Problem 34 - Algebra A Combined Function

Question

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{x^{2}}{16}-\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The equation represents a horizontally opening hyperbola centered at (0,0), with x-intercepts at (-4,0) and (4,0).

Step 1: Identify the Type of Conic Section

The given equation is \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \). This can be compared to the standard form of a hyperbola, which is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \( a^2 = 16 \) and \( b^2 = 4 \). Since the equation matches this form, it represents a hyperbola that opens horizontally along the \(x\)-axis.

Step 2: Determine Key Features of the Hyperbola

Identify \( a \), \( b \), and the center of the hyperbola. Here, \( a^2 = 16 \) thus \( a = 4 \), and \( b^2 = 4 \) thus \( b = 2 \). The hyperbola's center is at \((h, k) = (0, 0)\) given that there are no \(h\) or \(k\) terms and it's aligned to the origin.

Step 3: Sketch the Hyperbola

Draw a set of axes and mark the center at \((0,0)\). Plot the vertices at \((\pm 4, 0)\), since \(a = 4\). Draw the asymptotes, which have slopes of \( \pm \frac{b}{a} = \pm \frac{2}{4} = \pm \frac{1}{2} \), passing through the center. The transverse axis is on the \(x\)-axis from \(-4\) to \(4\), and sketch the hyperbola opening left and right.

Step 4: Label Key Features

For a hyperbola, the \(x\)-intercepts are the vertices which we found at \((-4, 0)\) and \((4, 0)\). There are no \(y\)-intercepts since the branches do not meet the \(y\)-axis.

Key Concepts

Hyperbola Equations

A hyperbola is one of four types of conic sections, defined by a particular equation form. A hyperbola's equation involves two squared terms, one subtracted from the other. In this form:

• The equation is generally written as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), or its counterpart with axes swapped.
• Here, \(a^2\) and \(b^2\) are positive constants, which help define the hyperbola's shape and orientation.
• The equation means the curve will open horizontally if \(x^2\) comes first and vertically if \(y^2\) comes first.

In the given example, \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \), this hyperbola's standard format verifies it opens along the horizontal direction. Variables \(a\) and \(b\) are crucial as they determine the stretch and direction the hyperbola will take. This form directly guides us during graphing by indicating precise vertex and intercept locations.

Graphing Conic Sections

Graphing conic sections can initially seem challenging, but knowing your conic type simplifies it incredibly. A hyperbola, given by its unique equation type, tells us how to draw its path around a center, which is typically at the origin unless translated. To plot a hyperbola:

• Identify the center: This is where the axes meet. For our hyperbola, it's at \((0,0)\).
• Locate the vertices: These are directly influenced by \(a\). Here, vertices lie on the x-axis at \((-4,0)\) and \((4,0)\) because \(a = 4\).
• Draw the asymptotes: Asymptotes aid in sketching a more accurate hyperbola. The asymptotes here are slanted lines with slopes \(\pm \frac{1}{2}\), cutting through the center.
• Sketch the hyperbola opening to each side of the center, guided by asymptotes.

This structured process breaks down the drawing into clear, manageable steps, making hyperbolas and other conic sections less intimidating to graph.

Identifying Conic Sections

To correctly identify conic sections, examine the equation provided. The equations reveal much about the underlying conic section type. Look for certain characteristics:

• Parabola: Defined by one squared term, e.g., \(y = ax^2 + bx + c\), opening in a single direction.
• Circle: Appears when both terms are squared and equally weighted, e.g., \(x^2 + y^2 = r^2\).
• Ellipse: Similar to a circle but with differing coefficients, e.g., \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), different from a circle by stretch differences along axes.
• Hyperbola: Defined when the squared terms are set apart by subtraction, like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

This step of identification is crucial as it governs the entire approach to graphing. By matching the pattern to one of the core conic forms, you streamline analysis, making sure you apply the right characteristics and plotting steps tailored to each type. Always start with this identification to ensure success in graphing conic sections.