Question

can someone please explain how to do this?
Describe the transformations to the graph of y=x^2 that result in the graph of y=3(x-2)^2+1

A. Shift left 2 units, down 1 units, then stretch by a factor of 3
B. Shift up 2 units, down 1 units, then reflect across the x-axis
C. Stretch by a factor of 3, shift down 2 units, then up 1 unit
D. Shift right 2 units, stretch by a factor of 3, then shift up 1 unit

Answer 1

Well, let's start with the easy part. Whenever you see something like (x-a), that means a translation (aka shift). The magnitude is a, but which direction? Think of the x value as representing how far you have to count to the right along the x-axis to get to your point. If the value of a is negative (such as in this case, a = -2), it means you count over fewer ticks to the right before drawing your point — you've shifted the entire curve to the left. If the value of a is positive (say we had (x+2) instead) that means you count over more ticks to the right, and the curve is shifted to the right.

Think of the starting equation, y = x^2, and make a table of values you might plot, starting from 0 and going to 5

x y=x^2
0 0
1 1
2 4
3 9
4 16
5 25

Now imagine sliding the x column up two places, so that x = 2 lined up with y = 0. If you plotted those points, you would have the same curve, shifted 2 points to the right. That's exactly the same effect as replacing x^2 with (x-2)^2 in the equation.

Next, we've multiplied the x^2 or (x-2)^2 term by 3. 3 is positive, so it just makes everything bigger. Our table above would look like this

x y = 3x^2
0 0
1 3
2 12
3 27
4 48
5 75

Same basic shape, but the sides of the parabola rise more quickly. If we multiply by a negative number, it has the same effect on the size, but flips the graph over along the x-axis.

Finally, we have that +1 tacked on the end of the expression. After we've shifted our parabola to the right, and stretched it by a factor of 3, it should be easy to see that adding 1 shifts the whole thing up 1 unit, just as subtracting 1 would shift the whole thing down 1 unit.

You can think of adding or subtracting inside the ^2 portion as relabeling your graph paper to move the location of (0,0). Say you had a circle x^2 + y^2 = 1. If you draw that circle shifted over 2 units to the right and up 1 unit (using the formula (x-2)^2 + (y-1)^2 = 1), the diagram would look identical to a diagram of x^2 + y^2 = 1 you had drawn on a piece of transparent graph paper with the circle at (0,0), and then shifted the center of the transparent graph so that it was over (2,1) on your paper. Does that make sense?

Answer 2

yes it does, thanks! the right answer is D.

Answer 3

@jigglygurl2 Here's a pretty picture :-)