Variation of a Function - Examples, Exercises and Solutions

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.

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.

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

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.

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.

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 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.

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 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.

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.

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 $(5, -1)$:

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

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

• 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.

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 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, 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 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.