Problem 42 - Intermediate Algebra

Question

Identify whether equation, when graphed, will be a parabola, circle, ellipse, or hyperbola. Sketch the graph of 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. \(x^{2}+y^{2}=16\)

Step-by-Step Solution

Answer

The graph is a circle with center (0, 0) and radius 4.

Step 1: Identify the Equation Type

The given equation is \(x^2 + y^2 = 16\). This equation takes the standard form \(x^2 + y^2 = r^2\), which is that of a circle. The equation of a circle is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Center and Radius

Based on the standard form, the equation \(x^2 + y^2 = 16\) implies \((x - 0)^2 + (y - 0)^2 = 4^2\). Hence, the center of the circle is \((0, 0)\) and the radius \(r\) is 4.

Step 3: Sketch the Graph

Draw a coordinate plane. Plot the center of the circle at \((0, 0)\) and use the radius to draw a circle that extends 4 units in all directions from the center. This gives the full circle.

Key Concepts

Parabola

A parabola is a U-shaped curve and is one of the most familiar conic sections. It can either open upwards or downwards if it's a vertical parabola, or to the left or right if it's horizontal. The standard form of a vertical parabola is given by \[ y = ax^2 + bx + c \] Whereas, the standard form of a horizontal parabola is \[ x = ay^2 + by + c \] Key features include:

• Vertex: The highest or lowest point of the parabola.
• Axis of symmetry: A vertical or horizontal line passing through the vertex, splitting the parabola into two mirror images.
• Focus and directrix: Each parabola is defined by a point called the focus and a line called the directrix, equidistant from each vertex point.

To identify if an equation represents a parabola, look for a squared term, and if only one variable is squared, likely indicating a parabola.

Circle

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The standard equation of a circle is written as: \[ (x - h)^2 + (y - k)^2 = r^2 \] Here,

• \((h, k)\) is the center of the circle. In the equation provided, \(x^2 + y^2 = 16\), the center is \((0, 0)\).
• \(r\) represents the radius. For \(x^2 + y^2 = 16\), the radius is 4, since \(r^2 = 16\), thus \(r = 4\).

When sketching the graph of a circle:

• Start by plotting the center on the coordinate plane.
• From the center, measure and draw points 4 units away in all directions to form the circle.

Understanding the circle equation is helpful for recognizing the symmetry and structure of this conic section.

Ellipse

An ellipse looks like a stretched circle, encompassing two focal points. The sum of the distances from these points to any point on the ellipse is constant. Its standard form is: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] Where,

• \((h, k)\) is the center of the ellipse.
• \(a\) and \(b\) are the distances from the center to the vertices along the x- and y-axes, respectively.

If \(a > b\), the ellipse stretches horizontally, and if \(b > a\), it stretches vertically. Important features:

• Axes: The major axis (longest diameter) and minor axis (shortest diameter).
• Foci: Two fixed points inside the ellipse.

Identifying ellipses in equations involves recognizing the two squared terms with similar signs but different coefficients beneath them.

Hyperbola

A hyperbola consists of two separate curves called branches, which mirror each other. Its standard form can either open left and right or up and down: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \] or \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \] Key components of hyperbolas include:

• Center: Identified at \((h, k)\).
• Vertices: Points where the branches cross the axes.
• Asymptotes: Lines that define the direction of the branches and intersect at the center.

To distinguish hyperbolas from other conics, note the minus sign between squared terms in its equation. This feature allows them to form two distinct curves that expand towards the asymptotes.