Rate of change represented by steps in the function graph

We can draw stairs on the graph of the function to see the rate of change.
The base of the step will represent the interval in the X X variables and the height will symbolize the interval in the Y Y .

The step will mark the "jump" from X X in relation to the "jump" in Y Y .
The bases of the steps will always be the same since we always choose fixed intervals in X X .

  • If the heights of the steps are increasing, it means that the rate of change is increasing.
  • If the heights of the steps are decreasing, it means that the rate of change is decreasing.
  • If the heights of the steps do not change, it means that the rate of change is constant.
Rate of change represented by steps in the function graph

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function

Practice Rate of change represented with steps in the graph of the function

Examples with solutions for Rate of change represented with steps in the graph of the function

Exercise #1

Look at the graph below and determine whether the function's rate of change is constant or not:

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Video Solution

Step-by-Step Solution

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.

Answer

Not constant

Exercise #2

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

111222333444555666777888999101010111111121212131313141414151515111222333444555666777888000

Video Solution

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

Exercise #3

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

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Video Solution

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

Exercise #4

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

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–3–3–3–2–2–2–1–1–1111222333444555666000

Video Solution

Step-by-Step Solution

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.

Answer

Uniform

Exercise #5

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

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Video Solution

Step-by-Step Solution

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 x there is a proportional and consistent change in y 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 y changes for each unit change in x 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

Answer

Non-uniform

Exercise #6

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

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Video Solution

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

Exercise #7

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

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Video Solution

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

Exercise #8

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

111222333444555666777888999101010111111121212111222333444555666000

Video Solution

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

Exercise #9

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

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Video Solution

Step-by-Step Solution

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.

Answer

Non-uniform

Exercise #10

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

111222333444555666111222333000

Video Solution

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

Exercise #11

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

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Video Solution

Step-by-Step Solution

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)(8, 2) and (9,4)(9, 4):
  • Rate of change=4298=21=2\text{Rate of change} = \frac{4 - 2}{9 - 8} = \frac{2}{1} = 2
  • For points (9,4)(9, 4) and (10,6)(10, 6):
  • Rate of change=64109=21=2\text{Rate of change} = \frac{6 - 4}{10 - 9} = \frac{2}{1} = 2
  • For points (10,6)(10, 6) and (11,8)(11, 8):
  • Rate of change=861110=21=2\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.

Answer

Uniform

Exercise #12

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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Video Solution

Step-by-Step Solution

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

  • Step 1: Calculate the slope between points:
    • For (2,3)(-2, 3) to (0,5)(0, 5):
    • m1=530+2=22=1 m_1 = \frac{5 - 3}{0 + 2} = \frac{2}{2} = 1
    • For (0,5) (0, 5) to (2,8) (2, 8):
    • m2=8520=32 m_2 = \frac{8 - 5}{2 - 0} = \frac{3}{2}
    • For (2,8) (2, 8) to (4,12) (4, 12):
    • m3=12842=42=2 m_3 = \frac{12 - 8}{4 - 2} = \frac{4}{2} = 2
  • Step 2: Compare these slopes:
    Since m1=1 m_1 = 1, m2=32 m_2 = \frac{3}{2}, and m3=2 m_3 = 2 are not equal, the rate of change is not constant.

Therefore, the rate of change is non-uniform.

Answer

Non-uniform

Exercise #13

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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Video Solution

Step-by-Step Solution

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)(1, -6) and (2,3)(2, -3):
    Δy=3(6)=3\Delta y = -3 - (-6) = 3
    Δx=21=1\Delta x = 2 - 1 = 1
    Slope =31=3= \frac{3}{1} = 3
  • Between (2,3)(2, -3) and (3,0)(3, 0):
    Δy=0(3)=3\Delta y = 0 - (-3) = 3
    Δx=32=1\Delta x = 3 - 2 = 1
    Slope =31=3= \frac{3}{1} = 3
  • Between (3,0)(3, 0) and (4,3)(4, 3):
    Δy=30=3\Delta y = 3 - 0 = 3
    Δx=43=1\Delta x = 4 - 3 = 1
    Slope =31=3= \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.

Answer

Uniform

Exercise #14

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

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Video Solution

Step-by-Step Solution

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)(0, -1) and (4,0)(4, 0).
    Rate of change=0(1)40=14 \text{Rate of change} = \frac{0 - (-1)}{4 - 0} = \frac{1}{4}
  • Step 2: Calculate between (4,0)(4, 0) and (8,1)(8, 1).
    Rate of change=1084=14 \text{Rate of change} = \frac{1 - 0}{8 - 4} = \frac{1}{4}
  • Step 3: Calculate between (8,1)(8, 1) and (12,2)(12, 2).
    Rate of change=21128=14 \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 14\frac{1}{4} between each pair of points, the rate of change is uniform.

Therefore, the solution to the problem is Uniform.

Answer

Uniform

Exercise #15

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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Video Solution

Step-by-Step Solution

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)(-2, 6) to (0,8) (0, 8): 860(2)=22=1. \frac{8 - 6}{0 - (-2)} = \frac{2}{2} = 1.

From (0,8)(0, 8) to (2,10) (2, 10): 10820=22=1. \frac{10 - 8}{2 - 0} = \frac{2}{2} = 1.

From (2,10)(2, 10) to (4,12) (4, 12): 121042=22=1. \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.

Answer

Uniform