# 2.5.1: Exercises 2.5

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## Terms and Concepts

Exercise $$\PageIndex{1}$$

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.

$$x=2$$ 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 $$x^2$$?

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 $$(x+a)^2+b$$?

$$a=2$$ 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 $$\PageIndex{5}$$

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

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

Exercise $$\PageIndex{6}$$

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

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

Exercise $$\PageIndex{7}$$

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

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

Exercise $$\PageIndex{8}$$

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

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

Exercise $$\PageIndex{9}$$

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

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

Exercise $$\PageIndex{10}$$

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

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

Exercise $$\PageIndex{11}$$

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

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

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 $$\PageIndex{12}$$

$$f(x)=9x^2-12x+12$$

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

Exercise $$\PageIndex{13}$$

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

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

Exercise $$\PageIndex{14}$$

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

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

Exercise $$\PageIndex{15}$$

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

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

Exercise $$\PageIndex{16}$$

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

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

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 $$\PageIndex{17}$$

$$f(x)=9x^2-12x+12$$

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

Exercise $$\PageIndex{18}$$

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

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

Exercise $$\PageIndex{19}$$

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

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

Exercise $$\PageIndex{20}$$

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

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

Exercise $$\PageIndex{21}$$

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

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

In exercises $$\PageIndex{22}$$ - $$\PageIndex{25}$$, complete the square and use your result to help you graph the function.

Exercise $$\PageIndex{22}$$

$$f(t)=t^2+2t+3$$

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

Exercise $$\PageIndex{23}$$

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

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

Exercise $$\PageIndex{24}$$

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

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

Exercise $$\PageIndex{25}$$

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

$$f(x) = (x-2)^2+2$$;
In exercises $$\PageIndex{26}$$ - $$\PageIndex{29}$$, expand and graph the function.
Exercise $$\PageIndex{26}$$
$$f(x) = (x-1)^2-2$$
$$f(x) = x^2-2x-1$$