1. **Identify the dependent and independent variables in the first practice problem:** - Problem: The distance $t$ depends on the speed $S$. - Explanation: The distance traveled depends on the speed, so the dependent variable is $t$. - The speed $S$ causes the change, so the independent variable is $S$. 2. **Find the pattern and write an equation for the first pattern problem:** - Given table: $$\begin{array}{c|cccc} x & 2 & 10 & 16 & 26 \\ y & 0 & 1 & 5 & ? \\\end{array}$$ - Step 1: Find the pattern in $y$ relative to $x$. - Step 2: Calculate differences: - From $x=2$ to $x=10$, $y$ changes from 0 to 1. - From $x=10$ to $x=16$, $y$ changes from 1 to 5. - Step 3: Try to find a linear relation $y = mx + b$. - Using points $(2,0)$ and $(10,1)$: $$m = \frac{1 - 0}{10 - 2} = \frac{1}{8}$$ $$b = y - mx = 0 - \frac{1}{8} \times 2 = -\frac{1}{4}$$ - Equation: $$y = \frac{1}{8}x - \frac{1}{4}$$ - Step 4: Check with $(16,5)$: $$y = \frac{1}{8} \times 16 - \frac{1}{4} = 2 - 0.25 = 1.75$$ This does not match 5, so the relation is not linear. - Step 5: Try a quadratic or piecewise pattern or check differences in $y$: Differences in $y$: 0 to 1 (1), 1 to 5 (4), difference increases by 3. - Step 6: Assume $y$ depends on $x$ in a nonlinear way; since the problem is incomplete, we cannot find exact rule here. - Step 7: For $x=26$, $y$ is unknown. 3. **Use the equation $y = 5x + 1$ to complete the table:** - Given $x$ values: 1, 2, 3, 4, 5 - Calculate $y$ values: 1. $y = 5 \times 1 + 1 = 6$ 2. $y = 5 \times 2 + 1 = 11$ 3. $y = 5 \times 3 + 1 = 16$ 4. $y = 5 \times 4 + 1 = 21$ 5. $y = 5 \times 5 + 1 = 26$ - Completed table: $$\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ y & 6 & 11 & 16 & 21 & 26 \\\end{array}$$