Variation of a Function: Calculate the rate of change from a table

To solve the problem, let's determine whether the rate of change between each pair of consecutive points is consistent:

First, calculate the rate of change between the first and second points, $(-1, 3) \rightarrow (5, 8)$:

$\frac{\Delta Y}{\Delta X} = \frac{8 - 3}{5 - (-1)} = \frac{5}{6}$.

Next, calculate the rate of change between the second and third points, $(5, 8) \rightarrow (11, 13)$:

$\frac{\Delta Y}{\Delta X} = \frac{13 - 8}{11 - 5} = \frac{5}{6}$.

Finally, calculate the rate of change between the third and fourth points, $(11, 13) \rightarrow (17, 18)$:

$\frac{\Delta Y}{\Delta X} = \frac{18 - 13}{17 - 11} = \frac{5}{6}$.

All calculated rates of change are $\frac{5}{6}$, indicating a constant rate of change.

Therefore, the rate of change is uniform.

To determine if the rate of change is uniform, follow these steps:

• Step 1: Calculate the slope between points:
• For $(-2, 3)$ to $(0, 5)$:
$m_1 = \frac{5 - 3}{0 + 2} = \frac{2}{2} = 1$
• For $(0, 5)$ to $(2, 8)$:
$m_2 = \frac{8 - 5}{2 - 0} = \frac{3}{2}$
• For $(2, 8)$ to $(4, 12)$:
$m_3 = \frac{12 - 8}{4 - 2} = \frac{4}{2} = 2$
• Step 2: Compare these slopes:
  Since $m_1 = 1$, $m_2 = \frac{3}{2}$, and $m_3 = 2$ are not equal, the rate of change is not constant.

Therefore, the rate of change is non-uniform.

To determine if the rate of change is uniform, we will follow these steps:

• Step 1: Calculate the rate of change between each pair of consecutive points.
• Step 2: Compare these rates to determine if they are consistent.

Let's work through the calculations:

Step 1: Calculate the rates of change (slopes) between consecutive points.

From $(-2, 6)$ to $(0, 8)$: $\frac{8 - 6}{0 - (-2)} = \frac{2}{2} = 1.$

From $(0, 8)$ to $(2, 10)$: $\frac{10 - 8}{2 - 0} = \frac{2}{2} = 1.$

From $(2, 10)$ to $(4, 12)$: $\frac{12 - 10}{4 - 2} = \frac{2}{2} = 1.$

Step 2: Compare the rates.

All calculated rates are equal to 1, indicating that the rate of change is uniform.

Therefore, the solution to the problem is the rate of change is Uniform.

To determine if the rate of change is uniform, we need to calculate the slope between each pair of consecutive points and check for consistency.

Let's compute the slopes:

• Between $(1, -6)$ and $(2, -3)$:
  $\Delta y = -3 - (-6) = 3$
  $\Delta x = 2 - 1 = 1$
  Slope $= \frac{3}{1} = 3$
• Between $(2, -3)$ and $(3, 0)$:
  $\Delta y = 0 - (-3) = 3$
  $\Delta x = 3 - 2 = 1$
  Slope $= \frac{3}{1} = 3$
• Between $(3, 0)$ and $(4, 3)$:
  $\Delta y = 3 - 0 = 3$
  $\Delta x = 4 - 3 = 1$
  Slope $= \frac{3}{1} = 3$

Since the slopes are all equal, the rate of change is the same between each pair of consecutive points.

Therefore, the rate of change is uniform.

To determine whether the rate of change is uniform for the function given by the table, follow these steps:

• Calculate the rate of change between each pair of consecutive points.
• Ensure the calculated rates are all equal to verify uniformity.

Let's calculate the rate of change between these points:

1. Between $(-7, 0)$ and $(-4, 1)$:
The rate of change is:
$\frac{1 - 0}{-4 - (-7)} = \frac{1}{3}.$

2. Between $(-4, 1)$ and $(-1, 2)$:
The rate of change is:
$\frac{2 - 1}{-1 - (-4)} = \frac{1}{3}.$

3. Between $(-1, 2)$ and $(2, 3)$:
The rate of change is:
$\frac{3 - 2}{2 - (-1)} = \frac{1}{3}.$

All calculated rates of change are equal to $\frac{1}{3}$, indicating the rate of change is uniform between each consecutive pair of points.

Therefore, the rate of change is Uniform.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the rate of change between each pair of consecutive points.
• Step 2: Compare the computed rates to determine if they are equal.
• Step 3: Conclude whether the rate of change is uniform or not.

Now, let's work through each step:
Step 1: Calculate the rate of change between each pair of consecutive points:
- Between $(2, 3)$ and $(4, 6)$:
$\text{Slope} = \frac{6 - 3}{4 - 2} = \frac{3}{2} = 1.5$
- Between $(4, 6)$ and $(6, 9)$:
$\text{Slope} = \frac{9 - 6}{6 - 4} = \frac{3}{2} = 1.5$
- Between $(6, 9)$ and $(8, 12)$:
$\text{Slope} = \frac{12 - 9}{8 - 6} = \frac{3}{2} = 1.5$

Step 2: Compare the computed rates:
- In all cases, the rate of change is $1.5$.

Step 3: Conclude that the rate of change is uniform across the expressed intervals.

