1YHyperbola

= Hyperbola =


 * A hyperbola is one of a group of shapes called conic sections (not in course).


 * A hyperbola has practical applications in Physics and Astronomy (not in course).


 * The Hyperbola is introduced in the VCE Maths Methods 34 course ( here ).

math . \qquad \text{If we consider the function } y=\dfrac{1}{x} math


 * We can see that as the value of x increases,
 * the value of y approaches zero but never reaches zero.


 * In symbols, as x → ¥, y → 0


 * y = 0 is called an asymptote because the y-value approaches but never reaches 0.


 * For the same function,
 * if we consider x-values as they approach 0,
 * the value of y increases to infinity.


 * At x = 0, y is undefined.


 * In symbols, as x + → 0, y → ¥


 * x = 0 is also an asymptote because the x-value approaches but never reaches 0.


 * Sketch Graph of the Standard Hyperbola **


 * If we plot the line y = x (in blue)
 * then consider the reciprocal of each of the points,
 * we get the standard hyperbola .(in red).

math . \qquad y = \dfrac{1}{x} math

.
 * The x-axis forms a horizontal asymptote
 * A vertical asymptote occurs where y = 0
 * The graphs intercept at y = 1 (and –1)


 * When y(blue) |y| < 1, y(red) becomes large |y| > 1
 * When y(blue) |y| > 1, y(red) becomes small |y| < 1


 * When y(blue) is negative, y(red) is negative


 * The asymptotes are along the axis,
 * so the equations of the asymptotes are
 * x = 0 and
 * y = 0


 * Domain: x Î ** R ** \{0}
 * Range: y Î ** R ** \{0}


 * There are no stationary points and no intercepts.


 * ** This is the Standard Hyperbola. **

Example

math \\ . \qquad \text{Sketch } y=x+2 \text{ and take the reciprocal} \\.\\ . \qquad \text{to sketch the hyperbola } y=\dfrac{1}{x+2} math


 * Solution: **


 * The line y = x + 2 is in blue


 * The hyperbola is in red


 * The x-axis forms a horizontal asymptote.
 * A vertical asymptote occurs where y = 0.


 * The graphs intercept at y = 1 (and –1)


 * When y(blue) |y| < 1, y(red) becomes large |y| > 1
 * When y(blue) |y| > 1, y(red) becomes small |y| < 1


 * When y(blue) is negative, y(red) is negative


 * The equations of the asymptotes are:
 * x = –2 and
 * y = 0


 * Domain: x Î ** R \ { ** –2}
 * Range: y Î ** R \ { ** 0}


 * There are no stationary points.


 * There is a y-intercept at y = ½


 * There are no x-intercepts.


 * ** This is the __standard hyperbola__ translated 2 units to the left. **

Return to Summary of Coordinate Geometry .