Question

daily work: jan. 27th
graph:
① ( f(x)=\begin{cases}2x - 4 & \text{if } x > -13end{cases} )
② ( f(x)=\begin{cases}|x + 1| & \text{if } -4 < x leq 23end{cases} )
③ ( f(x)=\begin{cases}-\frac{2}{3}x + 4 & \text{if } x leq 33end{cases} )
④ ( f(x)=\begin{cases}|x| - 7 & \text{if } -3 leq x leq 53end{cases} )
+finish 1/26 delta math

Explanation:

Step1: Analyze function 1: linear, x > -1

This is a linear function with slope $2$, y-intercept $-4$. For $x=-1$, $f(-1)=2(-1)-4=-6$; this point is open (since $x>-1$). Plot points like $(0,-4)$, $(1,-2)$ and draw a ray starting from the open point $(-1,-6)$ going right.

Step2: Analyze function 2: absolute value, -4 < x ≤ 2

Rewrite $|x+1|$ as a piecewise function:

• When $x+1\geq0$ (i.e., $x\geq-1$), $f(x)=x+1$
• When $x+1<0$ (i.e., $x<-1$), $f(x)=-x-1$

Key points:

• Open point at $x=-4$: $f(-4)=|-4+1|=3$
• Vertex at $x=-1$: $f(-1)=0$
• Closed point at $x=2$: $f(2)=|2+1|=3$

Plot these points and connect the segments for $-4

Step3: Analyze function 3: linear, x ≤ 3

This is a linear function with slope $-\frac{2}{3}$, y-intercept $4$. For $x=3$, $f(3)=-\frac{2}{3}(3)+4=2$; this point is closed (since $x\leq3$). Plot points like $(0,4)$, $(3,2)$ and draw a ray starting from the closed point $(3,2)$ going left.

Step4: Analyze function 4: absolute value, -3 ≤ x ≤5

Rewrite $|x|-7$ as a piecewise function:

• When $x\geq0$, $f(x)=x-7$
• When $x<0$, $f(x)=-x-7$

Key points:

• Closed point at $x=-3$: $f(-3)=|-3|-7=3-7=-4$
• Vertex at $x=0$: $f(0)=0-7=-7$
• Closed point at $x=5$: $f(5)=|5|-7=5-7=-2$

Plot these points and connect the segments for $-3\leq x\leq5$.

Answer:

1. For $f(x)=2x-4$ where $x>-1$: A ray starting at the open point $(-1, -6)$ with slope 2, extending to the right.
2. For $f(x)=|x+1|$ where $-4
3. For $f(x)=-\frac{2}{3}x+4$ where $x\leq3$: A ray starting at the closed point $(3,2)$ with slope $-\frac{2}{3}$, extending to the left.
4. For $f(x)=|x|-7$ where $-3\leq x\leq5$: A V-shaped graph with vertex at $(0,-7)$, closed left endpoint $(-3,-4)$, closed right endpoint $(5,-2)$.