Question
Which functions are linear?
Responses
A y = x + 3y = x + 3
B y = 2x +1
c. y=3^x+2
d. y = x/2+6
e y=3x^2-7
Responses
A y = x + 3y = x + 3
B y = 2x +1
c. y=3^x+2
d. y = x/2+6
e y=3x^2-7
Answers
GPT-4o mini
To determine which of the given functions are linear, we need to identify whether their equations can be represented in the standard form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Linear functions have a constant rate of change and graph as straight lines.
Let's analyze each option:
A. \( y = x + 3 \)
This can be rewritten in the form \( y = mx + b \) where \( m = 1 \) and \( b = 3 \).
**Linear**
B. \( y = 2x + 1 \)
In this case, \( m = 2 \) and \( b = 1 \).
**Linear**
C. \( y = 3^x + 2 \)
This function involves an exponent that makes it nonlinear.
**Not linear**
D. \( y = \frac{x}{2} + 6 \)
This can be rewritten as \( y = 0.5x + 6 \). Here, \( m = 0.5 \) and \( b = 6 \).
**Linear**
E. \( y = 3x^2 - 7 \)
This function has a \( x^2 \) term, making it a quadratic function, and hence not linear.
**Not linear**
**Summary: The linear functions from the list are: A, B, and D.**
Let's analyze each option:
A. \( y = x + 3 \)
This can be rewritten in the form \( y = mx + b \) where \( m = 1 \) and \( b = 3 \).
**Linear**
B. \( y = 2x + 1 \)
In this case, \( m = 2 \) and \( b = 1 \).
**Linear**
C. \( y = 3^x + 2 \)
This function involves an exponent that makes it nonlinear.
**Not linear**
D. \( y = \frac{x}{2} + 6 \)
This can be rewritten as \( y = 0.5x + 6 \). Here, \( m = 0.5 \) and \( b = 6 \).
**Linear**
E. \( y = 3x^2 - 7 \)
This function has a \( x^2 \) term, making it a quadratic function, and hence not linear.
**Not linear**
**Summary: The linear functions from the list are: A, B, and D.**