Answer :
Sure! Let's discuss how to draw the line represented by the equation [tex]\( y = 8x + 12 \)[/tex] on a graph. We will use specific points to plot the line accurately.
### Step-by-Step Solution
1. Identify the Equation Components:
- The equation of the line is [tex]\( y = 8x + 12 \)[/tex].
- This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\( m = 8 \)[/tex] (slope) and [tex]\( b = 12 \)[/tex] (y-intercept).
2. Plot the Y-Intercept:
- The y-intercept ([tex]\( b \)[/tex]) is the point where the line crosses the y-axis.
- For this equation, the y-intercept is [tex]\( (0, 12) \)[/tex].
- Plot the point [tex]\( (0, 12) \)[/tex] on the graph.
3. Determine Additional Points:
- To draw the line accurately, we need more points. We can choose different values of [tex]\( x \)[/tex] and calculate the corresponding values of [tex]\( y \)[/tex].
- We know from the given solution:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 20 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 28 \)[/tex]
- So, plot these points as well: [tex]\( (0, 12) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 28) \)[/tex].
4. Draw the Line:
- Once you have plotted the points [tex]\( (0, 12) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 28) \)[/tex], you can draw a straight line through these points.
- This line represents the equation [tex]\( y = 8x + 12 \)[/tex].
### Graph Description
- X-Axis (Horizontal Axis):
- Label this axis as "Hours" since [tex]\( x \)[/tex] represents the number of hours.
- Mark points 0, 1, 2, etc., along the axis.
- Y-Axis (Vertical Axis):
- Label this axis as "Total Cost" since [tex]\( y \)[/tex] represents the total cost.
- Mark points 12, 20, 28, etc., along the axis to correspond with the y-values.
### Example of Points Plotting:
1. Plot [tex]\( (0, 12) \)[/tex]: Start at the origin (0,0). Move 0 units on the x-axis (stay at 0) and 12 units up on the y-axis.
2. Plot [tex]\( (1, 20) \)[/tex]: Start at the origin. Move 1 unit right on the x-axis to 1 and 20 units up on the y-axis.
3. Plot [tex]\( (2, 28) \)[/tex]: Start at the origin. Move 2 units right on the x-axis and 28 units up on the y-axis.
### Connecting the Points
- Once the points are plotted, use a ruler to draw a straight line passing through all three points.
- Extend the line along both directions beyond the points plotted to cover more of the graph.
### Interpretation
- This line shows the relationship between the hours spent on lessons and the total cost. The slope (8) represents the rate at which the cost increases per hour, and the y-intercept (12) represents the initial cost or base fee.
This way, you can visually represent the cost structure on a graph, making it easier to understand how costs accumulate over hours.
### Step-by-Step Solution
1. Identify the Equation Components:
- The equation of the line is [tex]\( y = 8x + 12 \)[/tex].
- This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\( m = 8 \)[/tex] (slope) and [tex]\( b = 12 \)[/tex] (y-intercept).
2. Plot the Y-Intercept:
- The y-intercept ([tex]\( b \)[/tex]) is the point where the line crosses the y-axis.
- For this equation, the y-intercept is [tex]\( (0, 12) \)[/tex].
- Plot the point [tex]\( (0, 12) \)[/tex] on the graph.
3. Determine Additional Points:
- To draw the line accurately, we need more points. We can choose different values of [tex]\( x \)[/tex] and calculate the corresponding values of [tex]\( y \)[/tex].
- We know from the given solution:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 20 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 28 \)[/tex]
- So, plot these points as well: [tex]\( (0, 12) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 28) \)[/tex].
4. Draw the Line:
- Once you have plotted the points [tex]\( (0, 12) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 28) \)[/tex], you can draw a straight line through these points.
- This line represents the equation [tex]\( y = 8x + 12 \)[/tex].
### Graph Description
- X-Axis (Horizontal Axis):
- Label this axis as "Hours" since [tex]\( x \)[/tex] represents the number of hours.
- Mark points 0, 1, 2, etc., along the axis.
- Y-Axis (Vertical Axis):
- Label this axis as "Total Cost" since [tex]\( y \)[/tex] represents the total cost.
- Mark points 12, 20, 28, etc., along the axis to correspond with the y-values.
### Example of Points Plotting:
1. Plot [tex]\( (0, 12) \)[/tex]: Start at the origin (0,0). Move 0 units on the x-axis (stay at 0) and 12 units up on the y-axis.
2. Plot [tex]\( (1, 20) \)[/tex]: Start at the origin. Move 1 unit right on the x-axis to 1 and 20 units up on the y-axis.
3. Plot [tex]\( (2, 28) \)[/tex]: Start at the origin. Move 2 units right on the x-axis and 28 units up on the y-axis.
### Connecting the Points
- Once the points are plotted, use a ruler to draw a straight line passing through all three points.
- Extend the line along both directions beyond the points plotted to cover more of the graph.
### Interpretation
- This line shows the relationship between the hours spent on lessons and the total cost. The slope (8) represents the rate at which the cost increases per hour, and the y-intercept (12) represents the initial cost or base fee.
This way, you can visually represent the cost structure on a graph, making it easier to understand how costs accumulate over hours.
