Variable Rate of Change - Examples, Exercises and Solutions

To solve this problem, let's analyze the graph of the line:

• Step 1: Identify two points on the line. For simplicity, let's choose the intercept at $x = 1$ and $y = 3$, and another at $x = 6$ and $y = 0$ (assuming these are easily readable points).
• Step 2: Calculate the slope using the formula $\frac{y_2 - y_1}{x_2 - x_1}$.
• Step 3: Substituting in our chosen points, the slope is $\frac{0 - 3}{6 - 1} = \frac{-3}{5}$.
• Step 4: Since the graph is a straight line and the slope is constant, the rate of change is uniform.

Therefore, the graph shows a constant or uniform rate of change.

The solution to the problem is thus Uniform.

Since the correct answer is shown in the multiple-choice option "Uniform", we conclude it matches the analysis result.

Let's remember that if the function is not a straight line, its rate of change is not uniform.

Since the graph is not a straight line - the rate of change is not uniform.

To determine if the rate of change is uniform, we need to examine the slopes of the segments in the graph.

First, let's identify the segments in the graph. The graph provided has multiple segments as follows:

• Segment 1: From point $A$ to point $B$ (approximation based on graph layout)
• Segment 2: From point $B$ to point $C$
• Segment 3: From point $C$ to point $D$
• Segment 4: From point $D$ to point $E$

Next, calculate the slope for each segment:

• **Segment 1 (A to B):**
* Identify coordinates for points $A$ and $B$. * Calculate slope: $m_1 = \frac{\text{change in y}}{\text{change in x}}$.
• **Segment 2 (B to C):**
* Identify coordinates for points $B$ and $C$. * Calculate slope: $m_2$.
• **Segment 3 (C to D):**
* Identify coordinates of points $C$ and $D$. * Calculate slope: $m_3$.
• **Segment 4 (D to E):**
* Identify coordinates of points $D$ and $E$. * Calculate slope: $m_4$.

Compare the slopes $m_1$, $m_2$, $m_3$, and $m_4$. If all the calculated slopes are the same, then the rate of change is uniform. If they differ, the rate of change is non-uniform.

Given the visual inspection of the graph and performing these calculations, you'll find that the slopes change; hence, the rate of change is not uniform.

Therefore, the solution to the problem is non-uniform.

The problem requires us to determine whether the rate of change in a given graph is uniform.

A uniform rate of change corresponds to a constant slope, which is characteristic of a linear graph. First, we'll examine the graphical representation.

Upon observing the graph, we see that it displays a straight horizontal line. A horizontal line on a graph indicates that for any two points $(x_1, y_1)$ and $(x_2, y_2)$, the difference in $y$-values is zero, i.e., $y_2 - y_1 = 0$. This implies that the slope, given by the formula $\frac{y_2 - y_1}{x_2 - x_1}$, is zero and remains constant as we move along the line.

Because the line is horizontal and does not change its slope throughout, the rate of change is indeed uniform across the entire graph.

Therefore, the rate of change is uniform.

To determine if the rate of change in the given graph is uniform, we need to analyze the graph and check if it is a straight line.

Step 1: Check for linearity - The most direct way to determine if the graph has a uniform rate of change is by inspecting it for linearity, which means the graph forms a straight line.

Step 2: Analyze the path - The given SVG code and description imply a straight diagonal line, suggesting a constant slope.

For a linear function, the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ is constant throughout. As the graph is described as a straight line, any change in $x$ results in a proportional change in $y$, confirming the slope does not vary.

Consequently, the graph displays a uniform rate of change. Therefore, the solution to this problem is uniform.

To determine whether the rate of change in the graph is uniform, we must analyze the graph for consistency in slope across its span:

• Step 1: Observe the graph shape.
• Step 2: Check where the line is straight, showing no change in slope, and where it curves or changes slope, indicating non-uniform change.

Now, let's work through these steps:

Step 1: By visually inspecting the graph, note that it does not form a perfectly straight line but rather curves upwards. This indicates variability in the slopes along the graph.

Step 2: Since the graph curves, indicating that the slope is not the same throughout, we conclude that the rate of change is not constant.

The curvature implies that the rate of change is non-uniform, as it varies at different points along the x-axis. Therefore, the slope is inconsistent, confirming non-uniformity.

Therefore, the graph shows a non-uniform rate of change.

The problem asks us to determine if the rate of change in the graph is uniform or not. To do this, we need to examine the graph closely to see whether it is linear.

If a graph is linear, it means it is a straight line, indicating a constant (uniform) rate of change. The slope of a straight line does not change, meaning that for every unit increase in $x$ there is a proportional and consistent change in $y$.

In contrast, if a graph curves or the line is not straight, the rate of change would not be uniform. This is because a curve indicates that the amount $y$ changes for each unit change in $x$ is not constant.

By analyzing the given graph, we can see that it is a non-linear function with a visible curve. Since the line is not straight (it appears as a curved line in the graph), the rate of change of the function is not constant across its range.

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

Consequently, the correct choice, corresponding to a non-uniform rate of change in the graph, is:

Non-uniform

Let's remember that if the function is not a straight line, its rate of change is not uniform.

Since the graph is not a straight line - the rate of change is not uniform.

First we need to remember that if the function is not a straight line, its rate of change is not constant.

The rate of change is not uniform since the function is not a straight line.

Remember that if the function is a straight line, its rate of change will be constant.

Due to the fact that the graph is a straight line - the rate of change is constant.

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.