x) = (x - 3)^2 \] 3. Shift the graph of \( h(x) \) vertically upwards by 1 unit due to the \(+1\) at the end of the function. \[ g(x) = (x - 3)^2 + 1 \] Now, the function \( g(x) = (x - 3)^2 + 1 \) is a parabola that opens upwards, with its vertex at the point \((3, 1)\). The graph is a standard parabola moved 3 units to the right and 1 unit up.