Problem 18 - Algebra 2

Question

Find the foci of each hyperbola. Then draw the graph. $$ 14 y^{2}-28 x^{2}=448 $$

Step-by-Step Solution

Answer

The foci of the hyperbola are located at (0, 6.9) and (0, -6.9).

Step 1: Simplify the Equation

We first simplify the equation by dividing the entire equation by 448 to make it in the standard form, \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\). This gives us \(\frac{y^2}{32} - \frac{x^2}{16} = 1\) where a^2 = 32 and b^2 = 16.

Step 2: Find a and b

We can now find the values for a and b from the above equation. We have that a^2 = 32 and b^2 = 16, thus a = \(\sqrt{32}\) (approximately 5.66) is the semi-major axis length and b = \(\sqrt{16}\) (or 4) is the semi-minor axis length.

Step 3: Find the Foci

Next, we find the foci (h, k±c), where c is the distance from the center to each focus. We use the hyperbola formula \(c=\sqrt{a^2+b^2}\). By substituting the given a and b values, we find that \(c=\sqrt{32 + 16} = \sqrt{48}\), approximately 6.9. Because we know the hyperbola is centered at the origin and opens vertically, the foci occur at (0, ±6.9).

Step 4: Graph the Hyperbola

Finally, draw the hyperbola, including the center at (0,0), foci at (0,±6.9), and vertices that occur at the points (0,±5.66). The hyperbola intersects the y-axis at the vertices, then bends towards the x-axes and never intersects them.

Key Concepts

Foci of a Hyperbola

The foci are an essential part of understanding the structure of a hyperbola. To find them, you need to recognize their position relative to the hyperbola's center. In a hyperbola centered at the origin, the foci lie along the axis the hyperbola opens towards.

If the hyperbola is vertical, as it is in our example, the foci will be above and below the center along the y-axis. If horizontal, they will be to the left and right along the x-axis.

We find the foci's location using the formula \( c = \sqrt{a^2 + b^2} \) where \( c \) is the distance from the center to each focus, \( a \) and \( b \) are derived from the standard equation form. In our case, this calculation gives us foci at coordinates (0, ±6.9), indicating how far they sit from the origin on the y-axis.

Standard Form of Hyperbola Equation

To effectively work with and graph hyperbolas, you must convert their equations into the standard form. The standard form is written as: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), where \( h, k \) represent the center's coordinates, and \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes.

This form makes it easier to analyze the hyperbola's properties, such as its orientation, center, and axis lengths.

Converting an equation like our example begins by ensuring the left side equals 1, achieved by dividing all terms by the constant on the right. After this normalization, the equation clearly shows \( a^2 \) and \( b^2 \), which further helps in calculations involving the foci and graphing process.

Graphing Hyperbolas

Graphing hyperbolas can be simplified by understanding their structure. Hyperbolas have two axes: the transverse and the conjugate axis. The transverse axis slices through the center and vertices, while the conjugate axis does not.

The vertices and the foci lie along the transverse axis, which is why finding these points is crucial.

With our example, the graph centers at the origin (0,0), and the hyperbola opens vertically. You should plot the following:

• Center at (0,0)
• Vertices at (0, ±5.66)
• Foci at (0, ±6.9)

Once these are plotted, sketch the hyperbola, ensuring the lines bend towards the asymptotes which emerge from this equation. Remember, hyperbolas never intersect these guiding lines.