Math Problem Statement
What is the constant rate of change, or slope, between the values of x and y in the table?
Solution
To determine the constant rate of change (or slope) between the values of and in the table, we use the slope formula:
Given Table:
\hline x & 1 & 5 & 9 & 13 \\ \hline y & -6 & -3 & 0 & 3 \\ \hline \end{array}$$ ### Step 1: Pick any two points Let's use the first two points: $$(1, -6)$$ and $$(5, -3)$$. ### Step 2: Plug into the slope formula $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-6)}{5 - 1}$$ Simplify: $$m = \frac{-3 + 6}{4} = \frac{3}{4}$$ ### Step 3: Verify consistency To confirm the slope is constant, calculate between the next pair of points: $$(5, -3)$$ and $$(9, 0)$$. $$m = \frac{0 - (-3)}{9 - 5} = \frac{0 + 3}{4} = \frac{3}{4}$$ It remains $$\frac{3}{4}$$, showing the slope is consistent. ### Final Answer: The constant rate of change (slope) is: $$\frac{3}{4}$$ ### Correct Option: $$\text{The last choice: } \frac{3}{4}$$ --- ### Do you need any further clarification or a detailed explanation? ### Related Questions: 1. How do you calculate slope from a graph instead of a table? 2. What does a positive slope like $$\frac{3}{4}$$ indicate about the line's direction? 3. How would the slope change if the values of $$y$$ decreased instead of increasing? 4. What is the slope-intercept form of a line given the slope and one point? 5. How do you find the slope if given a vertical or horizontal line? ### Tip: The slope tells us how much $$y$$ changes for a unit increase in $$x$$. A positive slope means the line rises to the right.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Slope
Rate of Change
Linear Relationships
Formulas
m = (y2 - y1) / (x2 - x1)
Theorems
Slope Formula
Suitable Grade Level
Grades 8-10