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: - $y = af(x)$ vertically stretches the graph by a factor of $a$ if $|a|>1$, or compresses if $0<|a|<1$. - $y = f(x-h)$ shifts the graph horizontally right by $h$ units if $h>0$, left if $h<0$. - $y = f(x)+k$ shifts the graph vertically up by $k$ units if $k>0$, down if $k<0$. - $y = f(bx)$ horizontally compresses by factor $1/b$ if $|b|>1$, stretches if $0<|b|<1$. - $y = -f(x)$ reflects the graph about the x-axis. - $y = f(-x)$ 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 by $-1$: reflect 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 $f(x+3)$ 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 about y-axis. - Multiply by 4: vertical stretch by factor 4. - Subtract 4: shift graph down by 4 units. Final answers: a) Vertical stretch by 3, shift down 1. b) Shift right 2, shift up 3. c) Horizontal compression by 1/2, shift down 5. d) Horizontal stretch by 2, reflect about x-axis, shift down 2. e) Shift left 3, vertical compression by 2/3, shift up 1. f) Reflect about y-axis, vertical stretch by 4, shift down 4.