Problem 59
Question
Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=(x-2)^{2}+1$$
Step-by-Step Solution
Verified Answer
The function \(h(x)=(x-2)^2+1\) is a parabola with a vertex at (2,1), opened upwards. It is obtained by shifting the standard quadratic function \(f(x)=x^2\) two units to the right and one unit upwards.
1Step 1: Graph the standard function
Start by simply plotting the standard quadratic function \(f(x)=x^2\). This is a parabola with vertex at the origin (0,0) opening upwards.
2Step 2: Understand the transformation
The function \(h(x)=(x-2)^2+1\) takes the standard function \(f(x)=x^2\) and changes it in two ways. The term \((x-2)^2\) shifts the graph left or right and \(+1\) shifts the graph up or down. In this case, the graph moves 2 steps to the right and 1 step upwards.
3Step 3: Graph the transformed function
Now, transform the graph of \(f(x)=x^2\) by moving it two units to the right and one unit upwards. Plotting this new function \(h(x)=(x-2)^2+1\), results in a parabola with the vertex at the point (2,1) instead of (0,0).
Other exercises in this chapter
Problem 59
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