Problem 15 - Beginning and Intermediate Algebra

Question

Identify the center of each ellipse and graph the equation. $$x^{2}+\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

The center of the ellipse is at (0, 0), with a semi-major axis of 1 and a semi-minor axis of 2. The graph will have an ellipse centered at the origin, with a width of 2 units and a height of 4 units.

Step 1: Identify the center

The given equation is: \(x^{2}+\frac{y^{2}}{4}=1\) Comparing this to the general equation for an ellipse, we can see that the ellipse is centered at (h, k) = (0, 0), since there is no (x - h) or (y - k) term.

Step 2: Identify the semi-major and semi-minor axes

Now we need to find the semi-major axis (a) and the semi-minor axis (b). We can do this by identifying the coefficients in the given equation: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) Here, the coefficient of \(x^{2}\) is 1, meaning that \(a^{2} = 1\) and therefore \(a = 1\). The coefficient of \(y^{2}\) is \(\frac{1}{4}\), meaning that \(b^{2} = 4\), and therefore \(b = 2\).

Step 3: Sketch the ellipse

Now that we have the center (0, 0) and the lengths of the semi-major axis (a = 1) and semi-minor axis (b = 2), we can sketch the ellipse. To do this, we start at the center and draw the ellipse such that it extends 1 unit to the left and right (along the x-axis) and 2 units up and down (along the y-axis). The result should be an ellipse with the equation \(x^{2}+\frac{y^{2}}{4}=1\) centered at (0, 0) and with a width of 2 units and a height of 4 units.

Key Concepts

Center of an Ellipse

In mathematics, the center of an ellipse holds a special significance as it serves as the point around which the entire shape is symmetrical. To determine the center of an ellipse from its equation, you look for the

• (x - h)
• (y - k)

terms in its standard form equation \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]This general equation suggests that the center has coordinates (h, k). When these terms are not explicitly present, as in the equation \[x^2 + \frac{y^2}{4} = 1\],it implies that both h and k are zero.Thus, the center of this ellipse is at the origin, point (0, 0). This foundational concept is straightforward but vital for the accurate positioning of an ellipse on the coordinate plane. Knowing the center allows you to sketch the ellipse with symmetry in mind.

Semi-Major Axis

The semi-major axis of an ellipse is one of its most defining features. It represents the longest radius of the ellipse, stretching from the center to the perimeter. To find the semi-major axis in a given ellipse equation, analyze the denominators beneath each squared term in the standard form. In our exercise, this means evaluating the coefficients:\[\frac{x^2}{1} + \frac{y^2}{4} = 1\]Here, we identify that the larger denominator corresponds to the axis along the y-direction: \(b^2 = 4\), which gives us \(b = 2\).Thus, the semi-major axis length is 2 units. It extends vertically from the center (0, 0) to the points (0, 2) and (0, -2). Understanding the semi-major axis is crucial since it sets the primary stretch of the ellipse, dictating its length.

Semi-Minor Axis

The semi-minor axis of an ellipse complements its semi-major counterpart as the shorter radius extending from the center. It is perpendicular to the semi-major axis, providing the specific width of the ellipse. In the standard ellipse equation, the semi-minor axis relates to the smaller denominator:\[\frac{x^2}{1} + \frac{y^2}{4} = 1\]Here, examining the term, \(\frac{x^2}{1}\), the semi-minor axis, \(a\), is \(a = 1\) because \(a^2 = 1\).Thus, the ellipse stretches horizontally 1 unit to each side of the center, specifically to the points (1, 0) and (-1, 0). Appreciating the semi-minor axis reveals an ellipse's compactness, forming the opposite side of its symmetry.

Graphing Ellipses

Graphing ellipses can be a straightforward task once you understand its key components: the center, the semi-major axis, and the semi-minor axis. Combining all these elements allows you to visualize and sketch the ellipse accurately.To begin, plot the center of the ellipse, which for our example, is at (0, 0). From there, use the lengths of the semi-major axis and the semi-minor axis:

• Extend 2 units up and down from the center for the semi-major axis along the y-axis.
• Extend 1 unit left and right for the semi-minor axis along the x-axis.

This method results in key points that mark the shape's extremities.Lastly, smoothly connect these points with an oval curve, ensuring symmetry around the center point. This drawing method visually represents the equation \[x^2 + \frac{y^2}{4} = 1\].Graphing is a skill that reveals the elegance of mathematical relationships in visual form, often making complex concepts clearer and more tangible.