Question Type 2: Determining whether two transformations are the same effect wise Exercises

Show generally for $f(x)=x^n$ that a horizontal stretch by a factor of $k$ is equivalent to a vertical stretch by a factor of $k^{-n}$.

Determine whether a horizontal stretch by a factor of $5$ on $f(x)=|x|$ has the same effect as a vertical stretch by a factor of $\frac{1}{5}$.

Find the vertical scale factor equivalent to a horizontal stretch by $4$ on $f(x)=\sqrt{x}$.

For $f(x)=x^4$, find the horizontal stretch factor $k$ that yields the same effect as a vertical stretch by $16$.

For $f(x)=x^2$, is a horizontal compression by a factor of $\frac{1}{4}$ equivalent to a vertical stretch by a factor of $16$? Explain.

The height of a drone ascending vertically is modelled by $h(t)=t^2$. If the time axis is horizontally stretched by a factor of $3$, what vertical transformation on $h$ produces the same effect on its graph?

For $f(x)=x^3$, is a horizontal compression by a factor of $\frac{1}{3}$ equivalent to a vertical stretch by a factor of $27$? Justify your answer.

Find the vertical scale factor that makes a vertical stretch equivalent to a horizontal stretch of $f(x)=x^2$ by a factor of $5$.

Determine whether a horizontal stretch of $f(x)=x^2$ by a factor of $3$ has the same effect as a vertical stretch by a factor of $9$. Justify your answer.

Determine the vertical scale factor equivalent to a horizontal stretch by a factor of $2$ on $f(x)=x^3$.

For $f(x)=|x|$, is a horizontal stretch with scale factor $\frac{1}{2}$ equivalent to a vertical stretch by a factor of $2$? Explain your answer.

A transformation on $f(x)=x^2$ consists of a horizontal stretch with scale factor $2$ followed by a vertical stretch with scale factor $\frac{1}{4}$. Show that this is equivalent to a single transformation and identify it.