1. **Problem Statement:** Given the curve points A(0,2), B(1,0), C(4,4), D(6,0) for $y=f(x)$, find new points after transformations and explain effect on $y$. For the second part, given $y=f(x)$ with horizontal asymptote $y=2$ and vertical asymptote $x=1$, find transformed asymptotes and axis intersections. 2. **Part 1: Points transformations** - a) $y=f(x+1)$: Translation left 1 unit; coordinates move so new $x$ is old $x-1$. A: $(0,2)$ becomes $(-1,2)$ B: $(1,0)$ becomes $(0,0)$ C: $(4,4)$ becomes $(3,4)$ D: $(6,0)$ becomes $(5,0)$ - b) $y=f(x)-4$: $y$ values decrease by 4. A: $(0,2)$ to $(0,2-4)=(0,-2)$ B: $(1,0)$ to $(1,-4)$ C: $(4,4)$ to $(4,0)$ D: $(6,0)$ to $(6,-4)$ - c) $y=f(x+4)$: Translation left 4 units. A: $(0,2)$ to $(-4,2)$ B: $(1,0)$ to $(-3,0)$ C: $(4,4)$ to $(0,4)$ D: $(6,0)$ to $(2,0)$ - d) $y=f(2x)$: Horizontal compression by factor 2 (x-values scale by $1/2$). A: $(0,2)$ to $(0,2)$ B: $(1,0)$ to $(0.5,0)$ C: $(4,4)$ to $(2,4)$ D: $(6,0)$ to $(3,0)$ - e) $y=3f(x)$: Vertical stretch by factor 3. A: $(0,2)$ to $(0,6)$ B: $(1,0)$ to $(1,0)$ C: $(4,4)$ to $(4,12)$ D: $(6,0)$ to $(6,0)$ - f) $y=f(1/2 x)$: Horizontal stretch by factor 2. A: $(0,2)$ to $(0,2)$ B: $(1,0)$ to $(2,0)$ C: $(4,4)$ to $(8,4)$ D: $(6,0)$ to $(12,0)$ - g) $y=rac{1}{2}f(x)$: Vertical compression by factor 1/2. A: $(0,2)$ to $(0,1)$ B: $(1,0)$ to $(1,0)$ C: $(4,4)$ to $(4,2)$ D: $(6,0)$ to $(6,0)$ - h) $y=f(-x)$: Reflection about y-axis. A: $(0,2)$ to $(0,2)$ B: $(1,0)$ to $(-1,0)$ C: $(4,4)$ to $(-4,4)$ D: $(6,0)$ to $(-6,0)$ 3. **Part 2: Asymptotes and intersections transformations** (Original $y=f(x)$: passes origin $(0,0)$, horizontal asymptote $y=2$, vertical asymptote $x=1$) - a) $y=f(x)+2$: Horizontal asymptote increases by 2: $y=2+2=4$ Intersection shifts up by 2: $(0,0)$ to $(0,2)$ - b) $y=f(x+1)$: Horizontal asymptote unchanged: $y=2$ Vertical asymptote shifts left 1 unit: $x=1$ to $x=0$ Intersection shifts left: $(0,0)$ to $(-1,0)$ - c) $y=2f(x)$: Horizontal asymptote doubles: $y=2 imes 2=4$ Intersection doubles y: $(0,0)$ to $(0,0)$ (origin) - d) $y=f(x)-2$: Horizontal asymptote shifts down by 2: $y=2-2=0$ Intersection shifts down: $(0,0)$ to $(0,-2)$ - e) $y=f(2x)$: Horizontal asymptote unchanged: $y=2$ Vertical asymptote scales horizontally: $x=1$ to $x=0.5$ Intersection compress horizontally: $(0,0)$ remains $(0,0)$ - f) $y=f(\frac{1}{2} x)$: Horizontal asymptote unchanged: $y=2$ Vertical asymptote is stretched: $x=1$ to $x=2$ Intersection stretches horizontally: $(0,0)$ to $(0,0)$ - g) $y=\frac{1}{2} f(x)$: Horizontal asymptote halves: $y=2 \times \frac{1}{2}=1$ Intersection halves y: $(0,0)$ to $(0,0)$ - h) $y=-f(x)$: Horizontal asymptote negated: $y=-2$ Intersection negated y: $(0,0)$ to $(0,0)$ **Summary:** Transformations that add or subtract constants shift asymptotes and intersections vertically. Scaling y-values stretch/compress horizontal asymptote and intersections accordingly. Scaling x-values shift vertical asymptotes and stretch/compress intersections horizontally. Reflections negate y-values. Final transformed equations and coordinates as derived.