Question

does the table represent a linear, or an exponential function? explain your reasoning. show all of your work
a.

x	y
0	8
1	4
2	2
3	1

b.

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

Explanation:

Part a

Step1: Check for linearity (constant slope)

For a linear function, the slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \) should be constant. Let's calculate the slopes between consecutive points.

• Between \((0, 8)\) and \((1, 4)\): \( m_1=\frac{4 - 8}{1 - 0}=\frac{-4}{1}=-4 \)
• Between \((1, 4)\) and \((2, 2)\): \( m_2=\frac{2 - 4}{2 - 1}=\frac{-2}{1}=-2 \)

Since \( m_1
eq m_2 \), it's not linear. Now check for exponential (constant ratio).

Step2: Check for exponential (constant ratio)

The ratio of consecutive \( y \)-values:

• \( \frac{4}{8}=\frac{1}{2} \)
• \( \frac{2}{4}=\frac{1}{2} \)
• \( \frac{1}{2}=\frac{1}{2} \)

The ratio is constant (\( r = \frac{1}{2} \)), so it's exponential.

Step1: Check for linearity (constant slope)

Calculate slopes between consecutive points.

• Between \((-4, 1)\) and \((0, 0)\): \( m_1=\frac{0 - 1}{0 - (-4)}=\frac{-1}{4}=-\frac{1}{4} \)
• Between \((0, 0)\) and \((4, -1)\): \( m_2=\frac{-1 - 0}{4 - 0}=\frac{-1}{4} \)
• Between \((4, -1)\) and \((8, -2)\): \( m_3=\frac{-2 - (-1)}{8 - 4}=\frac{-1}{4} \)

The slope is constant (\( m = -\frac{1}{4} \)), so it's linear.

Answer:

Exponential (because the ratio of consecutive \( y \)-values is constant (\( \frac{1}{2} \)) while the slope is not constant)

Part b