Question
Match the characteristic to the function that is the same:
a) Initial value of 6
b) y-intercept of 7
c) Rate of change (common difference) of 6
d) Rate of change (common difference) of -4
a) Initial value of 6
b) y-intercept of 7
c) Rate of change (common difference) of 6
d) Rate of change (common difference) of -4
Answers
GPT-4o mini
To match the characteristics to the corresponding functions, we need to identify which type of function can represent each characteristic.
a) **Initial value of 6**: This characteristic typically refers to the starting point of a function, often associated with linear functions. It can be represented as \( y = mx + 6 \) where the function intersects the y-axis at 6.
b) **y-intercept of 7**: The y-intercept is where the function crosses the y-axis. This would indicate that the function can be expressed in the form \( y = mx + 7 \).
c) **Rate of change (common difference) of 6**: This characteristic applies to linear functions whose slope (rate of change) is 6. In a linear function, it would be represented as \( y = 6x + b \) where b is any constant.
d) **Rate of change (common difference) of -4**: Similar to (c), this characteristic indicates that the slope of the linear function is -4. It can be expressed as \( y = -4x + b \) where b is a constant.
To summarize:
- a) **Initial value of 6**: Represents the y-intercept, so it corresponds to "b) y-intercept of 7" if comparing with other functions with the same function forms.
- b) **y-intercept of 7**: Defined as such and corresponds with b).
- c) **Rate of change (common difference) of 6**: The function can be written as \( y = 6x + b \).
- d) **Rate of change (common difference) of -4**: The function can be expressed as \( y = -4x + b \).
In conclusion:
- a) Given its interpretation, it relates to the structure, rather than being directly matched.
- b) y-intercept of 7 is clearly defined.
- c) correlates with a function having a slope of 6.
- d) correlates with a function having a slope of -4.
So the characteristics described depend on their mathematical definitions and context instead.
a) **Initial value of 6**: This characteristic typically refers to the starting point of a function, often associated with linear functions. It can be represented as \( y = mx + 6 \) where the function intersects the y-axis at 6.
b) **y-intercept of 7**: The y-intercept is where the function crosses the y-axis. This would indicate that the function can be expressed in the form \( y = mx + 7 \).
c) **Rate of change (common difference) of 6**: This characteristic applies to linear functions whose slope (rate of change) is 6. In a linear function, it would be represented as \( y = 6x + b \) where b is any constant.
d) **Rate of change (common difference) of -4**: Similar to (c), this characteristic indicates that the slope of the linear function is -4. It can be expressed as \( y = -4x + b \) where b is a constant.
To summarize:
- a) **Initial value of 6**: Represents the y-intercept, so it corresponds to "b) y-intercept of 7" if comparing with other functions with the same function forms.
- b) **y-intercept of 7**: Defined as such and corresponds with b).
- c) **Rate of change (common difference) of 6**: The function can be written as \( y = 6x + b \).
- d) **Rate of change (common difference) of -4**: The function can be expressed as \( y = -4x + b \).
In conclusion:
- a) Given its interpretation, it relates to the structure, rather than being directly matched.
- b) y-intercept of 7 is clearly defined.
- c) correlates with a function having a slope of 6.
- d) correlates with a function having a slope of -4.
So the characteristics described depend on their mathematical definitions and context instead.