Question

Which functions are vertical stretches of the square root function? Check all that apply.

f(x) =
f(x) =
f(x) =
f(x) =
f(x) =
f(x) =

Answer 1

I don't see the function expressions — can you paste them? I need the actual formulas to check each one.

Meanwhile, here is how to tell whether a given function is a vertical stretch of y = sqrt(x):

- A vertical stretch means the graph is multiplied by a constant factor a with |a| > 1. So the stretched form is y = a·sqrt(x) (or more generally y = a·sqrt(x - h) + k with |a| > 1).
- If 0 < |a| < 1 it is a vertical compression, not a stretch.
- If a < 0 you have a reflection across the x-axis as well as a stretch/compression depending on |a|.
- Multiplying x inside the radical (e.g., sqrt(2x)) produces a horizontal compression, not a vertical stretch.
- Adding/subtracting outside the radical (e.g., sqrt(x) + 3) is a vertical shift, not a stretch.

Examples:
- y = 2√x → vertical stretch (a = 2 > 1)
- y = −3√x → vertical stretch plus reflection (|a| = 3 > 1)
- y = 1/2 √x → vertical compression (not a stretch)
- y = √(2x) → horizontal compression (not a vertical stretch)
- y = √x + 3 → vertical shift (not a stretch)

If you paste the six functions you have, I'll mark which are vertical stretches.