Math Problem Statement
The graph shows the relationship between the amount of change given from a $10 bill and the number of bagels purchased.
A graph shows number of bagels labeled 1 to 10 on the horizontal axis and change given (dollar sign) on the vertical axis. A line decreases from 0 to 10.
What function represents the amount of change given from a $10 bill, f(x), based on x, the number of bagels purchased?
f(x) = + 10 f(x) = + 10 f(x) = + 10 f(x) = + 10
Solution
To solve this, let’s break down the problem based on the information provided.
- The horizontal axis represents the number of bagels purchased, denoted as x.
- The vertical axis represents the amount of change given from a $10 bill, denoted as f(x).
- As the number of bagels increases, the amount of change decreases in a linear fashion.
This suggests that the price of one bagel is constant, and the amount of change depends directly on how many bagels are purchased.
Key Observations:
- When 0 bagels are purchased (x = 0), the amount of change is $10 (f(0) = 10), as the person is giving the full $10 bill.
- The amount of change decreases as the number of bagels increases.
If we assume that the price of each bagel is constant, let’s assume p is the price of one bagel.
The amount of change after purchasing x bagels is: where:
- is the original amount of money given,
- represents the total price of the bagels purchased.
Determining the Price of a Bagel:
From the graph description, it sounds like when 10 bagels are purchased, the change is 0, which suggests the total price of the 10 bagels is exactly $10. Therefore, the price per bagel is:
Thus, the function becomes:
This means for each bagel purchased, the amount of change decreases by $1.
Conclusion:
The function that represents the amount of change given from a $10 bill based on the number of bagels purchased is:
Do you need further clarification or any other details? Here are a few related questions you might find interesting:
- What happens to the function if the price of each bagel is increased to $2?
- How would the graph change if the person initially paid $20 instead of $10?
- Can you identify the slope and y-intercept in the function ?
- How can you graph this function step-by-step?
- What would the function look like if the number of bagels purchased was multiplied by a constant factor?
Tip: To interpret linear relationships, always look for the rate of change (the slope), which tells you how much the dependent variable (amount of change) changes for each unit change in the independent variable (number of bagels).
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Algebra
Slope-Intercept Form
Formulas
f(x) = 10 - x
Theorems
Linear relationships
Slope and intercept interpretation
Suitable Grade Level
Grades 7-9
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