Math Problem Statement
Look at this table: x y – 7 113 2 – 6 97 2 – 5 81 2 – 4 65 2 – 3 49 2 Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
Solution
Looking at the table, we have pairs of and values. Our goal is to determine whether the data fits a linear or exponential model. Let's break this down:
Data table:
\hline x & y \\ \hline 7 & 113 \\ 6 & 97 \\ 5 & 81 \\ 4 & 65 \\ 3 & 49 \\ \hline \end{array}$$ ### Step 1: Check if the data is linear. To check if the data fits a linear model $$y = mx + b$$, we need to see if the change in $$y$$ values is consistent for each unit change in $$x$$. - From $$x = 7$$ to $$x = 6$$, $$y$$ changes from 113 to 97: $$113 - 97 = 16$$ - From $$x = 6$$ to $$x = 5$$, $$y$$ changes from 97 to 81: $$97 - 81 = 16$$ - From $$x = 5$$ to $$x = 4$$, $$y$$ changes from 81 to 65: $$81 - 65 = 16$$ - From $$x = 4$$ to $$x = 3$$, $$y$$ changes from 65 to 49: $$65 - 49 = 16$$ Since the change in $$y$$ is consistent, the data fits a linear pattern. ### Step 2: Find the slope ($$m$$) and y-intercept ($$b$$) of the linear function. The slope $$m$$ is the rate of change between the two variables: $$m = \frac{{\Delta y}}{{\Delta x}} = \frac{{16}}{{1}} = 16$$ Now, we use one point (let's use $$(7, 113)$$) and the slope $$m = -16$$ to find the y-intercept ($$b$$). Using the linear equation $$y = mx + b$$: $$113 = -16(7) + b$$ $$113 = -112 + b$$ $$b = 113 + 112 = 225$$ ### Final Equation: Thus, the linear function that models the data is: $$y = -16x + 225$$ --- ### Exponential Model: The data does not suggest an exponential model since the change in $$y$$ is consistent, not multiplicative. Therefore, the linear model is the best fit. --- ### Would you like further details or have any questions on this solution? Here are some related questions to explore: 1. How would you solve for $$m$$ and $$b$$ if the data was not linear? 2. How do you determine if a dataset fits a linear or exponential function? 3. What happens to the graph of a linear function as the slope $$m$$ changes? 4. How can you use the linear function to predict values of $$y$$ for $$x = 2$$ or other points? 5. How do you test the accuracy of a model to check if it fits the data well? **Tip:** When checking if the data follows a linear trend, always look for constant differences between $$y$$-values as $$x$$-values change, which will indicate a linear relationship.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Exponential Functions
Slope
Intercept
Formulas
y = mx + b
y = a(b)^x
Theorems
Linearity Test: Constant change in y values indicates a linear relationship
Suitable Grade Level
Grades 7-9
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