Question

1. water freezes at 0 °celsius and 32 °fahrenheit. water boils at 100 °celsius and 212 °fahrenheit.

a. write degrees celsius as a linear function of degrees fahrenheit (1 point).
b. what is the slope of your linear equation? what does it mean? (1 point)
c. what is the y - intercept and what does it mean? (1 point)

Explanation:

Step1: Recall the linear - equation form

The linear - equation is $y = mx + b$, where $y$ is degrees Celsius ($C$), $x$ is degrees Fahrenheit ($F$), $m$ is the slope, and $b$ is the y - intercept. We have two points: $(F_1,C_1)=(32,0)$ and $(F_2,C_2)=(212,100)$.

Step2: Calculate the slope $m$

The slope formula is $m=\frac{C_2 - C_1}{F_2 - F_1}$. Substitute $C_1 = 0,F_1 = 32,C_2 = 100,F_2 = 212$ into the formula:
\[m=\frac{100 - 0}{212 - 32}=\frac{100}{180}=\frac{5}{9}\]

Step3: Find the y - intercept $b$

Substitute the point $(32,0)$ and $m=\frac{5}{9}$ into $C=mF + b$. We get $0=\frac{5}{9}\times32 + b$. Then $b=-\frac{160}{9}$.
So the linear function is $C=\frac{5}{9}(F - 32)$.

Step4: Interpret the slope

The slope $\frac{5}{9}$ means that for every 1 - degree increase in Fahrenheit, the Celsius temperature increases by $\frac{5}{9}$ degrees.

Step5: Interpret the y - intercept

The y - intercept is $-\frac{160}{9}$. When $F = 0$, $C=-\frac{160}{9}$. It represents the Celsius temperature when the Fahrenheit temperature is 0.

Answer:

a. $C=\frac{5}{9}(F - 32)$
b. The slope is $\frac{5}{9}$. It means that for every 1 - degree increase in Fahrenheit, the Celsius temperature increases by $\frac{5}{9}$ degrees.
c. The y - intercept is $-\frac{160}{9}$. It represents the Celsius temperature when the Fahrenheit temperature is 0.