Therefore, the rate of change for the given points is uniform.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the rate of change between the first pair of points $(1, 2)$ and $(2, 3)$.
• Step 2: Calculate the rate of change between the second pair of points $(2, 3)$ and $(3, 4)$.
• Step 3: Calculate the rate of change between the third pair of points $(3, 4)$ and $(4, 7)$.
• Step 4: Compare the calculated rates to determine uniformity.

Step 1: Calculate the slope between $(1, 2)$ and $(2, 3)$
$\text{Slope} = \frac{3 - 2}{2 - 1} = 1$

Step 2: Calculate the slope between $(2, 3)$ and $(3, 4)$
$\text{Slope} = \frac{4 - 3}{3 - 2} = 1$

Step 3: Calculate the slope between $(3, 4)$ and $(4, 7)$
$\text{Slope} = \frac{7 - 4}{4 - 3} = 3$

Step 4: Compare the slopes:
The slopes between the first two pairs of points are equal to 1, while the slope between the last pair of points is 3. Since these slopes are not equal, the rate of change is not uniform.

Therefore, the solution to the problem is that the rate of change is non-uniform.

To determine whether the rate of change is uniform, we calculate the slope between each consecutive pair of points provided in the table:

• Calculate the slope between $(-5, 3)$ and $(0, 2)$:
  - The change in $Y$ is $2 - 3 = -1$.
  - The change in $X$ is $0 - (-5) = 5$.
  - Thus, the slope is $\frac{-1}{5} = -0.2$.
• Calculate the slope between $(0, 2)$ and $(5, 0)$:
  - The change in $Y$ is $0 - 2 = -2$.
  - The change in $X$ is $5 - 0 = 5$.
  - Thus, the slope is $\frac{-2}{5} = -0.4$.
• Calculate the slope between $(5, 0)$ and $(10, -2)$:
  - The change in $Y$ is $-2 - 0 = -2$.
  - The change in $X$ is $10 - 5 = 5$.
  - Thus, the slope is $\frac{-2}{5} = -0.4$.

We observe that the slopes are not all the same: the first slope $-0.2$ differs from the others, which are both $-0.4$. Therefore, the rate of change is not uniform across the intervals.

Thus, the rate of change in the function represented by the table is non-uniform.

To determine if the rate of change is uniform, we will calculate it between consecutive points and compare them step-by-step:

• Step 1: Calculate between $(0, -1)$ and $(4, 0)$.
  $\text{Rate of change} = \frac{0 - (-1)}{4 - 0} = \frac{1}{4}$
• Step 2: Calculate between $(4, 0)$ and $(8, 1)$.
  $\text{Rate of change} = \frac{1 - 0}{8 - 4} = \frac{1}{4}$
• Step 3: Calculate between $(8, 1)$ and $(12, 2)$.
  $\text{Rate of change} = \frac{2 - 1}{12 - 8} = \frac{1}{4}$
• Step 4: Compare the rates from each step.

Since the rate of change is consistently $\frac{1}{4}$ between each pair of points, the rate of change is uniform.

Therefore, the solution to the problem is Uniform.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the rate of change between each consecutive pair of points.
• Step 2: Compare these rates to check for uniformity.

Now, let's work through each step:
Step 1: Calculate the rate of change for each consecutive pair of points:
- Between $(-2, 1)$ and $(-1, 3)$:
$\frac{3 - 1}{-1 - (-2)} = \frac{2}{1} = 2$
- Between $(-1, 3)$ and $(0, 5)$:
$\frac{5 - 3}{0 - (-1)} = \frac{2}{1} = 2$
- Between $(0, 5)$ and $(1, 7)$:
$\frac{7 - 5}{1 - 0} = \frac{2}{1} = 2$

Step 2: Compare the calculated rates of change.
We observe that the rate of change is constantly $2$ for each pair of points.

Therefore, the solution to the problem is that the rate of change is Uniform.

To determine if the rate of change is uniform, follow these steps:

• Step 1: Calculate the rate of change between consecutive points.
• Step 2: Compare these rates to see if they are consistent.

Let's work through each step:

Step 1: Calculate the rate of change.

• For points $(8, 2)$ and $(9, 4)$:
$\text{Rate of change} = \frac{4 - 2}{9 - 8} = \frac{2}{1} = 2$
• For points $(9, 4)$ and $(10, 6)$:
$\text{Rate of change} = \frac{6 - 4}{10 - 9} = \frac{2}{1} = 2$
• For points $(10, 6)$ and $(11, 8)$:
$\text{Rate of change} = \frac{8 - 6}{11 - 10} = \frac{2}{1} = 2$

Step 2: Comparing the rates of change, we find they are all equal to 2, indicating uniformity.

Therefore, the rate of change is Uniform.

To determine whether the rate of change is uniform, we will calculate the rate of change between each pair of consecutive points given.

• Step 1: Calculate the rate of change between consecutive points.

Calculate between $(3, -2)$ and $(5, -1)$:

$\frac{-1 - (-2)}{5 - 3} = \frac{1}{2}$

Calculate between $(5, -1)$ and $(7, 3)$:

$\frac{3 - (-1)}{7 - 5} = \frac{4}{2} = 2$

Calculate between $(7, 3)$ and $(9, 7)$:

$\frac{7 - 3}{9 - 7} = \frac{4}{2} = 2$

• Step 2: Analyze the calculated rates of change.

We observe that the calculated rates of change are $\frac{1}{2}$, $2$, and $2$. Since the first calculated rate of change is different from the others, the rate of change between the points is not consistent.

Therefore, the rate of change is non-uniform.