Problem 3 - A Graphical Approach to College Algebra

Question

Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Hyperbola; center \((2,4)\) B. Ellipse; foci \((\pm 2 \sqrt{3}, 0)\) C. Hyperbola; foci \((0, \pm 2 \sqrt{5})\) D. Hyperbola; center \((-2,4)\) E. Ellipse; center \((-2,4)\) F. Center \((0,0) ;\) horizontal transverse axis G. Ellipse; foci \((0, \pm 2 \sqrt{3})\) H. Vertical major axis; center \((2,-4)\) $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

The hyperbola is centered at (0,0) with a horizontal transverse axis.

Step 1: Identify the Conic Section

The given equation is \( \frac{x^{2}}{4} - \frac{y^{2}}{16} = 1 \). This equation has the form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), which is the standard form of a hyperbola. Hyperbolas are characterized by their 'minus' sign between the squared terms.

Step 2: Determine the Center of the Hyperbola

The standard form of a hyperbola \( \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \) has a center at \((h, k)\). Since our equation is \( \frac{x^2}{4} - \frac{y^2}{16} = 1 \), it implies \( h = 0 \) and \( k = 0 \), so the center is \((0,0)\).

Step 3: Identify Orientation of the Transverse Axis

In a hyperbola of the form \( \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \), the transverse axis is horizontal because the \( x \)-term is positive. This means the hyperbola opens horizontally.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that is formed when a plane intersects both nappes of a double cone. It's visually distinct with its two separate curves called "branches." Unlike ellipses and circles, hyperbolas are open-ended and extend infinitely.

What sets a hyperbola apart from other conic sections is its equation form. It has a characteristic 'minus' sign between the squared terms. For instance, the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is the standard equation of a hyperbola.

Some important properties of hyperbolas include:

• Each hyperbola has two transverse points called vertices.
• There are two axes: transverse and conjugate.
• The transverse axis contains the centers and vertices.

When analyzing or graphing a hyperbola, focus on identifying the terms and their relation, as well as observing the minus sign in its equation.

Center of Hyperbola

The center of a hyperbola is a critical point, as it helps in understanding and graphing the hyperbola effectively. This point is essentially the midpoint between the vertices and the foci.

In the standard form equation \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the coordinates \((h, k)\) represent the center of the hyperbola. From this equation, you can see that any shift in the values of \(h\) or \(k\) will move the entire hyperbola along the x-axis or y-axis, respectively.

To locate the center of the hyperbola:

• Identify \((h, k)\) from the equation.
• The center divides the hyperbola symmetrically along the transverse axis.

In our example equation \( \frac{x^2}{4} - \frac{y^2}{16} = 1 \), the center is found at \((0,0)\) because the expressions are not shifted from \(x\) or \(y\) (i.e., there's no \(h\) or \(k\) term).

Horizontal Transverse Axis

The transverse axis of a hyperbola indicates the direction in which its branches open. In hyperbolas, this axis is significant for understanding their orientation.

For a hyperbola given in the form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the transverse axis is horizontal. This means the equation begins with \(x\) and it "opens" along the x-axis. The transverse axis contains important points like the center and the vertices of the hyperbola.

Consider these features when dealing with a horizontal transverse axis:

• The standard horizontal equation ensures the hyperbola opens left and right.
• The distance \(2a\) represents the length of the transverse axis.
• The equation \( \frac{x^2}{4} - \frac{y^2}{16} = 1 \) in our example indicates a horizontal transverse axis since it starts with the \(x^2\) term.

This characteristic lets you predict how the hyperbola will look and help determine its vertices and foci orientation on a graph.