1. Problem: Find the rate of change in the amount of water in the pool from 8:00 A.M. to 11:00 A.M. 2. Given: Initial amount at 8:00 A.M. is 2000 gallons, amount at 11:00 A.M. is 500 gallons. 3. Formula for rate of change (slope) is $$\text{slope} = \frac{\text{change in amount}}{\text{change in time}} = \frac{y_2 - y_1}{x_2 - x_1}$$ where $y$ is amount of water and $x$ is time. 4. Calculate change in amount: $$500 - 2000 = -1500$$ gallons. 5. Calculate change in time: From 8:00 A.M. to 11:00 A.M. is 3 hours. 6. Calculate rate of change: $$\frac{-1500}{3} = -500$$ gallons per hour. 7. Interpretation: The pool is losing 500 gallons of water every hour. --- 1. Problem: Write the equation of a line with slope $-3$ and y-intercept $2$. 2. The slope-intercept form is $$y = mx + b$$ where $m$ is slope and $b$ is y-intercept. 3. Substitute values: $$y = -3x + 2$$. --- 1. Problem: Select points on the graph of $$y = -3x + 2$$. 2. Check points: - At $x=0$, $y = -3(0) + 2 = 2$ so $(0,2)$ is on the graph. - At $x=1$, $y = -3(1) + 2 = -3 + 2 = -1$ so $(1,-1)$ is on the graph. - $(0,0)$, $(1,1)$, and $(2,0)$ do not satisfy the equation. --- 1. Problem: Find slope of line through points $(3,4)$ and $(-7,4)$. 2. Use slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 4}{-7 - 3} = \frac{0}{-10} = 0$$. 3. Interpretation: The line is horizontal with slope 0. --- 1. Problem: Describe translation of $$g(x) = (x + 10) - 1$$ relative to parent function $f(x) = x$. 2. Translation rules: $x + h$ shifts graph $h$ units left if $h > 0$, and $k$ shifts graph $k$ units up if $k > 0$. 3. Here, $g(x)$ is shifted 10 units left and 1 unit down. --- 1. Problem: Describe translation of $$g(x) = |x - 2| + 7$$ relative to parent function $f(x) = |x|$. 2. Translation: $x - 2$ shifts right 2 units, $+7$ shifts up 7 units. 3. The graph is translated 2 units right and 7 units up. --- 1. Problem: Describe translation of $$g(x) = |x + 1| - 3$$ relative to parent function $f(x) = |x|$. 2. Translation: $x + 1$ shifts left 1 unit, $-3$ shifts down 3 units. 3. The graph is translated 1 unit left and 3 units down. --- Summary for number 13: The line with slope $-3$ and y-intercept $2$ passes through points $(0,2)$ and $(1,-1)$, confirming the equation $$y = -3x + 2$$.