Math Problem Statement
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:
From the table:
- Between and :
- Between and :
- Between and :
The rate of change for the table is consistently .
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 (). -
Option B:
The graph has a slope of 2, as it rises 2 units for every 1 unit it runs (). -
Option C ():
The slope is (). -
Option D ():
The slope is ().
3. Compare the slopes to :
The function in the table has a slope of (or 1.75). A lesser rate of change means the slope must be smaller than 1.75.
- Option A: Slope (lesser than 1.75).
- Option B: Slope (greater than 1.75).
- Option C: Slope (lesser than 1.75).
- Option D: Slope (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 .
Would you like a deeper breakdown of any step?
Related Questions:
- How can you determine the slope directly from a graph?
- What does it mean for a function to have a greater rate of change?
- Can you describe a real-life situation where slope represents a rate of change?
- How would the table change if the slope were instead of ?
- Why is slope important in comparing linear functions?
Tip:
Remember that the slope () 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