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Mathematics LibreTexts

24.11: E1.10- Section 6 Part 2

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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
122.png

Now we can notice that, when h=3, the lowest point on the graph is at x=3, and when h=-2, then the lowest point on the graph is at x=-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=0, h=4, and h=-3.

 

h=0

(leaving a=2 and k=4)

h=4

(leaving a=2 and k=4)

h=-3

(leaving a=2 and k=4)

 211.png 35.png  43.png

 

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=1, h=0, and explore the effect of changing k.

k=4

(leaving a=1 and h=0)

k=0

(leaving a=1 and h=0)

k=-7

(leaving a=1 and h=0)

 53.png 36.png  44.png

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=0 and k=0, and explore the effect of changing a.

a=1

(with h=0 and k=0)

a=3

(withh=0 and k=0)

a=-3

(with h=0 and k=0)

 82.png 91.png  101.png

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=3 are three times as large as those when a=1.

 

 

CC licensed content, Shared previously
  • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution

24.11: E1.10- Section 6 Part 2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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