Math Problem Statement
A table of values of a linear function is shown below. Find the output when the input is n.
Solution
To find the output when the input is for the given linear function, we first need to determine the pattern in the table.
Table:
\hline \text{Input} & 1 & 2 & 3 & 4 & n \\ \hline \text{Output} & -10 & -12 & -14 & -16 & ? \\ \hline \end{array}$$ ### Step 1: Identify the pattern - From $$\text{Input} = 1$$ to $$\text{Input} = 2$$, the output decreases by 2. - Similarly, from $$\text{Input} = 2$$ to $$\text{Input} = 3$$, and from $$3$$ to $$4$$, the output decreases by 2 each time. This suggests the function is linear with a constant rate of change (slope) of $$-2$$. ### Step 2: Find the function rule Given that the slope $$m$$ is $$-2$$, we can express the linear function as: $$y = mx + b$$ Where $$m = -2$$ and $$b$$ is the y-intercept. To find $$b$$, use one of the points from the table. Let's use the point $$(1, -10)$$: $$-10 = -2(1) + b$$ $$b = -10 + 2 = -8$$ So, the equation of the function is: $$y = -2x - 8$$ ### Step 3: Find the output when input is $$n$$ Substitute $$x = n$$ into the function: $$y = -2n - 8$$ Thus, the output when the input is $$n$$ is $$-2n - 8$$. Let me know if you'd like any additional details or have any questions! --- **5 Questions to Expand Understanding:** 1. How do you determine the slope of a linear function from a table of values? 2. What is the role of the y-intercept in a linear equation? 3. Can this method be applied to non-linear functions? Why or why not? 4. How would the output change if the slope were positive instead of negative? 5. How do you graph a linear function from an equation? **Tip:** Always check the difference between consecutive outputs to verify if a function is linear.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Linear Functions
Slope-Intercept Form
Formulas
y = mx + b
Slope formula: m = (change in y) / (change in x)
Theorems
Linear Equation Theorem
Suitable Grade Level
Grades 7-9