Table Analysis 691A97

1. **State the problem:** We are given a table of values for $x$ and $y$: $$\begin{array}{c|cccc} x & 0 & 5 & 10 & 15 \\ y & 4 & 8 & 12 & 15 \\\end{array}$$ We want to analyze the relationship between $x$ and $y$. 2. **Check if the relationship is linear:** A linear function has the form $$y = mx + b$$ where $m$ is the slope and $b$ is the $y$-intercept. 3. **Calculate the slope $m$ between points:** Between $(0,4)$ and $(5,8)$: $$m = \frac{8 - 4}{5 - 0} = \frac{4}{5} = 0.8$$ Between $(5,8)$ and $(10,12)$: $$m = \frac{12 - 8}{10 - 5} = \frac{4}{5} = 0.8$$ Between $(10,12)$ and $(15,15)$: $$m = \frac{15 - 12}{15 - 10} = \frac{3}{5} = 0.6$$ 4. **Interpretation:** The slope changes from $0.8$ to $0.6$, so the relationship is not perfectly linear. 5. **Find the equation for the first three points:** Using the slope $m=0.8$ and point $(0,4)$: $$y = 0.8x + 4$$ 6. **Check the last point:** For $x=15$, $$y = 0.8 \times 15 + 4 = 12 + 4 = 16$$ But the table shows $y=15$, which is less than $16$, confirming the slope change. **Final answer:** The data shows a mostly linear increase with slope approximately $0.8$ for $x$ from $0$ to $10$, then a smaller slope $0.6$ from $10$ to $15$. The $y$-intercept is $4$.