Math Problem Statement
Find the formula for a linear function f that models the data in the table exactly: x = {0, 1, 2, 3}, f(x) = {-2, 4, 10, 16}.
Solution
The problem asks for the formula of a linear function that models the data in the given table.
The table presents the following values:
\hline x & 0 & 1 & 2 & 3 \\ f(x) & -2 & 4 & 10 & 16 \\ \hline \end{array}$$ ### Step 1: Determine the slope Since this is a linear function, the slope $$m$$ can be calculated using the formula for the slope between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$: $$m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ Using the points $$(0, -2)$$ and $$(1, 4)$$: $$m = \frac{4 - (-2)}{1 - 0} = \frac{6}{1} = 6$$ So, the slope $$m = 6$$. ### Step 2: Find the y-intercept To find the y-intercept $$b$$, we can use the point-slope form of a linear equation $$f(x) = mx + b$$. Using the point $$(0, -2)$$, we substitute $$x = 0$$, $$f(x) = -2$$, and $$m = 6$$ into the equation: $$-2 = 6(0) + b$$ $$b = -2$$ ### Step 3: Write the linear equation Now that we have the slope $$m = 6$$ and the y-intercept $$b = -2$$, the linear equation is: $$f(x) = 6x - 2$$ This equation models the data exactly. Would you like any additional details or further explanations? Here are five related questions: 1. How do you find the slope between two points in a general case? 2. Can this method be applied to non-linear data, and how would the process differ? 3. How can you check if a function is linear based on a table of values? 4. What are other methods to find the equation of a line if given a graph instead of a table? 5. How does the y-intercept affect the graph of a linear function? **Tip:** When identifying a linear relationship from a table, always check if the rate of change (slope) between consecutive points is constant.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Slope-Intercept Form
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Linear equation: f(x) = mx + b
Theorems
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Suitable Grade Level
Grades 8-10