1. The problem asks to describe the transformations applied to the graph of $y=f(x)$ to obtain each new function. 2. Recall the general transformation rules: - Multiplying $f(x)$ by a constant $a$ vertically stretches ($|a|>1$) or compresses ($0<|a|<1$) the graph. - Adding/subtracting a constant outside $f(x)$ shifts the graph vertically. - Replacing $x$ by $x-h$ shifts the graph horizontally by $h$ units to the right if $h>0$, left if $h<0$. - Multiplying $x$ by a constant inside $f(ax)$ compresses ($|a|>1$) or stretches ($0<|a|<1$) the graph horizontally. - A negative sign outside $f(x)$ reflects the graph about the x-axis. - A negative sign inside $f(-x)$ reflects the graph about the y-axis. 3. Now, analyze each function: a) $y=3f(x)-1$ - Multiply $f(x)$ by 3: vertical stretch by factor 3. - Subtract 1: shift graph down by 1 unit. b) $y=f(x-2)+3$ - Replace $x$ by $x-2$: shift graph right by 2 units. - Add 3: shift graph up by 3 units. c) $y=f(2x)-5$ - Replace $x$ by $2x$: horizontal compression by factor $\frac{1}{2}$. - Subtract 5: shift graph down by 5 units. d) $y=-f(\frac{1}{2}x)-2$ - Replace $x$ by $\frac{1}{2}x$: horizontal stretch by factor 2. - Multiply $f$ by -1: reflect graph about x-axis. - Subtract 2: shift graph down by 2 units. e) $y=\frac{2}{3}f(x+3)+1$ - Replace $x$ by $x+3$: shift graph left by 3 units. - Multiply $f$ by $\frac{2}{3}$: vertical compression by factor $\frac{2}{3}$. - Add 1: shift graph up by 1 unit. f) $y=4f(-x)-4$ - Replace $x$ by $-x$: reflect graph about y-axis. - Multiply $f$ by 4: vertical stretch by factor 4. - Subtract 4: shift graph down by 4 units. Final answers summarize the transformations for each part.