Question

ladder lesson objective 2
directions: complete the examples with the teacher, then finish the guided practice with teacher feedback before starting the independent practice.
objective 2: determine and interpret the slope and y - intercept of a two - variable linear equation.
benchmark: ma.912.f.1.2 given a function represented in function notation, evaluate the function for an input in its domain. for a real - world context, interpret the output.
example 1: a car rental company charges a flat fee of $25 plus $0.15 per mile driven. the function can model the total cost: $f(x)=0.15x + 25$ where $x$ = number of miles driven and $f(x)$ = total cost in dollars.
what is the slope, and what does it mean in this context?
what is the y - intercept, and what does it mean in the context?
example 2: the table shows the total cost for different numbers of hours a moving truck is rented:
determine and interpret the slope and y - intercept of the linear equation that represents the total cost based on the number of hours the truck is rented.
hours rented (x) | total cost (y)
0 | 50
2 | 70
4 | 90
6 | 110
example 3: taylor is building a wall in a computer game. the graph shows the number of blocks taylor has on the wall as a function of time.
question: interpret the slope and y - intercept of the line that represents this situation. what does it tell you about taylor’s building activity?

Explanation:

Example 1

Step1: Identify the slope - formula form

The linear function is $f(x)=0.15x + 25$, which is in the form $y = mx + b$ where $m$ is the slope.

Step2: Determine the slope value

The slope $m = 0.15$. In this context, it means that for each additional mile driven, the total cost increases by $0.15$ dollars.

Step3: Identify the y - intercept

In the form $y=mx + b$, $b$ is the y - intercept. Here, $b = 25$. It means that even if no miles are driven (i.e., $x = 0$), there is a flat - fee of $25$ dollars.

Step1: Calculate the slope

The slope $m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(0,50)$ and $(2,70)$, $m=\frac{70 - 50}{2-0}=\frac{20}{2}=10$. It means that for each additional hour the truck is rented, the total cost increases by $10$ dollars.

Step2: Determine the y - intercept

When $x = 0$, $y = 50$. So the y - intercept is $50$. It represents the base cost (the cost when the truck is rented for $0$ hours), perhaps an initial processing or reservation fee.

Step1: Estimate the slope

Pick two points on the line, say $(0,50)$ and $(50,200)$. The slope $m=\frac{200 - 50}{50-0}=\frac{150}{50}=3$. It means that Taylor builds 3 blocks per minute.

Step2: Determine the y - intercept

The line crosses the y - axis at $y = 50$. It means that Taylor started with 50 blocks already on the wall at time $t = 0$.

Answer:

Slope: $0.15$, meaning cost per mile driven is $0.15$ dollars. Y - intercept: $25$, meaning flat - fee of $25$ dollars when no miles are driven.

Example 2