Final answer:
The focus of the parabola is at (3/4, 0), the directrix is the line x = -3/4, and the focal diameter is 3/2. The graph of the parabola can be plotted by finding y-values corresponding to different x-values and connecting the points.
Step-by-step explanation:
The given equation of the parabola is y² = 3x. To find the focus, directrix, and focal diameter of the parabola, we can compare the given equation to the standard form of a parabola, which is y² = 4ax. From the comparison, we can see that the value of a in the standard form is equal to 3 in the given equation.
To find the focus, we use the formula F = (a/4, 0), where F is the coordinates of the focus and a is the value from the standard form. Substituting the value of a = 3, we get F = (3/4, 0). Therefore, the focus of the parabola is at (3/4, 0).
Furthermore, the directrix of the parabola is a vertical line given by the equation x = -a/4, where a is the value from the standard form. Substituting the value of a = 3, we get x = -3/4. Therefore, the directrix of the parabola is the line x = -3/4.
The focal diameter of the parabola is the distance between the focus and any point on the directrix. Since the focus is at (3/4, 0) and the directrix is the line x = -3/4, the distance between them is 3/4 - (-3/4) = 6/4 = 3/2. Therefore, the focal diameter of the parabola is 3/2.
To sketch the graph of the parabola, we can plot the focus, directrix, and a few other points. Since the equation is y² = 3x, we can find the y-coordinate by taking the square root of x/3. By choosing different values of x and calculating the corresponding y values, we can plot the points and connect them to obtain the graph of the parabola.