Problem 10 - Intermediate Algebra

Question

Sketch the graph of each equation. \(x^{2}+\frac{y^{2}}{4}=1\)

Step-by-Step Solution

Answer

The graph is an ellipse centered at the origin with vertical major axis and vertices \((0, \pm 2)\) and \((\pm 1, 0)\).

Step 1: Identify the type of conic

The given equation is \(x^{2} + \frac{y^{2}}{4} = 1\). This can be recognized as the equation of an ellipse in standard form, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2 = 1\) and \(b^2 = 4\).

Step 2: Determine the major and minor axes

In this equation, \(a = 1\) and \(b = 2\). Because \(b > a\), the major axis is along the y-axis, making this a vertical ellipse.

Step 3: Find the center of the ellipse

The standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is centered at the origin \((0,0)\) because there are no terms with \(x\) or \(y\) translations (i.e., no added constant terms in \(x\) and \(y\)).

Step 4: Calculate the vertices

For an ellipse, vertices are located \(\pm a\) units from the center along the minor axis and \(\pm b\) units from the center along the major axis. The vertices along the y-axis are \((0, \pm 2)\) and along the x-axis are \((\pm 1, 0)\).

Step 5: Plot the key points and sketch the graph

Using the vertices found, plot the points \((0, 2), (0, -2), (1, 0), (-1, 0)\) on a coordinate plane. Draw a smooth, elliptical curve through these points to sketch the ellipse.

Key Concepts

Conic Sections

In geometry, conic sections are the curves obtained by intersecting a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas. The type of curve depends on the angle of the plane relative to the cone. When the intersection creates a closed curve, like an oval, it’s an ellipse. Conic sections are significant because they naturally appear in various fields like astronomy and physics. For example, the orbits of planets are elliptical. In mathematics, understanding conic sections helps in analyzing different real-world phenomena. They are defined by specific quadratic equations, which describe their shape and position on a plane.

Ellipse Equation

The equation of an ellipse in its standard form is given by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]This form is essential for graphing ellipses. The values of \(a\) and \(b\) determine its size and shape. These variables define the lengths of the major and minor axes, which are the longest and shortest diameters, respectively.

• If \(a > b\), the ellipse is wider (horizontal), and its major axis is parallel to the x-axis.
• If \(b > a\), like in our original exercise, the ellipse is taller (vertical), and its major axis aligns with the y-axis.

The center of an ellipse in this form is located at the origin \(0, 0\), unless the equation is altered by adding or subtracting constants to the x or y terms. Understanding this equation enables us to deduce key features like the vertices, co-vertices, and ellipse orientation easily.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to investigate geometric shapes. This method allows us to use algebraic equations to represent geometric figures like lines, circles, and ellipses. With coordinate geometry, it is easier to find important attributes of a shape, such as vertices, axes, and distances between points.
In solving the original exercise with the equation \(x^{2} + \frac{y^{2}}{4} = 1\), coordinate geometry aids in accurately plotting the graph. The center is at \(0, 0\), which forms the basis for placing other points. The vertices found by this method—along the y-axis at \(0, \pm 2\) and x-axis at \(\pm 1, 0\)—are critical in sketching the ellipse.

• It simplifies visualizations by converting complex algebraic equations into lines and curves on a plane.
• Enables detailed predictions about the behavior and properties of geometrical figures based on changes in their algebraic equations.

This aspect of geometry is a powerful tool in both pure and applied mathematics.