TMUA practice: Graph Transformations, problem 5

Track the equation through each step.

• Start. $\;y = f(x)$.
• Step 1 (reflect in $y$-axis). Replace $x$ by $-x$: the equation becomes $\;y = f(-x)$.
• Step 2 (translate $3$ upward). Add $3$ to the output: $\;y = f(-x) + 3$.
• Step 3 (vertical stretch by factor $2$). Multiply the current output by $2$. So the new equation is $\;y = 2\bigl(\,f(-x) + 3\,\bigr) = 2 f(-x) + 6$.

Hence the final equation is $\;y = 2 f(-x) + 6$. This is option E.

Why the other options fail.

• A, $y = 2 f(-x) + 3$, has stretched the $f$-part but not the constant. The stretch acts on the current output, which already contains the $+3$, so the $+3$ also gets multiplied.
• B, $y = 2 f(-x) - 6$, has the right factor $2$ but a sign error on the constant.
• C, $y = -2 f(x) + 6$, has confused the $y$-axis reflection $f(-x)$ with the $x$-axis reflection $-f(x)$.
• D, $y = f(-2x) + 6$, has applied the factor $2$ inside $f$ (a horizontal scaling) instead of outside (a vertical scaling).

The lesson: when transformations are applied in order, each new transformation acts on the current equation, not on the original $f$. Applying the stretch after the upward shift multiplies the shift by the stretch factor.