Identifying Constant Rate - Pre Algebra

1. How do you check a table for a constant rate when the $x$-values are not evenly spaced?

   Compute the change in $y$ divided by the change in $x$ for each pair of rows. The rate is constant only if every result is equal.

2. The points $(2, 10)$, $(4, 20)$ and $(6, 30)$ lie on a line. What is the rate of change?

   $5$. Each increase of $2$ in $x$ adds $10$ to $y$, and $\frac{10}{2} = 5$.

3. A car is $30$, $60$ and $90$ miles from home after $1$, $2$ and $3$ hours. What is its rate?

   A constant $30$ miles per hour. The distance grows by $30$ miles each hour.

4. A table has $x$-values $1$, $3$, $4$ and $y$-values $5$, $11$, $14$. What is the rate of change?

   A constant $3$. The rates are $\frac{11-5}{3-1} = 3$ and $\frac{14-11}{4-3} = 3$.

5. A plant is $2$, $4$, $8$ and $16$ cm tall at weeks $0$, $1$, $2$ and $3$. Why is its growth rate not constant?

   The weekly increases are $2$, $4$ and $8$ cm, which are not equal; the height doubles each week instead.

6. The cost of $1$, $2$ and $3$ tickets is 8, 16 and 24 dollars. What is the constant rate?

   8 dollars per ticket. Each added ticket adds 8 dollars.

7. A table has $x$-values $1$, $2$, $3$ and $y$-values $3$, $5$, $8$. Why is the rate not constant?

   The $y$-values increase by $2$ and then by $3$ for equal steps of $1$ in $x$.

8. On a straight-line graph, what is the constant rate of change called?

   The slope. It is the change in $y$ divided by the change in $x$.

9. What shape is the graph of a relationship with a constant rate of change?

   A straight line. A curve or a bend means the rate changes.

10. What makes a rate of change constant?

    Equal changes in one quantity always produce equal changes in the other, so the change in $y$ divided by the change in $x$ never varies.

11. Which has a constant rate: a car on cruise control at $55$ miles per hour, a ball dropped from a roof, or savings that double each year?

    The car on cruise control. A falling ball speeds up, and doubling savings grow by larger amounts each year.

12. A car's distance after hours $0$, $1$, $2$, $3$ and $4$ is $0$, $20$, $40$, $55$ and $70$ miles. In which hour does the rate first change?

    The third hour. The distance grows $20$ miles in each of the first two hours but only $15$ miles from hour $2$ to hour $3$.

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