Description
Problem 1. Write the following functions in the form f (x) = (x ± h)2 ± k by completing the square. Describe how x2 is shifted to obtain f(x). Graph f(x), label the vertex, label all axis intersections. An example of what I expect is given below.
(a) f(x)=x2+2x−1 (b) f(x)=x2 −7x+10
(c) f(x)=x2+x+1 (d) f(x)=x2 −8x+15 (e) f(x)=x2+3x
(f) f(x)=x2−4x+7 (g) f(x)=x2+23x+14 (h) f(x)=x2−x−1
(i) f(x)=x2+3x+17 36
Example. f(x) = x2 − 2x − 2
f(x)=x2 −2x−2
= (x2 −2x+(1)2 −(1)2)−2
= ((x−1)2 −1)−2
= (x − 1)2 − 3
Then f (x) = (x − 1)2 − 3 is the function x2 shifted right one unit, and shifted down 3 units. To find x-intercepts, we set f(x) = 0 and solve for x:
(x−1)2 −3=0 (x−1)2 =3
(x − 1)2 = ±√3 √
Note that 1 + f (0):
√√ 3 is positive and 1 −
x−1=± 3 √
x=1± 3
3 is negative. To find the y-intercept, we set x = 0 and find
f(0)=(0−1)2 −3 = (−1)2 − 3 =1−3
= −2
Note that if may have been easier to use the function as originally written to obtain this since
f(x)=x2 −2x−2 =⇒ f(0)=02 −2(0)−2=−2
In any case we have f(0) = −2 so that our y-intercept is at y = −2. Be sure that all intercepts are labeled and that the vertex is indicated as in the graph below.
