Math Problem Statement

Which relationship represents a function with a lesser rate of changeRate of Change than the function represented by the given table?

Solution

Let's solve this step by step:

The table provides a function, and we need to calculate its rate of change (slope). Then we’ll compare it to the slopes of the given options to determine which has a lesser rate of change.


1. Calculate the rate of change from the table:

The slope is calculated using the formula:
m=ΔyΔxm = \frac{\Delta y}{\Delta x}

From the table:

  • Between (0,3)(0, 3) and (4,10)(4, 10):
    m=10340=74m = \frac{10 - 3}{4 - 0} = \frac{7}{4}
  • Between (4,10)(4, 10) and (8,17)(8, 17):
    m=171084=74m = \frac{17 - 10}{8 - 4} = \frac{7}{4}
  • Between (8,17)(8, 17) and (12,24)(12, 24):
    m=2417128=74m = \frac{24 - 17}{12 - 8} = \frac{7}{4}

The rate of change for the table is consistently 74\frac{7}{4}.


2. Compare the slopes of the given options:

Now we calculate the slope for each option:

  • Option A:
    The graph has a slope of 1, as it rises 1 unit for every 1 unit it runs (Δy=1,Δx=1\Delta y = 1, \Delta x = 1).

  • Option B:
    The graph has a slope of 2, as it rises 2 units for every 1 unit it runs (Δy=2,Δx=1\Delta y = 2, \Delta x = 1).

  • Option C (y=32x4y = \frac{3}{2}x - 4):
    The slope is 32\frac{3}{2} (Δy=3,Δx=2\Delta y = 3, \Delta x = 2).

  • Option D (y=4x+1y = 4x + 1):
    The slope is 44 (Δy=4,Δx=1\Delta y = 4, \Delta x = 1).


3. Compare the slopes to 74\frac{7}{4}:

The function in the table has a slope of 74\frac{7}{4} (or 1.75). A lesser rate of change means the slope must be smaller than 1.75.

  • Option A: Slope =1= 1 (lesser than 1.75).
  • Option B: Slope =2= 2 (greater than 1.75).
  • Option C: Slope =32=1.5= \frac{3}{2} = 1.5 (lesser than 1.75).
  • Option D: Slope =4= 4 (greater than 1.75).

4. Answer:

The correct options are A and C, as their rates of change are less than the table’s slope of 74\frac{7}{4}.


Would you like a deeper breakdown of any step?


Related Questions:

  1. How can you determine the slope directly from a graph?
  2. What does it mean for a function to have a greater rate of change?
  3. Can you describe a real-life situation where slope represents a rate of change?
  4. How would the table change if the slope were 22 instead of 74\frac{7}{4}?
  5. Why is slope important in comparing linear functions?

Tip:

Remember that the slope (mm) of a line tells you how steep it is—smaller slopes mean the line is less steep!

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Rate of Change
Slope

Formulas

m = Δy / Δx

Theorems

Concept of Slope

Suitable Grade Level

Grade 8-10