Math Problem Statement
Which function below represents the table of x-values and corresponding y-values?
Solution
To determine which function represents the table of -values and corresponding -values, let's analyze the options by testing their validity with the table:
The given table is:
x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & 4 & 3 & 2 & 1 & 0 & -1 \\ \end{array}$$ --- ### Step 1: Observe the pattern From the table: - As $$x$$ increases by $$1$$, $$y$$ decreases by $$1$$. This suggests a **linear relationship** with a slope ($$m$$) of $$-1$$. --- ### Step 2: Write the general form of a linear equation A linear function is of the form: $$y = mx + c$$ where $$m$$ is the slope, and $$c$$ is the y-intercept. Given the slope $$m = -1$$, the equation becomes: $$y = -x + c$$ --- ### Step 3: Solve for $$c$$ using a point from the table Take any point from the table, for example $$(0, 2)$$: $$y = -x + c \quad \text{Substitute $$x = 0$$ and $$y = 2$$:}$$ $$2 = -0 + c \implies c = 2$$ Thus, the equation is: $$y = -x + 2$$ --- ### Step 4: Verify with the table We verify the equation $$y = -x + 2$$ using all $$x$$-values: - For $$x = -2$$: $$y = -(-2) + 2 = 2 + 2 = 4$$ ✓ - For $$x = -1$$: $$y = -(-1) + 2 = 1 + 2 = 3$$ ✓ - For $$x = 0$$: $$y = -(0) + 2 = 2$$ ✓ - For $$x = 1$$: $$y = -(1) + 2 = -1 + 2 = 1$$ ✓ - For $$x = 2$$: $$y = -(2) + 2 = -2 + 2 = 0$$ ✓ - For $$x = 3$$: $$y = -(3) + 2 = -3 + 2 = -1$$ ✓ The equation $$y = -x + 2$$ matches the table perfectly. --- ### Final Answer: The correct function is: $$\boxed{y = -x + 2}$$ --- ### Do you want me to explain any part in more detail, or do you have further questions? --- ### Related Questions: 1. How do you determine the slope of a linear relationship from a table? 2. What is the general form of a linear equation, and how do you calculate the y-intercept? 3. How can you verify if a table represents a linear function? 4. How do you find the equation of a line given two points? 5. What is the significance of slope and y-intercept in a linear equation? --- ### Tip: Always test your equation with all given points in a table to ensure accuracy!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope-Intercept Form
Formulas
y = mx + c
Theorems
Slope of a line is the change in y divided by the change in x
Suitable Grade Level
Grades 6-8