Problem 70 - College Algebra

Question

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Step-by-Step Solution

Answer

The ellipse graph will be centered at (0,0) with a semi-major axis length of 5 units along the x-axis and a semi-minor axis length of 4 units along the y-axis.

Step 1: Identify the Center

For the ellipse with the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), the center will always be at the origin (0,0) because there are no terms that subtract or add to \(x\) and \(y\). So, mark the point (0,0) on your graph as the center of the ellipse.

Step 2: Draw the Axes of the Ellipse

The length of the semi-major axis is given by the constant under \(x^{2}\) and the length of the semi-minor axis is given by the constant under \(y^{2}\). Draw a horizontal line passing through the center (0, 0) of the ellipse spanning from -5 to 5 along the x-axis and a vertical line also passing through the center of the ellipse spanning from -4 to 4 along the y-axis, using these axes lengths.

Step 3: Draw the Ellipse

Knowing the center of the ellipse and axes, you can now draw the ellipse. Keep your curve within the boundaries defined by the semi-major and semi-minor axes and make sure it encompasses the entire x and y span from -5 to 5 and -4 to 4 respectively.