When the temperature is 0 degrees Celsius, the Fahrenheit temperature is 32 degrees. When the Celsius temperature is 100 degrees, the corresponding Fahrenheit temperature is 212 degrees. Express the Fahrenheit temperature as a linear function of [tex]C[/tex], the Celsius temperature, [tex]F(C)[/tex].

1. Find the linear function [tex]F(C)[/tex]:
[tex]F(C) = \square[/tex]

2. Find the rate of change of Fahrenheit temperature for each unit change in Celsius temperature:
[tex]\qquad \text{Rate of change: } \square \text{ Fahrenheit degrees per Celsius degree}[/tex]

3. Find and interpret [tex]F(26)[/tex]. Round to one decimal place:
[tex]\qquad \text{At } 26 \text{ degrees Celsius, it is } \square \text{ degrees Fahrenheit.}[/tex]

4. Calculate [tex]F(-40)[/tex]:
[tex]\qquad F(-40) = \square[/tex]

Let's work through the problem step-by-step.

a. Find the rate of change of Fahrenheit temperature for each unit change in Celsius temperature.

The given data points are:
- When the Celsius temperature is [tex]$0$[/tex], the Fahrenheit temperature is [tex]$32$[/tex].
- When the Celsius temperature is [tex]$100$[/tex], the Fahrenheit temperature is [tex]$212$[/tex].

To find the rate of change (or slope), we use the formula for the slope between two points [tex]$(x_1, y_1)$[/tex] and [tex]$(x_2, y_2)$[/tex]:
[tex]\[
\text{Slope} \, m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]

Substitute the given points:
[tex]\[
m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8
\][/tex]

So, the rate of change of Fahrenheit temperature for each unit change in Celsius is [tex]$1.8$[/tex] degrees Fahrenheit per degree Celsius.

b. Express the Fahrenheit temperature as a linear function of Celsius temperature, [tex]$F(C)$[/tex].

The formula for a linear function is [tex]$F(C) = m \times C + b$[/tex], where [tex]$m$[/tex] is the slope and [tex]$b$[/tex] is the y-intercept. From part (a), we found [tex]$m = 1.8$[/tex].

Since we know [tex]$F(0) = 32$[/tex], this helps us determine the y-intercept:
[tex]\[
b = 32
\][/tex]

Thus, the function is:
[tex]\[
F(C) = 1.8 \times C + 32
\][/tex]

Find and interpret [tex]$F(26)$[/tex]:

To find [tex]$F(26)$[/tex] using this function, substitute [tex]$C = 26$[/tex]:
[tex]\[
F(26) = 1.8 \times 26 + 32 = 46.8 + 32 = 78.8
\][/tex]

So, when the Celsius temperature is [tex]$26$[/tex], the corresponding Fahrenheit temperature is approximately [tex]$78.8$[/tex] degrees.

c. Calculate [tex]$F(-40)$[/tex]:

Substitute [tex]$C = -40$[/tex] into the function:
[tex]\[
F(-40) = 1.8 \times (-40) + 32 = -72 + 32 = -40
\][/tex]

So, [tex]$F(-40)$[/tex] is [tex]$-40$[/tex] degrees Fahrenheit.

To summarize:
- The rate of change is [tex]$1.8$[/tex] degrees Fahrenheit per Celsius degree.
- [tex]$F(26)$[/tex] is approximately [tex]$78.8$[/tex] degrees Fahrenheit.
- [tex]$F(-40)$[/tex] is [tex]$-40$[/tex] degrees Fahrenheit.