Problem 42 - 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. $$ y^{2}=x^{2}+16 $$

Step-by-Step Solution

Answer

The equation represents a hyperbola centered at (0, 0) with vertices at (0, 4) and (0, -4).

Step 1: Identify the General Form of the Equation

The given equation is \(y^2 = x^2 + 16\). To identify the type of conic section, we compare it with standard conic forms. Here, notice that both \(y^2\) and \(x^2\) are present. However, the terms are not subtracted as in the ellipse equation, suggesting it might represent a hyperbola.

Step 2: Rearrange to Match Standard Hyperbola Form

A standard form of a hyperbola with a vertical transverse axis is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Rearrange \(y^2 = x^2 + 16\) into \( \frac{y^2}{16} - \frac{x^2}{16} = 1\), verifying this aligns with the standard hyperbola form, as both squared terms are set equal to 1.

Step 3: Identify and Label Key Features of the Hyperbola

The center of the hyperbola is at (0, 0) (since the equation has no \(x\) or \(y\) linear term). For \( \frac{y^2}{16} - \frac{x^2}{16} = 1\), we have \(a^2 = 16\), so \(a = 4\). The vertices are located at (0, 4) and (0, -4). The \(x\)-intercepts are imaginary, as there are no real solutions for \(y = 0\).

Step 4: Sketch the Graph of the Hyperbola

Center the graph on the origin. Plot vertices at (0, 4) and (0, -4), and draw the asymptotes, which will be diagonal lines passing through the origin since \(b^2 = a^2 = 16\). Sketch the hyperbola opening up and down along the y-axis.

Key Concepts

Parabola

A parabola is a conic section that appears as a U-shaped curve on the graph. It has many unique properties that distinguish it from other conic sections. The general equation of a parabola is of the form \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\). Here are a few key features of parabolas:

• Vertex: This is the highest or lowest point on the parabola, depending on its orientation. It can be easily identified and labeled on the graph.
• Axis of Symmetry: A line that runs vertically or horizontally through the vertex, dividing the parabola into two mirror-image halves.
• Focus and Directrix: Important for understanding the parabola's precise shape. The vertex is equidistant from the focus and directrix.

Let's consider the equation \(y^2 = 4px\). This implies a parabola opening either to the right or left, with its vertex at the origin. Remember, identifying a parabola from an equation involves recognizing that only one of the variables (either \(x\) or \(y\)) is squared.

Hyperbola

A hyperbola consists of two separate curves called branches. Its general form can appear as either \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). For hyperbolas:

• Center: This point is equidistant from the vertices and defines the midpoint of the hyperbola's transverse axis.
• Vertices: These are the closest points on each branch to the center.
• Asymptotes: Diagonal lines that guide the opening of the hyperbola, approaching the branches but never meeting them.
• Intersections: When hyperbolas have real solutions for either \(x\) or \(y\), these are important graph features.

In the exercise given, the equation \(\frac{y^2}{16} - \frac{x^2}{16} = 1\) represents a hyperbola centered at (0, 0) with vertical orientation. Since both \(a\) and \(b\) are equal, the asymptotes cross the origin at 45-degree angles.

Ellipse

Ellipses are like stretched or compressed circles. Their form is similar to a circle but with unequal radii. The standard ellipses equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the semi-major and semi-minor axes.

• Center: At the midpoint between its foci.
• Axes: It has a major and a minor axis. The major axis is longer than the minor.
• Foci (plural of Focus): These two fixed points define the stretch of the ellipse.
• Eccentricity: A measure of how much the ellipse is out of round, computed based on the distance between the foci and the center.

An example of an ellipse equation might be \(\frac{x^2}{36} + \frac{y^2}{16} = 1\). Here, the ellipse is centered at the origin with a major axis along the x-axis. Identifying an ellipse requires recognizing the plus sign between the squared terms and different denominators.

Circle

The simplest form of a conic section is a circle. A circle has a constant radius and is equidistant from a central point. The formula for a circle in its standard form is \((x-h)^2 + (y-k)^2 = r^2\), with \((h, k)\) being the center and \(r\) the radius.

• Center: The point \((h, k)\) from which all points on the circle are equidistant.
• Radius: The constant distance from the center to any point on the circle's edge.
• Equation: An easy way to recognize a circle is the absence of coefficients multiplying the squared terms, like \(x^2 + y^2 = r^2\).
• Symmetry: Circles have infinite lines of symmetry, all passing through the center.

When analyzing equations like \((x-3)^2 + (y+2)^2 = 25\), you can identify the circle's center at (3, -2) and a radius of 5.