Question

1. fractional slopes

table 1:

x	y
4	3
6	4

table 2:

x	y
3	2
5	3.5

a) calculate the slope of the function in table 1.
b) calculate the slope of the function in table 2.
c) describe the significance of fractional slopes and which table represents a function with a greater rate of change.

Explanation:

Step1: Recall slope formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope for Table 1

Let $(x_1,y_1)=(2,2)$ and $(x_2,y_2)=(4,3)$. Then $m_1=\frac{3 - 2}{4 - 2}=\frac{1}{2}$.

Step3: Calculate slope for Table 2

Let $(x_1,y_1)=(1,0.5)$ and $(x_2,y_2)=(3,2)$. Then $m_2=\frac{2 - 0.5}{3 - 1}=\frac{1.5}{2}=\frac{3}{4}$.

Step4: Explain significance of fractional slopes

Fractional slopes indicate the rate of change of a function. A larger - valued slope means a steeper line and a greater rate of change.

Step5: Compare slopes

Since $\frac{3}{4}>\frac{1}{2}$, Table 2 has a greater rate of change.

Answer:

a) $\frac{1}{2}$
b) $\frac{3}{4}$
c) Fractional slopes represent the rate of change of a function. A larger slope means a greater rate of change. Table 2 represents a function with a greater rate of change.