Let's determine which graph matches the equation [tex]\( y + 3 = 2(x + 3) \)[/tex].
1. Start by Expanding the Equation:
We need to distribute the 2 on the right side of the equation:
[tex]\[
y + 3 = 2(x + 3)
\][/tex]
Distribute the 2:
[tex]\[
y + 3 = 2x + 6
\][/tex]
2. Isolate y to put the equation in Slope-Intercept Form:
We want to express the equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Subtract 3 from both sides:
[tex]\[
y + 3 - 3 = 2x + 6 - 3
\][/tex]
Simplifying this gives:
[tex]\[
y = 2x + 3
\][/tex]
3. Identify the Slope and Y-Intercept:
From the equation [tex]\( y = 2x + 3 \)[/tex], we can see:
- The slope [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\( 3 \)[/tex]. This means the graph crosses the y-axis at the point (0,3).
4. Graph Characteristics:
- The line has a slope of 2, which means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
- The y-intercept is 3, so the line will intersect the y-axis at (0,3).
Thus, the graph that matches the equation [tex]\( y + 3 = 2(x + 3) \)[/tex] is the one that represents the line [tex]\( y = 2x + 3 \)[/tex]. This line will rise steeply with a slope of 2 and will cross the y-axis at the point (0, 3).
