Hyperbolas and Circles - navimath · 12/12/2013 · Hyperbola Characteristics The characteristics...
Transcript of Hyperbolas and Circles - navimath · 12/12/2013 · Hyperbola Characteristics The characteristics...
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Hyperbolas and Circles
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Learning Targets
To recognize and describe the characteristics of a hyperbola and circle.
To relate the transformations, reflections and translations of a hyperbola and circle to an equation or graph
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Hyperbola
A hyperbola is also known as a rational function and is expressed as
Parent function and Graph: 𝑓 𝑥 =1
𝑥
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Hyperbola Characteristics
The characteristics of a hyperbola are: • Has no vertical or
horizontal symmetry • There are both horizontal
and vertical asymptotes • The domain and range is
limited
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Locator Point
The locator point for this function is where the horizontal and vertical asymptotes intersect. Therefore we use the origin, (0,0).
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Standard Form
𝑓 𝑥 = −𝑎1
𝑥 − ℎ+ 𝑘
Reflects over x-axis when negative
Vertical Stretch or Compress Stretch: 𝑎 > 1
Compress: 0 < 𝑎 < 1
Horizontal Translation (opposite direction)
Vertical Translation
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Impacts of h and k
Based on the graph at the right what inputs/outputs can our function never produce? This point is known as the hyperbolas ‘hole’
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Impacts of h and k
The coordinates of this hole are actually the values we cannot have in our domain and range. Domain: all real numbers for 𝑥 ≠ ℎ Range: all real numbers for 𝑦 ≠ 𝑘
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Impacts of h and k
This also means that our asymptotes can be identified as: Vertical Asymptote: x=h Horizontal Asymptote: y=k
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Example #1
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What is the equation for this graph?
𝑓 𝑥 =1
𝑥 − 3− 2
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Example #2
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(-3,2)
You try:
𝑓 𝑥 =1
𝑥 + 4+ 1
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Impacts of a
Our stretch/compression factor will once again change the shape of our function. The multiple of the factor will will determine how close our graph is to the ‘hole’ The larger the a value, the further away our graph will be. The smaller the a value , the closer our graph will be.
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Example #3
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What is the equation for this function:
𝑓 𝑥 = 31
𝑥+2
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Circle
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The equation of a circle
What characterizes every point (x, y) on the circumference of a circle?
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Every point (x, y) is the same distance r from the center. Therefore, according to the Pythagorean distance formula for the distance of a point from the origin.
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Where r is the radius. The center of the circle, (0,0) is its
Locator Point.
𝑥2 + 𝑦2 = 𝑟2
Parent Function
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Examples
1) x² + y² = 64
2) (x-3)² + y² = 49
3) x² + (y+4)² = 25
4) (x+2)² + (y-6)² = 16
State the coordinates of the center and the measure of radius for each.
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x² + (y-3)² = 4²
Now let’s find the equation given the graph:
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(x-3)² + (y-1)² = 25
Now let’s find the equation given the graph:
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Homework
Worksheet #6 GET IT DONE NOW!!! ENJOY YOUR BREAK!!!