Problem 60
Question
Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=(x-1)^{2}+2$$
Step-by-Step Solution
Verified Answer
The graph of the function \(h(x)=(x-1)^{2}+2\) is a parabola that opens upward with its vertex at the point (1,2). This graph is obtained from the graph of the standard quadratic function \(f(x)=x^{2}\) by shifting it 1 unit to the right and 2 units upward.
1Step 1: Graph the Standard Quadratic Function
Start by graphing the standard quadratic function: \(f(x)=x^{2}\). This is a parabola that opens upward with its vertex at the origin (0,0).
2Step 2: Perform the Horizontal Shift
Next, apply the horizontal shift to the graph of \(f(x)=x^{2}\). This transformation is represented by \(x \rightarrow (x-1)\), suggesting a shift of 1 unit to the right. After this transformation, the vertex of the graph will be at (1,0).
3Step 3: Perform the Vertical Shift
Finally, add a vertical shift of 2 units upward to the graph, which is indicated by the \(+2\) at the end of the function. After this transformation, the vertex of the function will be located at (1,2). The function \(h(x)=(x-1)^{2}+2\) is now completely graphed.
Other exercises in this chapter
Problem 60
Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=5 x-2, g(x)=-x^{2}+4 x-1$$
View solution Problem 60
Let $$ \begin{array}{l} f(x)=2 x-5 \\ g(x)=4 x-1 \\ h(x)=x^{2}+x+2 \end{array} $$ Evaluate the indicated function without finding an equation for the function.
View solution Problem 60
a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. $$4 x+y
View solution Problem 61
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}-x+2 y+1=0$$
View solution