Question

use the data in each table to write a linear function using function notation.
7.

x	-3	1	7
y	-0.6	0.2	1.4

8.

x	-5	-2	4
y	-10	-1	17

9.

x	-5	-2	8
y	8	2	-18

Explanation:

Step1: Analyze Problem 7

We have the table with \( x \) values \(-3, 1, 7\) and \( y \) values \(-0.6, 0.2, 1.4\). First, find the slope \( m \) using two points. Let's take \((-3, -0.6)\) and \((1, 0.2)\). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). So \( m=\frac{0.2 - (-0.6)}{1 - (-3)}=\frac{0.8}{4}=0.2 \). Now use point - slope form \( y - y_1=m(x - x_1) \). Using the point \((1, 0.2)\), we get \( y - 0.2 = 0.2(x - 1) \), which simplifies to \( y=0.2x \). In function notation, \( f(x)=0.2x \).

Step2: Analyze Problem 8

The \( x \) values are \(-5, -2, 4\) and \( y \) values are \(-10, -1, 17\). Calculate the slope \( m \) using \((-5, -10)\) and \((-2, -1)\). \( m=\frac{-1-(-10)}{-2 - (-5)}=\frac{9}{3}=3 \). Using point - slope form with \((-2, -1)\), \( y - (-1)=3(x - (-2)) \), so \( y + 1=3(x + 2) \), which simplifies to \( y=3x + 5 \). In function notation, \( f(x)=3x + 5 \).

Step3: Analyze Problem 9

The \( x \) values are \(-5, -2, 8\) and \( y \) values are \( 8, 2, -18\). Calculate the slope \( m \) using \((-5, 8)\) and \((-2, 2)\). \( m=\frac{2 - 8}{-2-(-5)}=\frac{-6}{3}=-2 \). Using point - slope form with \((-2, 2)\), \( y - 2=-2(x - (-2)) \), so \( y - 2=-2(x + 2) \), which simplifies to \( y=-2x - 2 \). In function notation, \( f(x)=-2x - 2 \).

Answer:

• Problem 7: \( f(x)=0.2x \)
• Problem 8: \( f(x)=3x + 5 \)
• Problem 9: \( f(x)=-2x - 2 \)