2.5.1: Exercises 2.5
- Page ID
- 63337
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
How would completing the square on a quadratic function help you graph it?
- Answer
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After completing the square, you can quickly identify the horizontal and vertical shifts
Exercise \(\PageIndex{2}\)
After completing the square, you get \(f(x)=(x-2)^2+3\). Is \(x=2\) considered a root of \(f(x)\)? Explain.
- Answer
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\(x=2\) is not a root of \(f(x)\) because \(f(2)=3\), not \(0\).
Exercise \(\PageIndex{3}\)
One of the variations on completing the square gives you the form \((cx+a)^2+b\). Does \(c\) represent a vertical stretch/shrink or a horizontal stretch/shrink of the function \(x^2\)?
- Answer
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It represents a horizontal stretch/shrink because it is on the inside of the function.
Exercise \(\PageIndex{4}\)
After completing the square you get that \(g(t)=(t+2)^2-6\). What are the values of \(a\) and \(b\) if your goal form is \((x+a)^2+b\)?
- Answer
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\(a=2\) and \(b=-6\)
Problems
In exercises \(\PageIndex{5}\) - \(\PageIndex{11}\), write each function in the form \((x+a)^{2}+b\) and identify the values of \(a\) and \(b\).
Exercise \(\PageIndex{5}\)
\(f(x)=x^2-4x+6\)
- Answer
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\(f(x)=(x-2)^2+2\); \(a=-2\); \(b=2\)
Exercise \(\PageIndex{6}\)
\(g(x)=x^2+20x+40\)
- Answer
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\(g(x)=(x+10)^2-60\); \(a=10\); \(b=-60\)
Exercise \(\PageIndex{7}\)
\(h(x)=x^2-8x+5\)
- Answer
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\(h(x)=(x-4)^2-11\); \(a=-4\); \(b=-11\)
Exercise \(\PageIndex{8}\)
\(m(x)=x^2-22x-4\)
- Answer
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\(m(x)=(x-11)^2-125\); \(a=-11\); \(b=-125\)
Exercise \(\PageIndex{9}\)
\(n(x)=x^2-6x-2\)
- Answer
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\(n(x)=(x-3)^2-11\); \(a=-3\); \(b=-11\)
Exercise \(\PageIndex{10}\)
\(p(x)=x^2+11x+4\)
- Answer
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\(p(x)=(x+\frac{11}{2})^2-\frac{105}{4}\); \(a=\frac{11}{2}\); \(b=-\frac{105}{4}\)
Exercise \(\PageIndex{11}\)
\(p(x)=x^2+13x\)
- Answer
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\(p(x)=(x+\frac{13}{2})^2-\frac{169}{4}\); \(a=\frac{13}{2}\); \(b=-\frac{169}{4}\)
In exercises \(\PageIndex{12}\) - \(\PageIndex{16}\), write each function in the form \((cx+a)^{2}+b\) and identify the values of \(a,\: b,\) and \(c\).
Exercise \(\PageIndex{12}\)
\(f(x)=9x^2-12x+12\)
- Answer
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\(f(x)=(3x-2)^2+8\); \(a=-2\); \(b=8\); \(c=3\)
Exercise \(\PageIndex{13}\)
\(g(x)=x^2-2x+2\)
- Answer
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\(f(x)=(x-1)^2+1\); \(a=-1\); \(b=1\); \(c=1\)
Exercise \(\PageIndex{14}\)
\(h(x)=4x^2-4x-4\)
- Answer
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\(h(x)=(2x-1)^2-5\); \(a=-1\); \(b=-5\); \(c=2\)
Exercise \(\PageIndex{15}\)
\(w(x)=4x^2+4x+6\)
- Answer
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\(w(x)=(2x+1)^2+5\); \(a=1\); \(b=5\); \(c=2\)
Exercise \(\PageIndex{16}\)
\(y(x)=9x^2+18x+4\)
- Answer
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\(y(x)=(3x+3)^2-5\); \(a=3\); \(b=-5\); \(c=3\)
In exercises \(\PageIndex{17}\) - \(\PageIndex{21}\), write each function in the form \(c(x+a)^{2}+b\) and identify the values of \(a,\: b,\) and \(c\).
Exercise \(\PageIndex{17}\)
\(f(x)=9x^2-12x+12\)
- Answer
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\(f(x)=9(x-\frac{2}{3})^2+8\); \(a=-\frac{2}{3}\); \(b=8\); \(c=9\)
Exercise \(\PageIndex{18}\)
\(g(x)=x^2-2x+2\)
- Answer
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\(f(x)=(x-1)^2+1\); \(a=-1\); \(b=1\); \(c=1\)
Exercise \(\PageIndex{19}\)
\(h(x)=4x^2-4x-4\)
- Answer
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\(h(x)=4(x-\frac{1}{2})^2-5\); \(a=-\frac{1}{2}\); \(b=-5\); \(c=4\)
Exercise \(\PageIndex{20}\)
\(w(x)=4x^2+4x+6\)
- Answer
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\(w(x)=4(x+\frac{1}{2})^2+5\); \(a=\frac{1}{2}\); \(b=5\); \(c=4\)
Exercise \(\PageIndex{21}\)
\(y(x)=9x^2+18x+4\)
- Answer
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\(y(x)=9(x+1)^2-5\); \(a=1\); \(b=-5\); \(c=9\)
In exercises \(\PageIndex{22}\) - \(\PageIndex{25}\), complete the square and use your result to help you graph the function.
Exercise \(\PageIndex{22}\)
\(f(t)=t^2+2t+3\)
- Answer
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\(f(t)=(t+1)^2 +2\);
Exercise \(\PageIndex{23}\)
\(p(q)=q^2 -\frac{2}{3}q\)
- Answer
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\(p(q)=(q-\frac{1}{3})^2 - \frac{1}{9}\);
Exercise \(\PageIndex{24}\)
\(y(x) = x^2+4x+2\)
- Answer
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\(y(x) = (x+2)^2-2\);
Exercise \(\PageIndex{25}\)
\(f(x) = x^2-4x+6\)
- Answer
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\(f(x) = (x-2)^2+2\);
In exercises \(\PageIndex{26}\) - \(\PageIndex{29}\), expand and graph the function.
Exercise \(\PageIndex{26}\)
\(f(x) = (x-1)^2-2\)
- Answer
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\(f(x) = x^2-2x-1\)
Exercise \(\PageIndex{27}\)
\(g(x) = -(x+3)^2+4\)
- Answer
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\(g(x) = -x^2-6x-5\)
Exercise \(\PageIndex{28}\)
\(h(x) = (-x+3)^2+4\)
- Answer
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\(h(x) = x^2-6x+13\)
Exercise \(\PageIndex{29}\)
\(x(y) = (y+2)^2-1\)
- Answer
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\(x(y) = y^2+4y+3\)
