Math Problem Statement
Look at this table: x y – 2 – 8 – 1 – 9 0 – 10 1 – 11 2 – 12 Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
Solution
To determine a function that models the given data, we will first analyze the structure of the data points. The table provides the following pairs of and values:
\hline x & y \\ \hline -2 & 8 \\ -1 & 9 \\ 0 & 10 \\ 1 & 11 \\ 2 & 12 \\ \hline \end{array}$$ ### Step 1: Check for a **linear function** model $$y = mx + b$$. A linear function implies a constant rate of change in $$y$$ with respect to $$x$$, which means the difference between successive $$y$$-values should be constant. Let's calculate the differences between consecutive $$y$$-values: - $$9 - 8 = 1$$ - $$10 - 9 = 1$$ - $$11 - 10 = 1$$ - $$12 - 11 = 1$$ Since the differences are constant (all 1), this suggests the relationship between $$x$$ and $$y$$ is linear. ### Step 2: Find the slope ($$m$$) and the y-intercept ($$b$$). From the constant differences, we can deduce that the slope $$m = 1$$. To find the intercept $$b$$, use any of the data points. Let's use the point $$(0, 10)$$ since it is simple: $$y = mx + b \quad \Rightarrow \quad 10 = 1(0) + b \quad \Rightarrow \quad b = 10$$ ### Step 3: Write the linear equation. Thus, the linear function that models the data is: $$y = x + 10$$ ### Step 4: Check for an **exponential function** model $$y = a(b)^x$$. To test for an exponential model, observe how $$y$$ changes as $$x$$ increases. In an exponential model, the ratio between consecutive $$y$$-values should be constant, instead of the difference being constant as we saw in the linear case. Let's calculate the ratio of consecutive $$y$$-values: - $$\frac{9}{8} = 1.125$$ - $$\frac{10}{9} = 1.1111$$ - $$\frac{11}{10} = 1.1$$ - $$\frac{12}{11} = 1.0909$$ Since the ratios are not constant, an exponential function does not fit the data well. ### Conclusion: The **linear function** $$y = x + 10$$ best models the data. --- Would you like any further explanation or details on this process? Here are 5 related questions you can explore: 1. How do you determine the slope of a linear function? 2. How can you use two points to find the equation of a line? 3. What is the difference between linear and exponential functions? 4. How do you check if data fits an exponential model? 5. How would the equation change if the slope was different? **Tip:** When checking for a linear function, look for constant differences in the $$y$$-values. For an exponential function, check for constant ratios instead.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
Linear Function Theorem
Exponential Function Theorem
Suitable Grade Level
Grades 8-10
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