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

Determine the vertical stretch factor that produces the same graph as applying a horizontal stretch of scale factor $3$ to the graph of $y=x^2$.

Determine the horizontal stretch factor that produces the same graph as applying a vertical stretch of scale factor $25$ to the graph of $y=x^2$.

Determine the vertical stretch factor that produces the same graph as applying a horizontal stretch of scale factor $1/5$ to the graph of $y=x^2$.

When the graph of $y=x^2$ undergoes a horizontal stretch of scale factor $5$, the equation becomes $y=Kx^2$. Find $K$.

Verify that the graphs of $y=\bigl(\tfrac{x}{4}\bigr)^2$ and $y=\tfrac{1}{16}x^2$ represent the same transformation of $y=x^2$.

If the graph of $y=x^2$ is scaled horizontally by a factor of $k$ and the resulting graph has equation $y=\tfrac{1}{49}x^2$, find $k$.

The graph of $y=x^2$ is transformed to $y=ax^2$ by performing a horizontal compression of scale factor $1/6$. Find $a$ and describe the equivalent vertical transformation.

Determine the horizontal stretch factor that produces the same graph as applying a vertical shrink of scale factor $1/16$ to the graph of $y=x^2$.

On the graph of $y=x^2$, a horizontal stretch by factor $1/8$ is followed by a vertical stretch by factor $m$ to return the graph to $y=x^2$. Find $m$.

A transformation of $y=x^2$ consists of a vertical stretch by a factor of $9$ followed by a horizontal stretch by a factor of $k$, resulting in the same graph as a single horizontal stretch by a factor of $3$. Determine $k$.

Prove in general that a horizontal stretch by factor $k$ applied to $y=x^2$ is equivalent to a vertical stretch by factor $\tfrac{1}{k^2}$.

A function $y=x^2$ is first stretched vertically by a factor of $p$ and then horizontally by a factor of $q$, resulting in $y=9x^2$. Determine the relationship between $p$ and $q$.