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:

xxyy
5124
6149
7174
8199
9224

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 (mm), meaning the difference between consecutive yy-values should be the same.

Calculating the differences:

149124=25149 - 124 = 25

174149=25174 - 149 = 25

199174=25199 - 174 = 25

224199=25224 - 199 = 25

Since the differences are constant, the data follows a linear function.

Step 2: Find the Linear Equation

The linear equation is of the form:

y=mx+by = mx + b

where:

  • mm is the slope (rate of change), which we calculated as 2525.
  • bb is the y-intercept, which we determine by using one of the points.

Using the point (5,124)(5,124):

124=25(5)+b124 = 25(5) + b

124=125+b124 = 125 + b

b=1b = -1

Thus, the linear function is:

y=25x1y = 25x - 1

Step 3: Verify if the Function Fits

Checking for other points:

  • x=6x = 6: y=25(6)1=149y = 25(6) - 1 = 149
  • x=7x = 7: y=25(7)1=174y = 25(7) - 1 = 174
  • x=8x = 8: y=25(8)1=199y = 25(8) - 1 = 199
  • x=9x = 9: y=25(9)1=224y = 25(9) - 1 = 224

Since all values match, the equation y=25x1y = 25x - 1 correctly models the data.


Would you like me to explain this in more detail?

Here are five related questions to expand on this concept:

  1. How do you determine if data follows an exponential function?
  2. What does the slope represent in a linear function?
  3. How can you find the equation of a line given any two points?
  4. What happens when the y-intercept is negative?
  5. 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 yy-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