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Chapter 2: Problem 103

Solve each quadratic equation by the method of your choice. Use the graph of \(f(x)=x^{2}\) to graph \(g(x)=(x+3)^{2}+1\)

Short Answer

Expert verified
The graph of \(g(x)=(x+3)^{2}+1\) is a parabola that opens upwards with vertex at (-3, 1). It is the graph of \(f(x)=x^{2}\) shifted 3 units to the left and 1 unit up.

Step by step solution

01

Understand the base function

The base function here is \(f(x)=x^{2}\), which is a parabola that opens upwards with vertex at the origin (0, 0).
02

Apply the horizontal shift

Replace \(x\) with \(x+3\) in \(f(x)=x^{2}\). This results in \((x+3)^{2}\), which shifts the graph of \(f(x)\) to the left by 3 units because the shift is opposite the sign. The vertex of the graph is now at (-3, 0).
03

Apply the vertical shift

Add 1 to \((x+3)^{2}\) to get the function \(g(x)=(x+3)^{2}+1\). This shifts the graph of \((x+3)^{2}\) upwards by 1 unit. The vertex of \(g(x)\) is now at (-3, 1).

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Most popular questions from this chapter

Exercises \(103-105\) will help you prepare for the material covered in the next section. Let \(\left(x_{1}, y_{1}\right)=(7,2) \quad\) and \(\quad\left(x_{2}, y_{2}\right)=(1,-1) . \quad\) Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} .\) Express the answer in simplified radical form.

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=6 x-3, g(x)=\frac{x+3}{6}$$

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\frac{2}{x+3}, g(x)=\frac{1}{x}$$

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=\frac{1}{2 x-3}$$

Solve: $$5 x^{\frac{3}{4}}-15=0$$ (Section 1.6, \text { Example } 5).
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