To find which graph matches the given equation [tex]\( y + 3 = 2(x + 3) \)[/tex], we need to transform the equation into its simplest form:
1. Distribute the 2 on the right side:
[tex]\[
y + 3 = 2(x + 3)
\][/tex]
[tex]\[
y + 3 = 2x + 6
\][/tex]
2. Isolate [tex]\( y \)[/tex] on one side of the equation:
Subtract 3 from both sides of the equation to isolate [tex]\( y \)[/tex]:
[tex]\[
y + 3 - 3 = 2x + 6 - 3
\][/tex]
[tex]\[
y = 2x + 3
\][/tex]
Now, we have the equation in the form of [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Slope ([tex]\( m \)[/tex]): The coefficient of [tex]\( x \)[/tex] is 2, which means the slope of the line is 2. This tells us that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
- Y-intercept ([tex]\( b \)[/tex]): The constant term is 3, which means the graph crosses the y-axis at [tex]\( y = 3 \)[/tex].
### Graph Characteristics:
1. Y-intercept at (0, 3): The line will cross the y-axis at the point (0, 3).
2. Slope of 2: For every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] will increase by 2 units. This signifies a relatively steep incline.
### Conclusion:
The graph that matches the equation [tex]\( y = 2x + 3 \)[/tex] will have a line that:
- Crosses the y-axis at (0, 3).
- Rises 2 units for every 1 unit it runs to the right.
In summary, the graph matching the equation [tex]\( y + 3 = 2(x + 3) \)[/tex] is the one corresponding to the linear equation [tex]\( y = 2x + 3 \)[/tex], showing the described characteristics.
