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Mathematics LibreTexts

2.5.1: Exercises 2.5

  • Page ID
    63337
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    Terms and Concepts

    Exercise \(\PageIndex{1}\)

    How would completing the square on a quadratic function help you graph it?

    Answer

    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

    \(x=2\) 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

    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

    \(a=2\) 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

    \(f(x)=(x-2)^2+2\); \(a=-2\); \(b=2\)

    Exercise \(\PageIndex{6}\)

    \(g(x)=x^2+20x+40\)

    Answer

    \(g(x)=(x+10)^2-60\); \(a=10\); \(b=-60\)

    Exercise \(\PageIndex{7}\)

    \(h(x)=x^2-8x+5\)

    Answer

    \(h(x)=(x-4)^2-11\); \(a=-4\); \(b=-11\)

    Exercise \(\PageIndex{8}\)

    \(m(x)=x^2-22x-4\)

    Answer

    \(m(x)=(x-11)^2-125\); \(a=-11\); \(b=-125\)

    Exercise \(\PageIndex{9}\)

    \(n(x)=x^2-6x-2\)

    Answer

    \(n(x)=(x-3)^2-11\); \(a=-3\); \(b=-11\)

    Exercise \(\PageIndex{10}\)

    \(p(x)=x^2+11x+4\)

    Answer

    \(p(x)=(x+\frac{11}{2})^2-\frac{105}{4}\); \(a=\frac{11}{2}\); \(b=-\frac{105}{4}\)

    Exercise \(\PageIndex{11}\)

    \(p(x)=x^2+13x\)

    Answer

    \(p(x)=(x+\frac{13}{2})^2-\frac{169}{4}\); \(a=\frac{13}{2}\); \(b=-\frac{169}{4}\)

    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

    \(f(x)=(3x-2)^2+8\); \(a=-2\); \(b=8\); \(c=3\)

    Exercise \(\PageIndex{13}\)

    \(g(x)=x^2-2x+2\)

    Answer

    \(f(x)=(x-1)^2+1\); \(a=-1\); \(b=1\); \(c=1\)

    Exercise \(\PageIndex{14}\)

    \(h(x)=4x^2-4x-4\)

    Answer

    \(h(x)=(2x-1)^2-5\); \(a=-1\); \(b=-5\); \(c=2\)

    Exercise \(\PageIndex{15}\)

    \(w(x)=4x^2+4x+6\)

    Answer

    \(w(x)=(2x+1)^2+5\); \(a=1\); \(b=5\); \(c=2\)

    Exercise \(\PageIndex{16}\)

    \(y(x)=9x^2+18x+4\)

    Answer

    \(y(x)=(3x+3)^2-5\); \(a=3\); \(b=-5\); \(c=3\)

    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

    \(f(x)=9(x-\frac{2}{3})^2+8\); \(a=-\frac{2}{3}\); \(b=8\); \(c=9\)

    Exercise \(\PageIndex{18}\)

    \(g(x)=x^2-2x+2\)

    Answer

    \(f(x)=(x-1)^2+1\); \(a=-1\); \(b=1\); \(c=1\)

    Exercise \(\PageIndex{19}\)

    \(h(x)=4x^2-4x-4\)

    Answer

    \(h(x)=4(x-\frac{1}{2})^2-5\); \(a=-\frac{1}{2}\); \(b=-5\); \(c=4\)

    Exercise \(\PageIndex{20}\)

    \(w(x)=4x^2+4x+6\)

    Answer

    \(w(x)=4(x+\frac{1}{2})^2+5\); \(a=\frac{1}{2}\); \(b=5\); \(c=4\)

    Exercise \(\PageIndex{21}\)

    \(y(x)=9x^2+18x+4\)

    Answer

    \(y(x)=9(x+1)^2-5\); \(a=1\); \(b=-5\); \(c=9\)

    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

    \(f(t)=(t+1)^2 +2\);

    Exercise \(\PageIndex{23}\)

    \(p(q)=q^2 -\frac{2}{3}q\)

    Answer

    \(p(q)=(q-\frac{1}{3})^2 - \frac{1}{9}\);

    Exercise \(\PageIndex{24}\)

    \(y(x) = x^2+4x+2\)

    Answer

    \(y(x) = (x+2)^2-2\);

    Exercise \(\PageIndex{25}\)

    \(f(x) = x^2-4x+6\)

    Answer

    \(f(x) = (x-2)^2+2\);

    In exercises \(\PageIndex{26}\) - \(\PageIndex{29}\), expand and graph the function.

    Exercise \(\PageIndex{26}\)

    \(f(x) = (x-1)^2-2\)

    Answer

    \(f(x) = x^2-2x-1\)