Question
Identify whether the graph displays a linear or nonlinear function. Explain why.
Graph A
On a coordinate plane, graph A goes through points (0, negative 3) and (2, 1).
Graph A displays a
Graph A
On a coordinate plane, graph A goes through points (0, negative 3) and (2, 1).
Graph A displays a
Answers
GPT-4o mini
Graph A displays a **linear function**.
To explain why, let's consider the characteristics of linear functions. A linear function can be represented by a straight line on a coordinate plane. It has the general form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Since Graph A goes through two specific points: (0, -3) and (2, 1), we can determine the slope between these points.
1. To find the slope \( m \), we use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the points (0, -3) and (2, 1):
\[
m = \frac{1 - (-3)}{2 - 0} = \frac{1 + 3}{2} = \frac{4}{2} = 2
\]
2. Now, we can write the equation of the line using the slope and one of the points (let's use (0, -3)):
\[
y = 2x - 3
\]
This equation is linear in form, confirming that the relationship between \( x \) and \( y \) is linear.
Additionally, since the graph passes through only two points and maintains a consistent slope, it will remain straight; thus supporting the conclusion that Graph A represents a linear function.
To explain why, let's consider the characteristics of linear functions. A linear function can be represented by a straight line on a coordinate plane. It has the general form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Since Graph A goes through two specific points: (0, -3) and (2, 1), we can determine the slope between these points.
1. To find the slope \( m \), we use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the points (0, -3) and (2, 1):
\[
m = \frac{1 - (-3)}{2 - 0} = \frac{1 + 3}{2} = \frac{4}{2} = 2
\]
2. Now, we can write the equation of the line using the slope and one of the points (let's use (0, -3)):
\[
y = 2x - 3
\]
This equation is linear in form, confirming that the relationship between \( x \) and \( y \) is linear.
Additionally, since the graph passes through only two points and maintains a consistent slope, it will remain straight; thus supporting the conclusion that Graph A represents a linear function.