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
(h) f(x) = x2−x−1
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:
√ √
Note that 1+ 3 is positive and 1− 3 is negative. To find the y-intercept, we set x = 0 and find f(0):
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.
