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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 [latex]h=3[/latex], the lowest point on the graph is at [latex]x=3[/latex], and when [latex]h=-2[/latex], then the lowest point on the graph is at [latex]x=-2[/latex]. 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 [latex]h=0[/latex], [latex]h=4[/latex], and [latex]h=-3[/latex].  
[latex-display]h=0[/latex-display] (leaving [latex]a=2[/latex] and [latex]k=4[/latex]) [latex-display]h=4[/latex-display] (leaving [latex]a=2[/latex] and [latex]k=4[/latex]) [latex-display]h=-3[/latex-display] (leaving [latex]a=2[/latex] and [latex]k=4[/latex])
   
  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 [latex]a=1[/latex], [latex]h=0[/latex], and explore the effect of changing k.
[latex-display]k=4[/latex-display] (leaving [latex]a=1[/latex] and [latex]h=0[/latex]) [latex-display]k=0[/latex-display] (leaving [latex]a=1[/latex] and [latex]h=0[/latex]) [latex-display]k=-7[/latex-display] (leaving [latex]a=1[/latex] and [latex]h=0[/latex])
   
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 [latex]h=0[/latex] and [latex]k=0[/latex], and explore the effect of changing a.
[latex-display]a=1[/latex-display] (with [latex]h=0[/latex] and [latex]k=0[/latex]) [latex-display]a=3[/latex-display] (with[latex]h=0[/latex] and [latex]k=0[/latex]) [latex-display]a=-3[/latex-display] (with [latex]h=0[/latex] and [latex]k=0[/latex])
   
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 [latex]a=3[/latex] are three times as large as those when [latex]a=1[/latex].    

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  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.