Math Problem Statement
Look at this table: x y 5 124 5 6 149 5 7 174 5 8 199 5 9 224 5 Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
Solution
Let's analyze the given data:
5 | 124 |
6 | 149 |
7 | 174 |
8 | 199 |
9 | 224 |
We will determine whether the data follows a linear or exponential pattern.
Step 1: Check for Linearity
A function is linear if it has a constant rate of change (), meaning the difference between consecutive -values should be the same.
Calculating the differences:
Since the differences are constant, the data follows a linear function.
Step 2: Find the Linear Equation
The linear equation is of the form:
where:
- is the slope (rate of change), which we calculated as .
- is the y-intercept, which we determine by using one of the points.
Using the point :
Thus, the linear function is:
Step 3: Verify if the Function Fits
Checking for other points:
- : ✅
- : ✅
- : ✅
- : ✅
Since all values match, the equation correctly models the data.
Would you like me to explain this in more detail?
Here are five related questions to expand on this concept:
- How do you determine if data follows an exponential function?
- What does the slope represent in a linear function?
- How can you find the equation of a line given any two points?
- What happens when the y-intercept is negative?
- How do you apply linear equations in real-world scenarios?
Tip:
To quickly check if data follows an exponential pattern, calculate the ratios between consecutive -values instead of their differences.
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Slope
Y-intercept
Exponential Functions
Formulas
Linear function: y = mx + b
Exponential function: y = a(b)^x
Theorems
Constant rate of change (linear functions)
Exponential growth (exponential functions)
Suitable Grade Level
Grades 7-9