What is the rate of change of the table?

Question image

Understand the Problem

The question is asking for the rate of change based on the values presented in the table for x and y. We will determine the rate of change by examining how y changes in relation to x.

Answer

The rate of change of the table is $6$.
Answer for screen readers

The rate of change of the table is $6$.

Steps to Solve

  1. Identify the changes in x and y

We will calculate the change in $y$ for each change in $x$. The values from the table are:

  • For $x = 0 \rightarrow 1$, $y$ changes from $-4$ to $2$
  • For $x = 1 \rightarrow 2$, $y$ changes from $2$ to $8$
  • For $x = 2 \rightarrow 3$, $y$ changes from $8$ to $14$
  • For $x = 3 \rightarrow 4$, $y$ changes from $14$ to $20$
  1. Calculate each rate of change

The rate of change (slope) can be found using the formula:

$$ \text{Rate of Change} = \frac{\Delta y}{\Delta x} $$

where $\Delta y = y_{new} - y_{old}$ and $\Delta x = x_{new} - x_{old}$.

  1. Plug values into the formula
  • From $0$ to $1$: $$ \text{Rate}_1 = \frac{2 - (-4)}{1 - 0} = \frac{6}{1} = 6 $$

  • From $1$ to $2$: $$ \text{Rate}_2 = \frac{8 - 2}{2 - 1} = \frac{6}{1} = 6 $$

  • From $2$ to $3$: $$ \text{Rate}_3 = \frac{14 - 8}{3 - 2} = \frac{6}{1} = 6 $$

  • From $3$ to $4$: $$ \text{Rate}_4 = \frac{20 - 14}{4 - 3} = \frac{6}{1} = 6 $$

  1. Determine the overall rate of change

Since the rates of change between each interval are consistent, the overall rate of change for the table is:

$$ \text{Overall Rate of Change} = 6 $$

The rate of change of the table is $6$.

More Information

The rate of change represents how much $y$ increases for each unit increase in $x$. In this case, for every increase of 1 in $x$, $y$ consistently increases by 6.

Tips

  • Forgetting to calculate changes between the correct pairs of $x$ and $y$ values.
  • Confusing the order of values when calculating the rates of change.
  • Assuming the rate of change is constant without checking all intervals.

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