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Chapter 2: Problem 63

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=2(x-2)^{2}-1$$

Short Answer

Expert verified
The graph of \(h(x)=2(x-2)^{2}-1\) is a parabola that opens upwards with the vertex at (2,-1). It is derived from the graph of \(f(x)=x^{2}\) by shifting it 2 units to the right, stretching it vertically by a factor of 2 and shifting it 1 unit downwards.

Step by step solution

01

Graph the standard Quadratic Function

Start by sketching the graph of the standard quadratic function \(f(x)=x^{2}\). The graph is a U-shaped curve called a parabola, opening upwards, with a vertex at the origin (0,0).
02

Apply Horizontal Shift

The function \(h(x)=2(x-2)^{2}-1\) involves a horizontal shift of 2 units to the right from the graph of \(f(x)=x^{2}\). So, every point on the \(f(x)=x^{2}\) graph will be moved 2 units to the right on the x-axis. For instance, the vertex of the parabola will move from (0,0) to (2,0).
03

Apply Vertical Stretch

The number 2 in front of the squared term in \(h(x)=2(x-2)^{2}-1\), would cause the graph to stretch vertically by a factor of 2. In other words, the y-values for the quadratic function are doubled.
04

Apply Vertical Shape

Finally, the '-1' at the end of \(h(x)=2(x-2)^{2}-1\) represents a vertical shift, which means it would bring every point on the graph down by 1 unit. Therefore, the vertex will finally move from (2,0) to (2,-1).
05

Draw Final graph

With all the transformations applied, you can now sketch a final graph for \(h(x)=2(x-2)^{2}-1\), which is the graph of \(f(x)=x^{2}\) that is first horizontally shifted 2 units to right, then vertically stretched by a factor of 2 and lastly vertically shifted 1 unit down.

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