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Study Guides > Mathematics for the Liberal Arts

E1.10: Section 6 Part 2

Start exploring. How does changing h change the graph? Start by changing h to -2. That makes the spreadsheet look like the illustration below.
  A B C D E F G H
1 x y
2 -6 36 2 a
3 -5 22 -2 h
4 -4 12 4 k
5 -3 6
6 -2 4
7 -1 6
8 0 12
9 1 22
10 2 36
11 3 54
12 4 76
13 5 102
14 6 132
15
16
Now we can notice that, when h=3h=3, the lowest point on the graph is at x=3x=3, and when h=−2h=-2, then the lowest point on the graph is at x=−2x=-2. This suggests that maybe the value that is subtracted from x in the original formula is the one that determines where the lowest y-value is – that is, where the lowest point on the graph is.   Try h=0h=0, h=4h=4, and h=−3h=-3.  
h=0h=0 (leaving a=2a=2 and k=4k=4) h=4h=4 (leaving a=2a=2 and k=4k=4) h=−3h=-3 (leaving a=2a=2 and k=4k=4)
   
  Do these results support the conjecture we made in the previous sentence?   Answer: Yes.   Example 21.   Using the same formula and spreadsheet as in Example 18, use the values a=1a=1, h=0h=0, and explore the effect of changing k.
k=4k=4 (leaving a=1a=1 and h=0h=0) k=0k=0 (leaving a=1a=1 and h=0h=0) k=−7k=-7 (leaving a=1a=1 and h=0h=0)
   
We find that changing k alone changes how far up or down the lowest point on the graph is. It appears that the y-value of that lowest point is k. Example 22.   Using the same formula and spreadsheet as in Example 17, use h=0h=0 and k=0k=0, and explore the effect of changing a.
a=1a=1 (with h=0h=0 and k=0k=0) a=3a=3 (withh=0h=0 and k=0k=0) a=−3a=-3 (with h=0h=0 and k=0k=0)
   
We find that changing a from a positive to a negative number makes the graph change from opening upward to opening downward. Making a larger (from 1 to 3) changes how large the y-values are, so that the y-values for a=3a=3 are three times as large as those when a=1a=1.    

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