91Ó°ÊÓ

Understanding the standard form of an ellipse is essential for graphing it accurately. An ellipse in standard form is given by the equation \( (x - h)^2/a^2 + (y - k)^2/b^2 = 1 \), where \( h \) and \( k \) represent the coordinates of the ellipse's center. In this equation, \( a \) and \( b \) are the distances from the center to the ellipse's vertices along the major and minor axes, respectively. When transformed into standard form, the equation clearly indicates the size and position of the ellipse. For the given equation, \( 4x^2 + 25y^2 = 100 \), we start by dividing each term by 100 to rewrite it as \( x^2/25 + y^2/4 = 1 \), which now perfectly matches the standard form with the center at the origin.

The major and minor axes of an ellipse determine its shape. The length of the major axis (\( 2a \)) is always greater than that of the minor axis (\( 2b\)). They are not just distances but also directions; the major axis runs through the longest part of the ellipse, while the minor axis through the shortest. Identifying which is which is crucial for correct graphing. In our example, after converting to standard form, we identify \( a^2 = 25 \) and \( b^2 = 4 \) which means \( a = 5 \) and \( b = 2 \) giving us the major axis length of 10 and the minor axis length of 4. The major axis is horizontal since \( a^2 > b^2 \).

The center of an ellipse is analogous to the fulcrum of balance on a seesaw, dictating where the ellipse is situated on the coordinate plane. In standard form, the center is given by the coordinates \( (h, k) \). If the equation does not include \( (x - h) \) or \( (y - k) \) components, the center defaults to the origin, \( (0, 0) \). For the given equation \( 4x^2 + 25y^2 = 100\), there are no \( h \) or \( k \) values subtracted or added to \( x \) or \( y \) terms, so we can surmise that the ellipse is centered at the origin.

Algebraic manipulation is the tool we use to mould equations into recognizable forms—like a standard form of an ellipse—that reveal their geometric properties. This often involves dividing or multiplying terms, completing the square, or other algebraic operations. In our case, we divide the terms of \( 4x^2 + 25y^2 = 100 \) by 100, which unveils its standard form and allows us to discern the properties and graph of the ellipse. Doing these manipulations correctly is critical in identifying the axis lengths and the location of the center for accurate graphing.