1. Problem: Describe the transformations from $f(x)$ to $r(x)$ or $g(x)$ for each given function. 2. Recall the transformation rules: - Vertical stretch/shrink: multiplying $f(x)$ by a factor $a$ changes the vertical scale by $a$. - Horizontal stretch/shrink: replacing $x$ by $bx$ changes the horizontal scale by $1/b$. - Translation: adding/subtracting constants inside or outside the function shifts the graph. - Reflection: multiplying by $-1$ reflects over the x-axis; replacing $x$ by $-x$ reflects over the y-axis. 3. Analyze each case: 1. $f(x) = x + 2$, $r(x) = f(3x)$ - Horizontal shrink by factor $1/3$ (since $x$ replaced by $3x$) - No vertical stretch/shrink (factor 1) - No vertical translation - No horizontal translation - No reflection 2. $f(x) = 3x + 6$, $r(x) = \frac{1}{3} f(x)$ - Vertical shrink by factor $1/3$ - No horizontal stretch/shrink - No translation - No reflection 3. $f(x) = -3 - x$, $h(x) = f(-x)$ - Reflection over y-axis (replace $x$ by $-x$) - No vertical or horizontal stretch/shrink - No translation 4. $f(x) = \frac{1}{3} x + 1$, $h(x) = -f(x)$ - Reflection over x-axis (multiply by $-1$) - No stretch/shrink - No translation 5. $f(x) = 5x - 10$, $r(x) = f(\frac{2}{5} x)$ - Horizontal stretch by factor $\frac{5}{2}$ (since $x$ replaced by $\frac{2}{5} x$) - No vertical stretch/shrink - No translation 6. $f(x) = -\frac{1}{3} x + 2$, $r(x) = 6 f(x)$ - Vertical stretch by factor 6 - No horizontal stretch/shrink - No translation - Reflection over x-axis included in $f(x)$ (due to negative slope) 7. $f(x) = -3x + 5$, $g(x) = f(x - 3)$ - Horizontal translation right by 3 - No stretch/shrink - No vertical translation 8. $f(x) = -2x + 6$, $g(x) = f(\frac{4}{3} x)$ - Horizontal shrink by factor $\frac{3}{4}$ - No vertical stretch/shrink - No translation 9. $f(x) = 4x - 3$, $g(x) = \frac{1}{2} f(x)$ - Vertical shrink by factor $\frac{1}{2}$ - No horizontal stretch/shrink - No translation 10. $f(x) = -2x$, $g(x) = f(x) + 3$ - Vertical translation up by 3 - No stretch/shrink - No horizontal translation Final answers: 1. Vertical stretch: 1, Vertical shrink: 1/3, Horizontal stretch: 1, Horizontal shrink: 3, Translate up/down: 0, Translate left/right: 0, Reflect: none 2. Vertical stretch: 1/3, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: 0, Reflect: none 3. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: 0, Reflect: over y axis 4. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: 0, Reflect: over x axis 5. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 5/2, Horizontal shrink: 2/5, Translate up/down: 0, Translate left/right: 0, Reflect: none 6. Vertical stretch: 6, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: 0, Reflect: over x axis (from $f$) 7. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: right 3, Reflect: none 8. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 3/4, Translate up/down: 0, Translate left/right: 0, Reflect: none 9. Vertical stretch: 1/2, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: 0, Translate left/right: 0, Reflect: none 10. Vertical stretch: 1, Vertical shrink: 1, Horizontal stretch: 1, Horizontal shrink: 1, Translate up/down: up 3, Translate left/right: 0, Reflect: none