Problem 45 - 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. \(9 x^{2}-4 y^{2}=36\)

Step-by-Step Solution

Answer

The equation is a hyperbola with center at (0,0); x-intercepts are (2,0) and (-2,0).

Step 1: Identify the Type of Conic Section

The given equation is \(9x^2 - 4y^2 = 36\). Notice that both \(x^2\) and \(y^2\) terms are present and they have opposite signs, which indicates that this equation represents a hyperbola.

Step 2: Rewrite the Equation in Standard Form

Divide the entire equation by 36 to get it in standard form: \[ \frac{x^2}{4} - \frac{y^2}{9} = 1 \]This is the standard form of a hyperbola centered at the origin \((0,0)\) with the transverse axis along the x-axis.

Step 3: Identify Key Features of the Hyperbola

The equation \(\frac{x^2}{4} - \frac{y^2}{9} = 1\) tells us the hyperbola is centered at \((0,0)\) with semi-major axis squared (\(a^2 = 4\)) and semi-minor axis squared (\(b^2 = 9\)). The vertices are located at \((\pm a, 0)\) or \((\pm2, 0)\).

Step 4: Sketch the Hyperbola

Draw the coordinate axes. Mark the center at the origin. Plot the vertices at \((2,0)\) and \((-2,0)\). The hyperbola opens along the x-axis, and its asymptotes can be drawn by connecting \((0,\pm3)\) with dashed lines, which give the lines \(y=\pm\frac{3}{2}x\).

Key Concepts

Hyperbola

A hyperbola is a type of conic section that is formed by the intersection of a double cone with a plane in such a way that the angle between the plane and the cone's axis is less than that made by the plane with the generator of the cone. Graphically, a hyperbola consists of two separate curves, known as branches, each resembling a mirrored parabola. This unique shape occurs due to the subtraction of squared terms in its equation. Hyperbolas are often associated with concepts of reflection, as they can explain certain types of reflective properties.

• The transverse axis connects the vertices and is the primary center of the hyperbola.
• The center is a crucial point that acts as the midpoint between the two foci.
• Hyperbolas have two asymptotes that act as guide lines for its shape, though they never intersect the actual curve.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation can be critical to understanding its geometric properties. For a hyperbola centered at the origin, the equation can be expressed as:

• Horizontal orientation: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Here, the transverse axis is along the x-axis.
• Vertical orientation: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] This implies the transverse axis is along the y-axis.

The denominators in these equations, \(a^2\) and \(b^2\), correspond to the semi-axis lengths of the hyperbola.

• The center of the hyperbola for these forms is typically at the origin, \(0,0\).
• These equations allow for straightforward identification of the vertices, asymptotes, and the slopes.

Vertices of Hyperbola

The vertices of a hyperbola are crucial points where these curves make their closest or widest departure from the center. In hyperbola equations, vertices can be derived from the terms associated with \(a\) and \(b\).
For a hyperbola defined by \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \], the vertices are given by \( (\pm a, 0) \).

• In the example, since \(a^2 = 4\), we find \(a = 2\), indicating vertices at \((2,0)\) and \((-2,0)\).
• These represent the turning points of the hyperbola along its transverse axis.
• Understanding the vertices helps in sketching the graph accurately.

Asymptotes of Hyperbola

Asymptotes are lines that hyperbolas approach but never actually reach. They play a critical role in defining the overall shape and orientation of hyperbolas.
For hyperbolas represented by \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \], the asymptotes can be derived as linear equations:

• The formula to find asymptotes is \[ y = \pm \frac{b}{a} x \].
• For the given hyperbola, this translates to \[ y = \pm \frac{3}{2} x \].
• These lines guide the slope and direction of the hyperbola's branches.

Asymptotes help visualize how the hyperbola extends to infinity in both directions, hugging these lines increasingly closely as it moves away from the center.