Question
The table below represents a linear function.
xx yy
00 55
44 1111
88 1717
1212 2323
Which relationship represents a function with a lesser rate of change than the function represented above?
A
x
y
By, equals, 6, x, minus, 1
y=6x−1
C
x
y
Dy, equals, 3, x, minus, 1
y=3x−1
Answer
xx yy
00 55
44 1111
88 1717
1212 2323
Which relationship represents a function with a lesser rate of change than the function represented above?
A
x
y
By, equals, 6, x, minus, 1
y=6x−1
C
x
y
Dy, equals, 3, x, minus, 1
y=3x−1
Answer
Answers
Answered by
GPT-4o mini
To determine which relationship represents a function with a lesser rate of change than the function represented in the given table, we first need to find the rate of change of the function described in the table.
The points in the table suggest the following pairs:
- (0, 5)
- (4, 11)
- (8, 17)
- (12, 23)
To find the rate of change, we can take two points and calculate the slope (rate of change).
Let's take the first two points: (0, 5) and (4, 11).
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using (0, 5) and (4, 11):
\[
m = \frac{11 - 5}{4 - 0} = \frac{6}{4} = \frac{3}{2}
\]
Now, let's analyze the given options:
A. \( y = 6x - 1 \)
- The slope is 6.
B. \( y = 3x - 1 \)
- The slope is 3.
C. \( y = 6x - 1 \)
- The slope is 6 (same as option A).
D. \( y = 3x - 1 \)
- The slope is 3 (same as option B).
The function represented by the table has a rate of change of \( \frac{3}{2} \) (which is equal to 1.5). Now we need to find an option with a slope less than \( \frac{3}{2} \):
- The slope of A (6) is greater than \( \frac{3}{2} \).
- The slope of B (3) is also greater than \( \frac{3}{2} \).
- The slope of C (6) is greater than \( \frac{3}{2} \).
- The slope of D (3) is greater than \( \frac{3}{2} \).
However, I see that the rates of change in the provided options are greater than \( \frac{3}{2} \). None of them provide a slope that is less than \( \frac{3}{2} \).
Thus, the options provided do not have a relationship which represents a function with a lesser rate of change than the function represented in the original table. If the problem contains errors such that other choices were meant to be evaluated or potential miscommunication in the question, the interpretation will vary. As per the given choices, none fulfill the requirement of being less than \( \frac{3}{2} \).
The points in the table suggest the following pairs:
- (0, 5)
- (4, 11)
- (8, 17)
- (12, 23)
To find the rate of change, we can take two points and calculate the slope (rate of change).
Let's take the first two points: (0, 5) and (4, 11).
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using (0, 5) and (4, 11):
\[
m = \frac{11 - 5}{4 - 0} = \frac{6}{4} = \frac{3}{2}
\]
Now, let's analyze the given options:
A. \( y = 6x - 1 \)
- The slope is 6.
B. \( y = 3x - 1 \)
- The slope is 3.
C. \( y = 6x - 1 \)
- The slope is 6 (same as option A).
D. \( y = 3x - 1 \)
- The slope is 3 (same as option B).
The function represented by the table has a rate of change of \( \frac{3}{2} \) (which is equal to 1.5). Now we need to find an option with a slope less than \( \frac{3}{2} \):
- The slope of A (6) is greater than \( \frac{3}{2} \).
- The slope of B (3) is also greater than \( \frac{3}{2} \).
- The slope of C (6) is greater than \( \frac{3}{2} \).
- The slope of D (3) is greater than \( \frac{3}{2} \).
However, I see that the rates of change in the provided options are greater than \( \frac{3}{2} \). None of them provide a slope that is less than \( \frac{3}{2} \).
Thus, the options provided do not have a relationship which represents a function with a lesser rate of change than the function represented in the original table. If the problem contains errors such that other choices were meant to be evaluated or potential miscommunication in the question, the interpretation will vary. As per the given choices, none fulfill the requirement of being less than \( \frac{3}{2} \).