When temperature is 0 degrees Celsius, the Fahrenheit...

When temperature is 0 degrees Celsius, the Fahrenheit temperature is $32 .$ When the Celsius temperature is $100,$ the corresponding Fahrenheit temperature is $212 .$ Express the Fanrenheit temperature as a linear function of $C,$ the Celsius temperature, $F(C) .$ a. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius. b. Find and interpret $F(28)$ . c. Find and interpret $F(-40)$ .

When temperature is 0 degrees Celsius, the Fahrenheit temperature is $32 .$ When the Celsius temperature is $100,$ the corresponding Fahrenheit temperature is $212 .$ Express the Fanrenheit temperature as a linear function of $C,$ the Celsius temperature, $F(C) .$
a. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.
b. Find and interpret $F(28)$ .
c. Find and interpret $F(-40)$ .

Key Concepts

Intercept in a Linear Function

The intercept, often represented by b in the equation f(x) = mx + b, is the function's value when the independent variable is zero. This value provides a baseline or starting point from which changes dictated by the rate of change can be calculated, and it often has a meaningful interpretation in practical applications.

Linear Function

A linear function represents a relationship between two variables where the rate of change is constant, and is typically written in the form f(x) = mx + b. This form allows for direct calculation and interpretation of the output value for any given input, making it a staple in modelling conversion formulas and physical relationships.

Rate of Change

The rate of change, or slope, represents how much the output of a function changes with a unit change in the input. In the context of a linear function, it quantifies the constant relationship between the two variables, indicating how quickly one variable increases or decreases relative to the other.

Transcript

00:01 So for this problem, i went ahead and use the information that we want to write a fahrenheit as a function of c to help me establish that the fahrenheit is going to correspond to our y variable, and the celsius will correspond to our x variable.

00:16 And so with this, i made our information that we're given about various degrees celsius corresponding to their counterparts in fahrenheit to write these two points here, 032, and 100 to 12.

00:32 And so in order to write our equation for fahrenheit as a function of celsius, we're going to first want to find our slope between our two points.

00:43 So we know that slope is equal to y2 minus y1, which will be 212 minus 32 over x2 minus x1, which is 100 minus 0.

00:56 And so when we this it's actually going to reduce very nicely to 9 over 5 and now that we have our slope we can go ahead and plug in our slope and one of our points to solve for our y intercept and so when we do this we i can go ahead and plug in point 1 here so we know that y is equal to 32 and this is equal to m which is 9 over 5 times x which is 0 plus b so since this cancels out that tells us that our b or our y intercept will just be 32.

01:32 So that allows us to write our equation in slope intercept form as y is equal to m, or excuse me, we're not actually going to use y.

01:41 We're going to use f of c because we know that we want to find fahrenheit in terms of celsius.

01:47 And this is going to be equal to 9 over 5 times our celsius plus 32.

01:53 And so for part a of our problem, we want to find the rate of change of fahrenheit temperature.

01:59 So the rate of change is just another way of saying slope...

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