Variation of a Function - Examples, Exercises and Solutions

Understanding Variation of a Function

Complete explanation with examples

Different Types of Rates of Change of a Function

The rate of change of a function describes the pace of modification that the variables Y Y experience with respect to the change in the variables X X .
The rate of change can be:

  • b> b> Constant Rate of Change - fixed
    Describes a situation in which we will see constant intervals of variables Y Y at constant intervals of variables X X .
    In a linear function (a straight line), the rate of change is constant and will be the slope of the function.
  • Non-constant Rate of Change - not fixed
    Describes a situation in which we will see different intervals of variables Y Y at constant intervals of variables X X .
    In a function that is not linear (not a straight line), the rate of change will not be constant. Each part of the function will have a different slope.
Different Types of Rates of Change of a Function

We can see the rate of change represented in a graph, a table of values, or by creating steps or stairs in the graph of the function.

Detailed explanation

Practice Variation of a Function

Test your knowledge with 9 quizzes

Given a table showing points on the graph of a function, determine whether or not the rate of change is uniform.

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Examples with solutions for Variation of a Function

Step-by-step solutions included
Exercise #1

Given the following graph, determine whether the rate of change is uniform or not

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Step-by-Step Solution

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 x = 1 and y=3 y = 3 , and another at x=6 x = 6 and y=0 y = 0 (assuming these are easily readable points).
  • Step 2: Calculate the slope using the formula y2y1x2x1\frac{y_2 - y_1}{x_2 - x_1}.
  • Step 3: Substituting in our chosen points, the slope is 0361=35\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.

Answer:

Uniform

Video Solution
Exercise #2

Given the following graph, determine whether the rate of change is uniform or not

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Step-by-Step Solution

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.

Answer:

Non-uniform

Video Solution
Exercise #3

Given the following graph, determine whether the rate of change is uniform or not

111222333444555666777888999101010111111121212111222333444555666000

Step-by-Step Solution

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 AA to point BB (approximation based on graph layout)
  • Segment 2: From point BB to point CC
  • Segment 3: From point CC to point DD
  • Segment 4: From point DD to point EE

Next, calculate the slope for each segment:

  • **Segment 1 (A to B):**
  • * Identify coordinates for points AA and BB. * Calculate slope: m1=change in ychange in xm_1 = \frac{\text{change in y}}{\text{change in x}}.
  • **Segment 2 (B to C):**
  • * Identify coordinates for points BB and CC. * Calculate slope: m2m_2.
  • **Segment 3 (C to D):**
  • * Identify coordinates of points CC and DD. * Calculate slope: m3m_3.
  • **Segment 4 (D to E):**
  • * Identify coordinates of points DD and EE. * Calculate slope: m4m_4.

Compare the slopes m1m_1, m2m_2, m3m_3, and m4m_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.

Answer:

Non-uniform

Video Solution
Exercise #4

Given the following graph, determine whether the rate of change is uniform or not

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Step-by-Step Solution

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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the difference in yy-values is zero, i.e., y2y1=0y_2 - y_1 = 0. This implies that the slope, given by the formula y2y1x2x1 \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.

Answer:

Uniform

Video Solution
Exercise #5

Given the following graph, determine whether the rate of change is uniform or not

–3–3–3–2–2–2–1–1–1111222333444–1–1–1111222333000

Step-by-Step Solution

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=y2y1x2x1 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 x results in a proportional change in y y , confirming the slope does not vary.

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

Answer:

Uniform

Video Solution

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