Slope & Intercept - Pre Algebra

1. What is the slope of the line through $(2, 5)$ and $(6, 13)$?

   2. $\frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$.

2. A pool's water is $g = 300 - 20t$ gallons after $t$ minutes. What does $-20$ mean?

   The pool loses 20 gallons of water per minute.

3. A rental plan costs $C = 0.5m + 12$ dollars for $m$ movies. What does 0.5 represent?

   The rate: each movie adds 0.5 dollars (50 cents) to the cost.

4. What formula gives the slope through points $(x_1, y_1)$ and $(x_2, y_2)$?

   $m = \frac{y_2 - y_1}{x_2 - x_1}$, the change in $y$ divided by the change in $x$.

5. What are the slope and $y$-intercept of $y = -4x + 7$?

   Slope $-4$ and $y$-intercept 7, the point $(0, 7)$.

6. What does the $y$-intercept of a line represent?

   The value of $y$ when $x = 0$, which in a context is often the starting or initial value.

7. A rental plan costs $C = 0.5m + 12$ dollars for $m$ movies. What does 12 represent?

   The fixed charge of 12 dollars, the cost when $m = 0$ (the starting value).

8. If $y$ is distance in miles and $x$ is time in hours, what unit does the slope have?

   Miles per hour. Slope units are $y$-units per $x$-unit.

9. What does a negative slope mean in a real-world context?

   The quantity decreases as the input increases, such as a balance going down each month.

10. A bike's value drops at a constant rate from 900 dollars to 600 dollars in 3 years. What is the slope?

    $-100$ dollars per year. $\frac{600 - 900}{3} = -100$.

11. What does the slope of a line represent?

    The rate of change: how much $y$ changes for each 1-unit increase in $x$.

12. What is the slope of a horizontal line, and what does it mean in context?

    0. The quantity stays constant as the input changes.

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