Math Problem Statement
Solution
To determine whether represents a linear function, we need to check if the changes in are consistent for each unit change in . In other words, if has a constant rate of change (slope), then it could represent a linear function.
Let's examine the values given in the table:
\hline x & 0 & 1 & 2 & 3 \\ \hline f(x) & -7 & -2 & 3 & 8 \\ \hline \end{array}$$ ### Step 1: Calculate the differences in $$f(x)$$ for each successive $$x$$-value - From $$x = 0$$ to $$x = 1$$: $$f(1) - f(0) = -2 - (-7) = 5$$ - From $$x = 1$$ to $$x = 2$$: $$f(2) - f(1) = 3 - (-2) = 5$$ - From $$x = 2$$ to $$x = 3$$: $$f(3) - f(2) = 8 - 3 = 5$$ The differences are consistent (all equal to 5), indicating that $$f(x)$$ has a constant rate of change, so it is likely a linear function. ### Step 2: Write $$f(x)$$ in the form $$f(x) = mx + b$$ Since the slope $$m$$ is the constant change in $$f(x)$$ per unit increase in $$x$$, we have: $$m = 5$$ Now, we need to find $$b$$, the $$y$$-intercept, by substituting a known point, such as $$(0, -7)$$, into the equation $$f(x) = 5x + b$$. $$-7 = 5(0) + b \Rightarrow b = -7$$ Thus, the equation for $$f(x)$$ is: $$f(x) = 5x - 7$$ ### Final Answer Yes, $$f(x)$$ could represent a linear function, and it can be written as: $$f(x) = 5x - 7$$ Would you like further details or have any questions? Here are some related questions: 1. What is the general method to check if a function is linear? 2. How does the slope influence the graph of a linear function? 3. Why is a constant rate of change indicative of linearity? 4. How would we determine linearity with a larger dataset? 5. How does changing the $$y$$-intercept affect the linear function's graph? **Tip:** For any function table, if you notice equal differences in $$y$$-values for each step in $$x$$, it suggests a linear function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Algebra
Slope-Intercept Form
Formulas
f(x) = mx + b
Slope = (change in y) / (change in x)
Theorems
Constant rate of change theorem
Suitable Grade Level
Grade 8
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