The graph of straight line `y = sqrt(3)x + 2sqrt(3)` is :

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying the graph of the straight line given by the equation \( y = \sqrt{3}x + 2\sqrt{3} \), we will follow these steps:

### Step 1: Identify the slope and y-intercept
The equation of the line is in the form \( y = mx + c \), where:
- \( m \) is the slope,
- \( c \) is the y-intercept.

From the equation \( y = \sqrt{3}x + 2\sqrt{3} \):
- The slope \( m = \sqrt{3} \).
- The y-intercept \( c = 2\sqrt{3} \).

### Step 2: Determine the angle of inclination
The slope \( m \) can also be expressed in terms of the angle \( \theta \) that the line makes with the positive x-axis:
\[ m = \tan(\theta) \]
Given \( m = \sqrt{3} \), we find \( \theta \):
\[ \tan(\theta) = \sqrt{3} \implies \theta = 60^\circ \]

### Step 3: Plot the y-intercept
The y-intercept \( c = 2\sqrt{3} \) is the point where the line crosses the y-axis. We can calculate \( 2\sqrt{3} \):
\[ 2\sqrt{3} \approx 3.464 \]
This means the line crosses the y-axis at approximately \( (0, 3.464) \).

### Step 4: Use the slope to find another point
Using the slope \( \sqrt{3} \), we can find another point on the line. The slope indicates that for every 1 unit increase in \( x \), \( y \) increases by \( \sqrt{3} \):
- If \( x = 1 \):
\[ y = \sqrt{3}(1) + 2\sqrt{3} = \sqrt{3} + 2\sqrt{3} = 3\sqrt{3} \approx 5.196 \]
Thus, another point on the line is \( (1, 3\sqrt{3}) \).

### Step 5: Sketch the graph
Now we can plot the points:
- The y-intercept at \( (0, 2\sqrt{3}) \).
- The point \( (1, 3\sqrt{3}) \).

Draw a straight line through these points. The line will have a positive slope and will rise steeply due to the value of \( \sqrt{3} \).

### Conclusion
By analyzing the slope and y-intercept, we can determine the correct graph based on the characteristics of the line. The correct option will be the one that shows a line with a slope of \( \sqrt{3} \) and a y-intercept of \( 2\sqrt{3} \).