a) Complete the chart below to compare y=-3(x+4)^2-2 to y=a[k(x-d)]n +c.
b) Describe the effects of a, k, d and c on the original curve y=x^2.
CHART (doing my best to make a chart)
y=x^2 / y=-3x^2 / y=-3(x+4)^2-2
------ / ------- / -------------
/ /
(-2,4) / /
--------/-------------/-----------------
(-1,1) / /
--------/-------------/-----------------
(0,0) / /
--------/-------------/-----------------
(1,1) / /
--------/-------------/-----------------
(2,4) / /
--------/-------------/-----------------
(3,9) / /
--------/-------------/-----------------
I suggest you visit wolframalpha.com and play around with the parameters, comparing the graphs. By saying
plot y=x^2,y=(x-2)^2+5 where x=-10..10
you can show multiple curves, and specify the domain.
Or, you can visit
http://rechneronline.de/function-graphs/
where you can plot up to three curves at once.
a) To complete the chart comparing y=-3(x+4)^2-2 to y=a[k(x-d)]n +c, we will substitute the x-values of (-2, -1, 0, 1, 2, 3) into the equation y=-3(x+4)^2-2.
For y=-3(x+4)^2-2:
- For x = -2:
y = -3(-2+4)^2-2 = -32
- For x = -1:
y = -3(-1+4)^2-2 = -11
- For x = 0:
y = -3(0+4)^2-2 = -2
- For x = 1:
y = -3(1+4)^2-2 = -13
- For x = 2:
y = -3(2+4)^2-2 = -32
- For x = 3:
y = -3(3+4)^2-2 = -59
To compare with y=a[k(x-d)]n +c, we need to determine the values of a, k, d, n, and c for the three equations.
For y=x^2:
- a = 1, k = 1, d = 0, n = 2, c = 0
For y=-3x^2:
- a = -3, k = 1, d = 0, n = 2, c = 0
For y=-3(x+4)^2-2:
- a = -3, k = 1, d = -4, n = 2, c = -2
b) The effects of a, k, d, and c on the original curve y=x^2 are as follows:
- a: The value of a affects the vertical stretch or compression of the parabola. If a is greater than 1, the parabola is vertically stretched. If a is between 0 and 1, the parabola is vertically compressed. If a is negative, the parabola is reflected across the x-axis.
- k: The value of k affects the horizontal stretch or compression of the parabola. If k is greater than 1, the parabola is horizontally compressed. If k is between 0 and 1, the parabola is horizontally stretched. If k is negative, the parabola is reflected across the y-axis.
- d: The value of d affects the horizontal shift of the parabola. Adding a positive value to x moves the parabola to the left, while subtracting a positive value from x moves the parabola to the right.
- c: The value of c affects the vertical shift of the parabola. Adding a positive value to y moves the parabola upward, while subtracting a positive value from y moves the parabola downward.
In the case of y=-3(x+4)^2-2 compared to y=x^2, we see that the original parabola is reflected across the x-axis, compressed horizontally by a factor of 1, shifted 4 units to the left, and shifted 2 units downward.