To match the graph to the given equation [tex]\( y + 3 = 2(x + 3) \)[/tex], follow these steps to rewrite the equation in its slope-intercept form, [tex]\( y = mx + b \)[/tex], which clearly shows the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex]:
1. Start with the given equation:
[tex]\[
y + 3 = 2(x + 3)
\][/tex]
2. Distribute the 2 on the right-hand side:
[tex]\[
y + 3 = 2x + 6
\][/tex]
3. Subtract 3 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 6 - 3
\][/tex]
4. Simplify the right-hand side:
[tex]\[
y = 2x + 3
\][/tex]
So, the equation in slope-intercept form is:
[tex]\[
y = 2x + 3
\][/tex]
The key features of this linear equation are:
- The slope ([tex]\( m \)[/tex]) is 2, which means the line rises 2 units for every 1 unit it runs to the right.
- The y-intercept ([tex]\( b \)[/tex]) is 3, which means the line crosses the y-axis at (0, 3).
To match the graph to this equation, look for a graph where:
- The line crosses the y-axis exactly at the point (0, 3).
- The line has a consistent upward slope of 2.
For every point [tex]\( x \)[/tex] on the line, corresponding to a point [tex]\( y \)[/tex] can be calculated as [tex]\( y = 2x + 3 \)[/tex]. This positive slope indicates that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases at twice the rate.
Therefore, the correctly matching graph will be a straight line that intersects the y-axis at (0, 3) and has a steepness or slope where for each one-unit increase in [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value increases by 2 units.
