f(x)=x^2 Create a function g(x) that represents a vertical stretch of 2 and a shift 3 units right.

Question

Help, please been struggling on this for 2 weeks

Answer 1

First, we recognize that the first (original) equation f(x) = x² is the parent of the quadratic function, and with what's given, we are going to use those transformations to transform/change that parent function into the new function, g(x).

The first transformation listed, the vertical stretch (by a factor of 2) will take a specific position in the equation and represent a specific change in the graph - that is, it will change in the vertical orientation, and with a factor larger than 1 for this orientation, it will be a stretch vertically. I recommend using Desmos to test this out to see in comparing the original function to the added transformation of vertical stretch.

So in adding the vertical stretch by 2, we see the first difference in our new function: g(x) = 2(x)²

The second transformation, shift 3 units right, tells us to simply move the graph/function to the right by 3 units. This horizontal change in the graph position will be positioned with the 'x' inside the parentheses, like so: g(x) = 2(x - 3)²

Oddly enough, as you've surely learned at this point, while it shows "negative 3" in the equation, we know it to show up as positive 3 on the graph. One way to look at why this is, is by experimenting with plugging zero in place of g(x) or y and then solving for x, which would lead to the positive 3.

With those placements (and understandings) we get the answer: g(x) = 2(x - 3)²

I hope you found this helpful! And I encourage you to go check out your textbook for your class or research in other math books, online sites/videos dedicated to the area to get more diverse or even deeper understanding if interested. :) Peace