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

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

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

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

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

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 the value of $k$.

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

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

Verify that the graphs of $y=\left(\frac{x}{4}\right)^2$ and $y=\frac{1}{16}x^2$ represent the same graph, and describe the transformation of $y=x^2$ that each represents.

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 shrink of scale factor $1/16$ to the graph 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 $y=\frac{1}{49}x^2$, find $k$.