Question

show all of your work
ential function? explain your reasoning.

b.

x	-4	0	4	8
y	1	0	-1	-2

Explanation:

Step1: Recall linear function form

A linear function has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate the slope between first two points

Take \( (x_1,y_1)=(-4,1) \) and \( (x_2,y_2)=(0,0) \). Then \( m_1=\frac{0 - 1}{0-(-4)}=\frac{- 1}{4}=-\frac{1}{4}\).

Step3: Calculate the slope between second and third points

Take \( (x_1,y_1)=(0,0) \) and \( (x_2,y_2)=(4,-1) \). Then \( m_2=\frac{-1 - 0}{4 - 0}=\frac{-1}{4}=-\frac{1}{4}\).

Step4: Calculate the slope between third and fourth points

Take \( (x_1,y_1)=(4,-1) \) and \( (x_2,y_2)=(8,-2) \). Then \( m_3=\frac{-2-(-1)}{8 - 4}=\frac{-1}{4}=-\frac{1}{4}\).

Step5: Recall exponential function property

For an exponential function \( y = ab^x \), the ratio of consecutive \( y \) - values (when \( x \) increases by a constant amount) should be constant. Let's check the ratios.

The \( x \) - values increase by 4 each time. The \( y \) - values are 1, 0, - 1, - 2. The ratio of \( 0\div1 = 0 \), \( - 1\div0 \) is undefined, so the ratios are not constant.

Since the slope between consecutive points is constant (\( m=-\frac{1}{4} \)) and the ratio of consecutive \( y \) - values for exponential is not constant, the function is linear, not exponential.

Answer:

The function represented by the table is not an exponential function. It is a linear function because the slope between consecutive points (where \( x \) increases by 4 each time) is constant (\( m =-\frac{1}{4}\)), while for an exponential function, the ratio of consecutive \( y \) - values (for a constant increase in \( x \)) should be constant, which is not the case here (the ratio of consecutive \( y \) - values is not constant and one of the ratios is undefined).