Question

1. graph both functions and visually decide what happened to the second function. $y_1=|x|$ to $y_2=|x-1|-3$.
2. graph both functions and visually decide what happened to the second function. $y_1=2^x$ to $y_2=2^{x-3}+1$.

Explanation:

For Problem 6

Step1: Plot parent function $y_1=|x|$

Key points: $(-2,2), (-1,1), (0,0), (1,1), (2,2)$; this forms a V-shape with vertex at $(0,0)$.

Step2: Plot transformed function $y_2=|x-1|-3$

Key points: $(-1, -1), (0,-2), (1,-3), (2,-2), (3,-1)$; this is a V-shape with vertex at $(1,-3)$.

Step3: Identify transformation

Compare vertices: $(0,0) \to (1,-3)$.

Step1: Plot parent function $y_1=2^x$

Key points: $(-2,\frac{1}{4}), (-1,\frac{1}{2}), (0,1), (1,2), (2,4)$; this is an increasing exponential curve with y-intercept at $(0,1)$.

Step2: Plot transformed function $y_2=2^{x-3}+1$

Key points: $(1,\frac{1}{4}+1=\frac{5}{4}), (2,\frac{1}{2}+1=\frac{3}{2}), (3,1+1=2), (4,2+1=3), (5,4+1=5)$; this is an increasing exponential curve with a horizontal shift and vertical shift.

Step3: Identify transformation

Compare corresponding points: $(x,y) \to (x+3, y+1)$ for $y_1$ to $y_2$.

Answer:

The graph of $y_2=|x-1|-3$ is the graph of $y_1=|x|$ shifted 1 unit to the right and 3 units down.

---

For Problem 7