Math Problem Statement
Look at this table:
x | y
-5 | 25
-4 | 18
-3 | 11
-2 | 4
-1 | -3
Write a linear function (y = mx + b) or an exponential function (y = a(b)^x) that models the data.
Solution
The table in the image you provided lists the following pairs of and values:
\hline x & y \\ \hline -5 & 25 \\ -4 & 18 \\ -3 & 11 \\ -2 & 4 \\ -1 & -3 \\ \hline \end{array}$$ To determine whether a linear or exponential function models the data, let's check the differences in $$y$$ values. ### Checking for a Linear Function: A linear function follows the form $$y = mx + b$$, where the difference between consecutive $$y$$ values should be constant. Let’s compute the differences between consecutive $$y$$ values: - $$18 - 25 = -7$$ - $$11 - 18 = -7$$ - $$4 - 11 = -7$$ - $$-3 - 4 = -7$$ Since the differences are constant (-7), this indicates that the data follows a **linear** relationship. ### Finding the Linear Function: We can now find the equation of the line using the slope-intercept form $$y = mx + b$$. The slope $$m$$ is $$-7$$, and we can use one of the points to find $$b$$. Let’s use the point $$(-5, 25)$$: $$25 = -7(-5) + b$$ $$25 = 35 + b$$ $$b = 25 - 35 = -10$$ Thus, the linear function that models this data is: $$y = -7x - 10$$ If you have any further questions or need more details, feel free to ask! ### Further questions to explore: 1. How would you graph the equation $$y = -7x - 10$$? 2. What does the slope of the line represent in the context of the data? 3. Can you find the value of $$y$$ for $$x = 0$$ using this equation? 4. How would the equation change if the differences were not constant? 5. What is the significance of the y-intercept ($$b = -10$$) in this context? ### Tip: Always check the differences in $$y$$ when trying to identify if a relationship is linear or exponential. For linear, the difference is constant; for exponential, the ratios between consecutive $$y$$ values are constant.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Exponential Functions
Algebra
Formulas
y = mx + b
y = a(b)^x
Theorems
Constant difference in y-values indicates a linear function
Suitable Grade Level
Grades 7-9