Help with Transformations?

Question

So I've got two transformations questions, both of which wanting the answer given in the y= a f(bx+c) + d format. The starting equation is y= f(x). The first one is:

1. Reflect across the x-axis
2. Shift vertically down 1 unit
3. Shift the graph to the right 3 units
4. Horizontally stretch by a factor of 2

The second one is:

1. Shift vertically down 1 unit
2. Reflect across the x-axis
3. Horizontally stretch by a factor of 2
4. Shift the graph to the right 3 units

The question also asks: Suppose the point (2,4) is on the graph y=f(x). After these graph transformations what will be the corresponding point of (2,4) on the new graph y = a f(bx+c) +d?

Thanks so much!

Answer 1

Hi Caitlin,

Using the form y = aƒ(b(x+c) + d allows us to complete all transformations the question is asking for all at once:

1. a is the amount of vertical stretch (if |a| > 1) or compression (if |a| < 1), or can flip ƒ(x) across the x-axis if negative
2. b is the amount of horizontal stretch or compression, and the rules are reversed from a. |b| < 1 will stretch the function in the x-direction, whereas |b| > 1 will squish it
3. c will shift the entire function to the left (if c > 0) or right (if c < 0)
4. d will shift the entire function up (d > 0) or down (d < 0). This is equivalent to b in y = mx + b

So, with these in mind, a good way to approach this question is to break down each transformation into a value, then follow these transformations one at a time to obtain the new point.

1. For the first one, we have a reflection across the x-axis, a vertical shift down by one unit, a shift to the right 3 units, and a horizontal stretch by a factor of 2. Using the definitions above, we can build the equation like so:

1. The function is flipped across the x-axis but is not stretched or compressed in this direction. Thus, we let a = -1 as it will flip the function while still preserving the shape of the function
2. The function is vertically shifted down one unit, so we can let d = -1
3. The function is shifted right by 3 units, so we let c = -3. It is easy to confuse the signs here, so be careful to remember that negative numbers move the function to the right and positive moves it left
4. A horizontal stretch by a factor of 2 means we can let b = 2

Now with all of the letters done, we can build the new equation by plugging them in! This new function is now y = -ƒ(2(x-3)) - 1. If we follow the transformations, we can track the point (2,4) at each step:

1. Flip across the x-axis will flip the sign of the y-value: (2,4) –> (2,-4)
2. Moving down one unit means we subtract 1 from the y value: (2,-4) –> (2,-5)
3. Moving right three units means we add 3 to the x value: (2,-5) –> (5,-5)
4. Finally, stretching in the x-direction means we need to multiply the x-value by b, which is 2: (5,-5) –> (10,-5)

So the answer to part one is y = -ƒ(2(x-3)) - 1 and the point (2,4) is shifted to (10,-5).

You can follow this same process to find the answer to part two, which is y = -ƒ(2(x-3)) - 1. Overall the equation is the same, but it is important to note that changing the order of transformations will change where the original point goes:

The point (2,4) in the second part will travel like this: (2,4) –> (2,3) –> (2,-3) –> (4,-3) –> (7,-3)

Hope this helps!