Cost of Driving The monthly cost of driving a car depends on...

cost of Driving The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May her driving cost was $\$ 380$ for 480 mi and in June her cost was $\$ 460$ for 800 mi. Assume that there is a linear relationship between the monthly cost $C$ of driving a car and the distance $x$ driven. (a) Find a linear function $C$ that models the cost of driving $x$ miles per month. (b) Draw a graph of $C .$ What is the slope of this line? (c) At what rate does Lynn's cost increase for every additional mile she drives?

cost of Driving The monthly cost of driving a car depends
on the number of miles driven. Lynn found that in May her
driving cost was $\$ 380$ for 480 mi and in June her cost was
$\$ 460$ for 800 mi. Assume that there is a linear relationship
between the monthly cost $C$ of driving a car and the distance
$x$ driven.
(a) Find a linear function $C$ that models the cost of driving
$x$ miles per month.
(b) Draw a graph of $C .$ What is the slope of this line?
(c) At what rate does Lynn's cost increase for every additional mile she drives?

Key Concepts

Linear Functions

A linear function models a relationship where the change between two quantities is constant. It is generally expressed in the form y = mx + b, where m is the slope and b is the y-intercept. This model is useful in understanding costs or other relationships that change at a fixed rate over time or quantity.

Slope

The slope in a linear function represents the constant rate of change of the dependent variable with respect to the independent variable. It tells us how much the value of the dependent variable increases or decreases for every unit increase in the independent variable, making it crucial for problems involving rates such as cost per mile.

Y-intercept

The y-intercept is the value of the dependent variable when the independent variable is zero. In practical applications, it often signifies a fixed cost or starting value that exists regardless of how much the independent variable changes.

Graphing Linear Functions

Graphing a linear function involves plotting a straight line on the coordinate plane that represents the relationship between the two variables. The slope determines the steepness and direction of the line, while the y-intercept tells where the line crosses the y-axis. This visualization helps in understanding how changes in one variable affect the other.

Transcript

0:00 All right.

00:01 So with this problem, this person drives 480 miles and it costs $380 or 800 miles cost $460.

00:14 I went ahead and i set up this t chart over here.

00:18 So that can help me find the rate of change.

00:22 And you know the rate of change formula is the change in y over the change in x.

00:28 And if if this is normally the miles is normally the x and the money is normally the y, then i'm going to say this is one, this is two.

00:41 This is the first x value.

00:42 This is the second.

00:43 So i can use the formula and i can do for 60 minus 380 divided by x2 is 800 divided or minus 480.

01:07 All right.

01:08 And so this is 460 minus 380.

01:14 It's going to be 20 would take it to 400 and another 60.

01:21 So 20 and 60, that's 80 over 20 would take it to 500, so 320.

01:32 And this reduces to one -fourth.

01:36 So my rate of change i know is one -fourth.

01:42 Now, my function, on the other hand, i have to figure out kind of and find out like, how can i get a function if i need y -equals a x plus b? so i know the a value now.

02:06 I just found this rate right here.

02:09 So i have y is equal to one fourth x plus b.

02:15 But i'm really not sure about this b value.

02:18 I do not know what the y intercept is.

02:21 But what i could do to figure the y intercept out is i can use this coordinate right here because i know this is x miles and y intercept.

02:32 So i could replace that to figure out what my y intercept is going to be.

02:39 So 380 is equal to one -fourth times the x value, which is 480 plus the b value.

02:51 And again, we're going to look for what that b value is.

02:55 So first things first, i'm going to work on this fraction.

02:59 1 4 .80, it reduces down to 120, okay, plus b.

03:11 Now i'm going to subtract 120 from each side of the equal sign.

03:21 And so these cancel out here...