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Problem 15 Medium Difficulty

The relationship between the Fahrenheit $ (F) $ and Celsius $ (C) $ temperature scales is given by the linear function $ F = \frac{9}{5} C + 32 $.

(a) Sketch a graph of this function.
(b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?

Answer

(a)
(b) The slope of $\frac{2}{5}$ means that $F$ increases $\frac{3}{5}$ degrees for each increase
of $1^{\circ} \mathrm{C}$. (Equivalently, $F$ increases by 9 when $C$ increases by 5
and $F$ decreases by 9 when $C$ decreases by $5 .$ ) The $F$ -intercept of
32 is the Fahrenheit temperature corresponding to a Celsius
temperature of 0.

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Video Transcript

all right. So here we have the function, which represents the temperature and degrees Celsius related to the temperature in degrees Fahrenheit Fahrenheit as a function of Celsius, and we want to sketch the graph of it. So I'm going to start by finding a couple points that would be on the graph, and then we'll draw a line through them. So if the temperature in degrees Celsius was zero, we would have 9/5 time zero plus 32. So that would be 32 F. And I'm going to choose a convenient number for C as a second point. I'm going to choose C equals 25 because it's easy to multiply by 9/5. And when we multiply 25 by 9/5 we get 45 plus 32 is 77. Okay, so we could plot these two points 0 32 and 25 77 and then we can draw the line through these points. And that represents the relationship between the two temperature scales in Part B. We want to talk about the slope. So what's the slope and what does it represent? When you look at the equation, you can see that the slope is 9/5 and if you think about the units on slope, the slope is changing. Why over change in X. So the units on why were degrees Fahrenheit and the units on on X or C? In this case, we're degrees Celsius, and so the slope is showing you the change in degrees Fahrenheit for a change in degree Celsius. So it's telling us that it would go up 9 F for every five degrees it goes up Celsius, so goes up 9 F for every five degree Celsius rise. Now, how about the y intercept or in this case will call it the F intercept, since our Y axis is known as the F axis in this problem. So the point where the graph intersects the Y axis is the 0.0.0 32 and that would be the point where it is zero degrees Celsius. So when it zero degrees Celsius, it's 32 F. Zero degree Celsius is 32 F, and we think of that as the temperature of freezing for water. That's the water's freezing temperature

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