Question

match the characteristics of the equation to the graph using a slope of \\(\frac{3}{2}\\) and passes through the point (-2, 0).

Explanation:

Step1: Recall Point - Slope Form

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Here, $m = \frac{3}{2}$ and $(x_1,y_1)=(-2,0)$.
Substitute these values into the point - slope form: $y - 0=\frac{3}{2}(x - (-2))$, which simplifies to $y=\frac{3}{2}(x + 2)$.

Step2: Simplify the Equation

Expand the right - hand side: $y=\frac{3}{2}x+3$.
To find the $x$ - intercept, set $y = 0$: $0=\frac{3}{2}x + 3$. Subtract 3 from both sides: $-3=\frac{3}{2}x$. Multiply both sides by $\frac{2}{3}$: $x=-2$ (which we already know from the given point).
To find the $y$ - intercept, set $x = 0$: $y=\frac{3}{2}(0)+3=3$. So the line has a $y$ - intercept at $(0,3)$ and a slope of $\frac{3}{2}$ (rise of 3, run of 2).
To match the graph, we can look for a line that goes through $(-2,0)$ and $(0,3)$ (since from $(-2,0)$, if we move 2 units to the right (run = 2) and 3 units up (rise = 3), we reach $(0,3)$), and has a slope of $\frac{3}{2}$.

Answer:

The graph of the line with equation $y=\frac{3}{2}x + 3$ (derived from the point - slope form using slope $\frac{3}{2}$ and point $(-2,0)$) will have a $y$ - intercept at $(0,3)$, pass through $(-2,0)$, and have a slope of $\frac{3}{2}$. To identify the correct graph, look for a line that passes through $(-2,0)$ and $(0,3)$ (or follows the slope - rise over run - of $\frac{3}{2}$).