Question

part ii. write equations for the piece - wise functions whose graphs are shown below. assume that the units are 1 for every tick mark. 7. 8. 9. 10. 11.

Explanation:

Step1: Analyze graph 9

For \(x < - 1\), use two - point form \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) with \((x_1,y_1)=(-1,1)\) and \((x_2,y_2)=(0,2)\). The slope \(m = 1\), and the equation is \(y=x + 2\). For \(-1\leq x<1\), use two - point form with \((x_1,y_1)=(-1,3)\) and \((x_2,y_2)=(1,2)\). The slope \(m=-\frac{1}{2}\), and using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=( - 1,3)\) gives \(y=-\frac{1}{2}x+\frac{5}{2}\). For \(x\geq1\), use two - point form with \((x_1,y_1)=(1,2)\) and \((x_2,y_2)=(2,4)\). The slope \(m = 2\), and the equation is \(y=2x\).

Step2: Write piece - wise function for graph 9

\[f(x)=

$$\begin{cases}x + 2&x<-1\\-\frac{1}{2}x+\frac{5}{2}&-1\leq x<1\\2x&x\geq1\end{cases}$$

\]

Step3: Analyze graph 10

For \(x < - 2\), use two - point form with \((x_1,y_1)=(-3,0)\) and \((x_2,y_2)=(-2,2)\). The slope \(m = 2\), and the equation is \(y=2x + 6\). For \(-2\leq x<2\), \(y = 2\). For \(x\geq2\), use two - point form with \((x_1,y_1)=(2,2)\) and \((x_2,y_2)=(3,0)\). The slope \(m=-2\), and the equation is \(y=-2x+6\).

Step4: Write piece - wise function for graph 10

\[f(x)=

$$\begin{cases}2x + 6&x<-2\\2&-2\leq x<2\\-2x + 6&x\geq2\end{cases}$$

\]

Step5: Analyze graph 11

For \(x<0\), \(y = 0\). For \(0\leq x<1\), \(y = 1\). For \(1\leq x<2\), \(y = 2\). For \(x\geq2\), \(y = 3\).

Step6: Write piece - wise function for graph 11

\[f(x)=

$$\begin{cases}0&x<0\\1&0\leq x<1\\2&1\leq x<2\\3&x\geq2\end{cases}$$

\]

Answer:

For graph 9: \(f(x)=

$$\begin{cases}x + 2&x<-1\\-\frac{1}{2}x+\frac{5}{2}&-1\leq x<1\\2x&x\geq1\end{cases}$$

\)
For graph 10: \(f(x)=

$$\begin{cases}2x + 6&x<-2\\2&-2\leq x<2\\-2x + 6&x\geq2\end{cases}$$

\)
For graph 11: \(f(x)=

$$\begin{cases}0&x<0\\1&0\leq x<1\\2&1\leq x<2\\3&x\geq2\end{cases}$$

\)