Question

From the table below, determine whether the data shows an exponential function. Explain why or why not. x31-1-3y1234a.No; the domain values are at regular intervals and the range values have a common sum 1.b.No; the domain values are not at regular intervals.c.Yes; the domain values are at regular intervals and the range values have a common factor 2.d.Yes; the domain values are at regular intervals and the range values have a common sum 1.

Answer 1

Given:

The table of values is given:

From the table,

We see the data is a linear function. This is because a linear function has domain values at regular intervals.

Also, the linear equation can be formed as shown below, indicating it is a linear function.

Considering two points, (3,1) and (1,2)

where,

`\begin{gathered} x_1=3 \\ y_1=1 \\ x_2=1 \\ y_2=2 \\ \\ \text{Then,} \\ \text{slope, m is given by;} \\ m=(y_2-y_1)/(x_2-x_1) \\ \\ \text{Substituting the values into the formula above,} \\ m=(2-1)/(1-3) \\ m=(1)/(-2) \\ m=-(1)/(2) \end{gathered}`

A linear equation is of the form;

`\begin{gathered} y=mx+b \\ \text{where m is the slope} \\ b\text{ is the y-intercept} \\ \\ To\text{ get the value of the y-intercept, we use any given point} \\ U\sin g\text{ point (3,1)} \\ y=mx+b \\ 1=-(1)/(2)(3)+b \\ 1=-(3)/(2)+b \\ 1+(3)/(2)=b \\ 1+1.5=b \\ b=2.5 \\ \\ \\ \text{Thus, the linear equation is;} \\ y=-(1)/(2)x+2.5 \end{gathered}`

From the above, has confirmed it is a linear function and not an exponential function, we can deduce that;

a) The function is not an exponential function.

b) The domain values (x-values) are at regular intervals

c) The range values (y-values) have a common difference of 1

Therefore, the correct answer is OPTION A