91Ó°ÊÓ

The standard form of an ellipse equation is critical because it allows us to graph the ellipse accurately. The general form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center of the ellipse. A standard form provides a clear way to identify the lengths of the axes and the center without additional calculations.
The given equation \(9x^2 + 4y^2 = 36\) must be rewritten to match this standard form. We achieve this by dividing all terms by 36, simplifying it to \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). Now, the equation resembles the standard ellipse form, where the ellipse is centered at the origin (0,0).
Recognizing the standard form enables easier identification of the semi-major and semi-minor axes, crucial for graph plotting. Thus, always aim to convert the equation of the ellipse to its standard form to simplify the visualization process.

Understanding the semi-major axis in ellipses is essential as it determines the ellipse's longest radius and primarily influences its vertical or horizontal stretch. In our standard form \( \frac{x^2}{4} + \frac{y^2}{9} = 1\), the terms beneath the denominators indicate the square of the axes' lengths.
The semi-major axis is associated with the larger value beneath the \(x\) or \(y\) terms — in this case, it is y with a bigger denominator (9). Therefore, the semi-major axis lies along the y-axis. To find its length, we take the square root of 9, resulting in 3. This tells us the ellipse is longer vertically.
Graphically, once you find the semi-major axis, plot additional points equal to its length from the center along the specified axis — here, the points are (0, 3) and (0, -3). Always starting at the center allows for an accurate representation of the ellipse's expansive nature.

The semi-minor axis characterizes the ellipse's shorter radius, indicating how wide it appears horizontally or vertically. Referring again to the standard form \( \frac{x^2}{4} + \frac{y^2}{9} = 1\), the term beneath the smaller denominator identifies the semi-minor axis.
In this problem, the smaller denominator is 4 underneath the \(x\)-term. This signifies that the semi-minor axis is parallel to the x-axis. Calculate the semi-minor axis by taking the square root of 4, revealing a length of 2.
With the semi-minor axis known, plot points equal to this length from the center along the specified axis — here (2, 0) and (-2, 0).
The knowledge of both axes' lengths helps create the ellipse's shape, ensuring the graph reflects the true proportions as dictated by its equation.