Question 1199243: The graph of a basic cubic function k(x)=x^3 is shown. Suppose that p(x)=k(x+3). Use reference points and symmetry to complete the table of values for p(x). Then graph p(x) on the same coordinate plane as k(x) and label it.
\Reference points k(x) given
(0,0)
(1,1)
(2,8)
I am having a hard time with the substitution -I can do the graph once I have the substitution done
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Here's the graph of y = x^3 through the three reference points mentioned.
Imagine this green curve is etched in stone on a wall or the ground.
Being etched in stone means the green curve itself physically cannot be moved.
Imagine that what can move however is the red xy axis.
Think of this axis as part of the camera looking at the curve.
We can think of this as the crosshairs of the camera.
Since you're holding the camera, you can of course move the camera left, right, up, or down.
If you moved the camera's crosshairs 3 units to the right, then each x input is now 3 units larger.
In other words: the old input x is now x+3
Example: if x = 1 was the old input, then x+3 = 1+3 = 4 is the new input.
After the crosshairs move 3 units to the right, it gives the illusion the green curve (remember that it's etched in stone) has moved 3 units to the left.
It's all a matter of perspective more or less.
Effectively, the jump from k(x) = x^3 to p(x) = k(x+3) will move this green curve 3 units to the left.
We'll arrive at the blue curve as shown below.
All points on this curve also move 3 units to the left. After all, a curve is simply a collection of points.
The point (0,0) moves to (-3,0)
The point (1,1) moves to (-2,1)
The point (2,8) moves to (-1,8)
We subtract 3 from the x coordinate.
Keep the y coordinate the same.
Here's what this shifting looks like:
k(x) = x^3 in green
p(x) = k(x+3) = (x+3)^3 in blue
You can use free graphing software tools like Desmos or GeoGebra to confirm the above graphs.
-----------------------------------------------
In short:
The jump from k(x) to k(x+3) means we shift the entire curve 3 units to the left.
A point like (0,0) moves to (-3,0) after such a shift. Use this idea for the other reference points as well.
Let me know if you have any questions.
|
|
|
