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]h=0[/latex]

(leaving [latex]a=2[/latex] and [latex]k=4[/latex])

[latex]h=4[/latex]

(leaving [latex]a=2[/latex] and [latex]k=4[/latex])

[latex]h=-3[/latex]

(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]k=4[/latex]

(leaving [latex]a=1[/latex] and [latex]h=0[/latex])

[latex]k=0[/latex]

(leaving [latex]a=1[/latex] and [latex]h=0[/latex])

[latex]k=-7[/latex]

(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]a=1[/latex]

(with [latex]h=0[/latex] and [latex]k=0[/latex])

[latex]a=3[/latex]

(with[latex]h=0[/latex] and [latex]k=0[/latex])

[latex]a=-3[/latex]

(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].