Math Problem Statement
The table above shows x and y values that have a linear relationship to each other. Which of the graphs below could illustrate the relationship between these variables?
Solution
To solve this, let's examine the and values given in the table and determine the relationship, then match that with the correct graph.
Step 1: Check for Linear Relationship
The table shows:
x & y \\ \hline 1 & 9 \\ 4 & 3 \\ 7 & -3 \\ 10 & -9 \\ 13 & -15 \\ \end{array}$$ Each increase of 3 in $$x$$ corresponds to a decrease of 6 in $$y$$. This suggests a negative linear relationship. ### Step 2: Determine the Slope Calculating the slope between consecutive points: $$\text{slope} = \frac{\Delta y}{\Delta x} = \frac{-6}{3} = -2$$ So, the equation for the line could be of the form $$y = -2x + b$$, showing a consistent decrease in $$y$$ as $$x$$ increases. ### Step 3: Match with the Graphs Since the slope is negative, the correct graph should show a line that slopes downward from left to right. Among the options: - **Graph A** has a positive slope (upward). - **Graph B** has an undefined slope (vertical). - **Graph C** has a negative slope (downward). - **Graph D** also has an undefined slope (vertical). Thus, **Graph C** correctly illustrates the relationship between these variables, showing a downward linear trend. **Answer: Graph C** Would you like more details on how to derive the equation, or do you have other questions? ### Related Questions 1. How do you calculate the equation of a line from a table of values? 2. What does a negative slope represent in real-world contexts? 3. How can you verify if a set of points has a linear relationship? 4. Why do some graphs appear as vertical or horizontal lines? 5. How does the slope affect the direction of a line in a graph? **Tip:** When identifying linear relationships, always check the consistency of changes in $$x$$ and $$y$$ values; this can help determine the slope and direction.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Relationships
Coordinate Graphs
Slope
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Linear equation: y = mx + b
Theorems
Slope-Intercept Form
Suitable Grade Level
Grades 8-10