Problem 27 - Algebra 2

Question

Describe how the translation of a hyperbola affects the equations of its asymptotes.

Step-by-Step Solution

Answer

Translation of a hyperbola affects the equations of its asymptotes by shifting the lines in the xy-plane. The equations change from \(y = \pm \frac{b}{a}x\) to \(y = k \pm \frac{b}{a}(x - h)\) depending on whether hyperbola is shifting vertically or horizontally.

Step 1: Define the Standard Equation of a Hyperbola

The standard equation of a hyperbola centered at the origin (0,0) opening left-right can be given as \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\); and for a hyperbola opening up-down it is \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\) where 'a' and 'b' are distances from the center to vertices and asymptotes respectively.

Step 2: Understanding Asymptotes of Hyperbola

The equations of the asymptotes of a hyperbola centered at the origin are \(y = \frac{b}{a}x\) and \(y = - \frac{b}{a}x\). For a hyperbola that opens left-right the lines pass through the origin, and for a hyperbola that opens upwards and downwards the lines also cross the origin, but with a different slope.

Step 3: Translation of the Hyperbola

When translating a hyperbola centered at the origin to a new location at (h,k), the standard equations of the hyperbola become \(\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1\) and \(\frac{(y-k)^{2}}{a^{2}} - \frac{(x-h)^{2}}{b^{2}} = 1\).

Step 4: Effect on Asymptote Equations

With the translation of the hyperbola, the equations of the asymptotes also change. Given the translation, the asymptotes of the hyperbola become \(y = k + \frac{b}{a}(x - h)\) and \(y = k - \frac{b}{a}(x - h)\). This describes how the equations of the asymptotes are affected by the translation of the hyperbola in the xy-plane.

Key Concepts

Standard Equation of a Hyperbola

A hyperbola is a type of conic section that can open either horizontally (left-right) or vertically (up-down). Understanding its standard equation is crucial. For a hyperbola centered at the origin, the equations are different based on the direction it opens:

• For a hyperbola that opens left-right, the equation is given by: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]where \(a\) is the distance from the center to the vertices horizontally, and \(b\) is the distance from the center to the points where the asymptotes intersect the hyperbola.
• For a hyperbola that opens up-down, the equation is:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]Here, \(a\) is vertically oriented from the center to the vertices, while \(b\) relates similarly to the asymptotes' interaction.

Knowing which form to use depends on how the hyperbola is oriented, which helps in predicting its behavior and deducing its properties like the vertices and asymptotes.

Asymptotes of Hyperbola

Asymptotes are lines that a hyperbola approaches but never actually meets. These help define the hyperbola's shape. For a hyperbola centered at the origin, the asymptotes are represented by the following linear equations:

• For a hyperbola that opens left-right, the asymptote equations are:\[ y = \frac{b}{a}x \quad \text{and} \quad y = -\frac{b}{a}x\]These equations indicate that the asymptotes pass through the origin with slopes determined by \( \pm \frac{b}{a} \).
• For a hyperbola that opens up-down, the asymptotes still have the same slope but are flipped vertically, remaining as:\[ y = \frac{b}{a}x \quad \text{and} \quad y = -\frac{b}{a}x\]

Understanding these asymptote equations is essential for sketching the hyperbola and predicting its direction and steepness based on \(a\) and \(b\). They represent a boundary that the hyperbola gets infinitely closer to as it extends outward.

Effect of Translation on Hyperbola

When a hyperbola is translated, it shifts from being centered at the origin to a new center at \((h, k)\). This translation affects the hyperbola's equation and its asymptotes. The translated standard equations of the hyperbola are:

• If the hyperbola opens left-right, the equation becomes:\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
• If it opens up-down, it becomes:\[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]

When the hyperbola shifts to a new center, its asymptote equations also adjust accordingly:

• The equations change to reflect the translation by becoming:\[ y = k + \frac{b}{a}(x - h) \quad \text{and} \quad y = k - \frac{b}{a}(x - h)\]
• This shows how the asymptotes have shifted vertically by \(k\) units and horizontally by \(h\) units.

Overall, translating the hyperbola affects its central position, moving the entire graph, including the asymptotes, all while keeping the hyperbola’s overall shape consistent.