Question

25: graph the function using a combination of transformations applied to the graph of a basic function.

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

Explanation:

Identify the parent function

The given function is \(g(x) = 2(x - 3)^2 + 1\). The basic parent function is the quadratic function:

$$ f(x) = x^2 $$

Apply horizontal translation

Using the Horizontal Translations knowledge point

$$ \text{Shift } f(x) = x^2 \text{ right by 3 units to get } f_1(x) = (x - 3)^2 $$

Apply vertical stretching

We apply a vertical stretch by a factor of \(2\) to the function \(f_1(x)\). This multiplies all \(y\)-values of the shifted parent function by \(2\):

$$ f_2(x) = 2(x - 3)^2 $$

Apply vertical translation

Using the Vertical Translations knowledge point

$$ \text{Shift } f_2(x) \text{ up by 1 unit to get } g(x) = 2(x - 3)^2 + 1 $$

Determine key points for graphing

Using the Graphing Parabolas knowledge point

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Answer:

To graph the function \(g(x) = 2(x - 3)^2 + 1\), apply the following transformations to the parent function \(f(x) = x^2\):

1. Horizontal Shift: Shift the graph of \(f(x) = x^2\) to the right by \(3\) units.
2. Vertical Stretch: Stretch the graph vertically by a factor of \(2\).
3. Vertical Shift: Shift the graph upward by \(1\) unit.

The resulting parabola has its vertex at \((3, 1)\) and passes through the points \((2, 3)\), \((4, 3)\), \((1, 9)\), and \((5, 9)\).