Linear Functions and Their Graphs - Free Printable

Problem Analysis:

The image shows a straight line on a Cartesian coordinate grid. The task is to determine the equation of the line. To solve this, we need to identify two key components:
1. Slope (m): The steepness of the line.
2. y-intercept (b): The point where the line crosses the y-axis.

Step-by-Step Solution:

#### Step 1: Identify Points on the Line
From the graph, we can identify at least two points that lie on the line. Let's choose the following points:
- Point 1: \( (x_1, y_1) = (-3, -2) \)
- Point 2: \( (x_2, y_2) = (3, 4) \)

#### Step 2: Calculate the Slope (m)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the coordinates of the points:
\[
m = \frac{4 - (-2)}{3 - (-3)} = \frac{4 + 2}{3 + 3} = \frac{6}{6} = 1
\]
So, the slope \( m \) is:
\[
m = 1
\]

#### Step 3: Determine the y-Intercept (b)
The equation of a line in slope-intercept form is:
\[
y = mx + b
\]
We already know \( m = 1 \). To find \( b \), we can substitute the coordinates of one of the points into the equation. Let's use the point \( (-3, -2) \):
\[
y = mx + b
\]
Substitute \( x = -3 \), \( y = -2 \), and \( m = 1 \):
\[
-2 = 1(-3) + b
\]
Simplify:
\[
-2 = -3 + b
\]
Solve for \( b \):
\[
b = -2 + 3 = 1
\]
So, the y-intercept \( b \) is:
\[
b = 1
\]

#### Step 4: Write the Equation of the Line
Now that we have the slope \( m = 1 \) and the y-intercept \( b = 1 \), we can write the equation of the line in slope-intercept form:
\[
y = mx + b
\]
Substitute \( m = 1 \) and \( b = 1 \):
\[
y = 1x + 1
\]
Simplify:
\[
y = x + 1
\]

Final Answer:

The equation of the line is:
\[
\boxed{y = x + 1}
\]