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Answer:
- The y-intercept is 800 and represents the initial amount of money borrowed ($800).
- The slope is -40 and represents the rate at which James is paying back the loan per month ($40 per month).
- The function is linear because it is a first-degree polynomial with a constant rate of change.
Step-by-step explanation:
James borrowed money from his father to buy a used car. He is paying his father back $40 per month until the loan is paid back.
The function that models the given situation is m(x) = -40x + 800, where m(x) is the amount owed and x is the number of months since James borrowed the money.
The function is in slope-intercept form:
[tex]y = mx + b[/tex]
where:
- m is the slope.
- b is the y-intercept.
Therefore, the y-intercept of function m(x) is 800. This represents the initial amount of money borrowed, which is $800.
The slope is -40, which represents the rate at which James is paying back the loan per month. Since the slope is -40, it means he is reducing the amount owed by $40 per month.
The function m(x) = -40x + 800 is linear because it is a first-degree polynomial with a constant rate of change.
Answer:
Y-intercept: [tex] (0, 800) [/tex] represents the initial loan amount.
Slope: -40 represents the rate at which James pays back the loan each month.
Linearity: Yes, it is a linear function because it follows the form [tex] y = mx + b [/tex].
Step-by-step explanation:
Let's analyze the given function [tex] m(x) = -40x + 800 [/tex]:
1. Y-intercept:
- The y-intercept occurs when [tex] x = 0 [/tex]. Substitute [tex] x = 0 [/tex] into the function:
[tex] m(0) = -40 \times 0 + 800 = 800 [/tex]
- The y-intercept is [tex] (0, 800) [/tex].
In this context, it represents the initial amount of money borrowed by James, which is $800.
2. Slope:
- The coefficient of [tex] x [/tex] in the function is -40.
The slope is -40.
- In this context, the slope represents the rate at which James is paying back the loan each month. It's $40, but with a negative sign indicating that he is reducing the amount owed.
3. Linearity:
Yes, this is a linear function because it is of the form:
[tex] y = mx + b [/tex],
where
- [tex] m [/tex] is the slope and
- [tex] b [/tex] is the y-intercept.
The function describes a straight line on a graph, and the variables are only raised to the first power.
In summary:
Y-intercept: [tex] (0, 800) [/tex] represents the initial loan amount.
Slope: -40 represents the rate at which James pays back the loan each month.
Linearity: Yes, it is a linear function because it follows the form [tex] y = mx + b [/tex].