To determine which graph represents the equation [tex]\( y = 12 + 2x \)[/tex], let's break down the equation and analyze its components step-by-step.
1. Understand the Equation:
- The equation of the line is given by [tex]\( y = 12 + 2x \)[/tex].
- This is a linear equation in slope-intercept form, [tex]\( y = mx + c \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the y-intercept, which is the point where the line crosses the y-axis.
2. Find the Slope and Y-Intercept:
- The slope ([tex]\( m \)[/tex]) of the line is 2.
- The y-intercept ([tex]\( c \)[/tex]) is 12. This means when [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex].
3. Plotting Key Points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 12 \)[/tex]. So, one point on the graph is (0, 12).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 12 + 2(1) = 14 \)[/tex]. Another point is (1, 14).
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 12 + 2(2) = 16 \)[/tex]. Another point is (2, 16).
4. Characteristics of the Graph:
- The line will have a positive slope, indicating it rises as it moves from left to right.
- It will cross the y-axis at 12.
- For each increase of 1 in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2.
Now, let's visualize how the graph should look based on this analysis:
- The starting point is at (0, 12).
- The line rises with a slope of 2, meaning for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
- If the graph has more points plotted, they should consistently follow this pattern, maintaining the linearity and slope we calculated.
To select the correct graph, look for the graph that:
- Crosses the y-axis at 12.
- Shows a straight line moving upward with a slope that reflects an increase of 2 in [tex]\( y \)[/tex] for each increase of 1 in [tex]\( x \)[/tex].
By matching these criteria, you can identify the correct graph representing the equation [tex]\( y = 12 + 2x \)[/tex].
