Question

If the graph of $y=f\left(x\right)$ is transformed into the graph of $2y-6=-4f(x-3)$ point $(a,b)$ on the graph of $y=f\left(x\right)$ becomes point $(A,B)$ on the graph of $2y-6=-4f(x-3)$ where $A$ and $B$ are given by

• A. $A=a-3$, $B=b$
• B. $A=a-3$, $B=b-2$
• C. $A=a+3$, $B=-2b$
• D. $A=a+3$, $B=-2b+3$

Answer 1

The correct option is D $A=a+3$, $B=-2b+3$

We first solve $2y-6=-4f(x-3)$ for $y$ as

$y=-2f(x-3)+3$

The graph of $y=-2f(x-3)+3$ is that of $y=f\left(x\right)$ shifted $3$ units to the right, stretched vertically by a factor of $2$, reflected on the $x$ axis and shifted up by $3$ units. A point of $y=f\left(x\right)$ will undergo the same transformations.

Hence,

Point $(a,b)$ on the graph of $y=f\left(x\right)$

Becomes $(a+3,b)$ on the graph of $f(x-3)$ : shifted $3$ units to the right.

Becomes $(a+3,2b)$ on the graph of $2f(x-3)$ : stretched vertically by $2$.

Becomes $(a+3,-2b)$ on the graph of $-2f(x-3)$ : reflected on $x$ axis.

Becomes $(a+3,-2b+3)$ on the graph of $-2f(x-3)+3$ : shifted up $3$ units.

Hence, $A=a+3$ and $B=-2b+3$.