The task involves graphing linear equations. The provided image shows two linear equations and their corresponding graphs:
1.
Equation 1: \( y = -2x - 8 \)
2.
Equation 2: \( y = 4x + 5 \)
Explanation of Graphing Linear Equations
To graph a linear equation, we follow these steps:
#### Step 1: Identify the slope (\( m \)) and the y-intercept (\( b \))
The general form of a linear equation is:
\[
y = mx + b
\]
- \( m \) is the slope of the line.
- \( b \) is the y-intercept (where the line crosses the y-axis).
#### Step 2: Plot the y-intercept
The y-intercept is the point where \( x = 0 \). So, substitute \( x = 0 \) into the equation to find \( y \).
#### Step 3: Use the slope to find another point
The slope \( m \) is the ratio of the change in \( y \) to the change in \( x \) (i.e., \( m = \frac{\Delta y}{\Delta x} \)). Starting from the y-intercept, use the slope to find another point on the line.
#### Step 4: Draw the line
Connect the points with a straight line to complete the graph.
---
Detailed Solution for Each Equation
#### Equation 1: \( y = -2x - 8 \)
1.
Identify the slope and y-intercept:
- Slope (\( m \)): \( -2 \)
- Y-intercept (\( b \)): \( -8 \)
2.
Plot the y-intercept:
- When \( x = 0 \), \( y = -8 \).
- So, the y-intercept is at the point \( (0, -8) \).
3.
Use the slope to find another point:
- Slope \( m = -2 \) means for every 1 unit increase in \( x \), \( y \) decreases by 2 units.
- Starting from \( (0, -8) \):
- If \( x = 1 \), then \( y = -2(1) - 8 = -10 \). So, another point is \( (1, -10) \).
4.
Draw the line:
- Connect the points \( (0, -8) \) and \( (1, -10) \) with a straight line.
#### Equation 2: \( y = 4x + 5 \)
1.
Identify the slope and y-intercept:
- Slope (\( m \)): \( 4 \)
- Y-intercept (\( b \)): \( 5 \)
2.
Plot the y-intercept:
- When \( x = 0 \), \( y = 5 \).
- So, the y-intercept is at the point \( (0, 5) \).
3.
Use the slope to find another point:
- Slope \( m = 4 \) means for every 1 unit increase in \( x \), \( y \) increases by 4 units.
- Starting from \( (0, 5) \):
- If \( x = 1 \), then \( y = 4(1) + 5 = 9 \). So, another point is \( (1, 9) \).
4.
Draw the line:
- Connect the points \( (0, 5) \) and \( (1, 9) \) with a straight line.
---
Final Answer
The graphs of the given equations are already provided in the image. Here is a summary:
1. For \( y = -2x - 8 \):
- Y-intercept: \( (0, -8) \)
- Another point: \( (1, -10) \)
- The line slopes downward (negative slope).
2. For \( y = 4x + 5 \):
- Y-intercept: \( (0, 5) \)
- Another point: \( (1, 9) \)
- The line slopes upward (positive slope).
Thus, the graphs are correctly drawn as shown in the image.
\[
\boxed{\text{Graphs are correct as shown in the image.}}
\]
Parent Tip: Review the logic above to help your child master the concept of graphing a linear equation worksheet.
