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Cheryl Corbett, Ivy Cook, and Erin Koch5th Period
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http://itech.pjc.edu/falzone/handouts/parent_functions.pdf
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http://itech.pjc.edu/falzone/handouts/parent_functions.pdf
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Asymptotes
• Definition: A line such that the distance between the curve and the line approaches zero as they tend to infinity.
• Types: Horizontal, Vertical, and Oblique
www.mathnstuff.com/.../300/fx/library/lines.htm www.sparknotes.com/.../section2.rhtml
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Horizontal Asymptotes
1. higher exponent on top—no horizontal asymptote
2. higher exponent on bottom—y=0
3. Exponents same—ratio of coefficients
• webgraphing.com/algebraictricksoftrade.jsp
Example: y= x/(x2+1)1/2
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Vertical Asymptotes • Definition: a vertical line (perpendicular the to
x-axis) near which the function rose without bound.
• How to find it: The zeros of the function are the vertical asymptotes.
www.sparknotes.com/.../section2.rhtml
Example: f(x)= 1 / (x-5)
X-5=0
X=5
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Oblique Asymptotes • Definition: are diagonal lines so that the difference
between the curve and the line approaches zero as x tends positive or negative infinity.
• How to find it: If the power of the numerator is bigger than the denominator then you must do long division to find the asymptote.
www.mathwords.com/a/asymptote.htm
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Symmetry • y-axis f(-x)=f(x)• x-axis f(-x) = - f(x) • origin f(-x,-y)=f(x,y); origin symmetry contains
x and y symmetry
Example: f(x)=x2
Solution:f(-x)= (-x)2 or f(-x)=x2This problem has y-axis symmetry
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Quadratic Formula
• Example: y=x2+5x+2
2 42
b b acx
a
5 ± √17 2
5 ± √17 2
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Trig Identities
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Example: cotx*secx*sinx=1
Step 1: cosx * 1 * sinx = 1sinx cosx 1
Step 2: cosx * 1 * sinx = 1sinx cosx 1
Step 3: 1 * sinx = 1sinx 1
Step 4: 1 * sinx = 1sinx 1
Answer: 1 = 11
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Unit Circle
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Example of Unit Circle Problems
1. sin(∏/3)=2. cos(5∏/4)=3. cos(∏/2)=4. sin(11∏/6)=5. sin(7∏/4)=6. cos(∏/6)=7. sin(∏)=
1. √(3)/22. -√(2)/23. 04. -1/25. -√(2)/26. √(3)/27. -1
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Logarithmic Rules
• Logarithmic Rule 1:
• Logarithmic Rule 2: • Logarithmic Rule 3:
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Expand the following:
log2(8x)
Simplify the following: log4(3) - log4(x)
Solve the following:
log3(x)5
log2(8x) = log2(8) + log2(x)
log4(3) – log4(x) = log4(3/x)
log3(x)5 = 5log3 (x)
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Conic SectionsConic Sections are defined as the intersection of a plane and a cone (and, at a simple level, we just consider the four shapes obtained when the plane does not pass thru the vertex of the cone).
http://www.mathwords.com/c/conic_sections.htm
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• Types of conic sections:– Circle– Parabola– Hyperbola– Ellipse
Standard Equations for each conic section:
– Circle: x2 + y2 = r2
– Parabola: y2 = 4ax– Hyperbola: x2 - y2 = 1
a2 b2
– Ellipse: x2 + y2 = 1 a2 b2
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Circles Standard form: (x – h)2 + (y – k)2 = r2 Center: (h, k)Radius: r
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_circle.xml
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Graph a circle with the following formula:
The vertex is (2,-1)
The radius is √9=3
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Ellipses
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Graph the ellipse:
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Parabolas
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Graph the following parabola: x2=8y
Focus
Directrix
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Hyperbolas
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Graph the following:
Center: (0,0)Foci: (2,0), (-2,0)Asymptotes: y=7/2x, y=-7/2x
Foci
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Change of BaseFormula
logb(x) = logd(x)
logd(b)
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Change of Base Examples
• Example 1: log3(6) = ln(6) = 1.791759 = 1.63093
ln(3) 1.098612
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Limits• What is a limit?
– Holes, jump discontinuity, removable discontinuity
• Ways to find a limit…– Plug in h value and see if you get a y value– L’Hospital’s Rule
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Limit Example –plugging in method
• Lim x²-4 = Lim (x-2)(x+2) =x→2 x+2 x→2 (x-2)
Lim (x+2) = Lim (2+2)=4 x→2 x→2
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L’Hospitals Rule
• Part one: – If Lim f(x) = 0 , then lim f¹(x) =
x→a g(x) 0 x→a g¹(x)– Can be used as many times as needed until you get an
answer
• Part two: – lim 1 = 0 x→∞ x– Find the highest power and divide every term by that
number or use rules for finding asymptotes
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Limit Example-L’hospital’s Rule
lim 4x⁵ - 7x⁴ + ∏x³ + ex² - 1000 = 0x→∞ 6x¹⁷ - 8x¹² + (1/x)
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1975 AB 1
Given the function f defined by f(x) = ln(x²-9).a. Describe the Symmetry of the graph of f.b. Find the domain of f.c. Find all values of x such that f(x) = 0d. Write a formula for f¯¹(x), the inverse
function of f, for x>3
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FRQa. f(-x) = ln((-x)²-9)
f(-x) = ln(x²-9)– This function has y-axis
symmetry.b. x² -9 ≥ 0
x² ≥ 9 IxI ≥ 3
c. 0 = ln(x²-9)1 = x²-910 = x²±√10 = x
d. x = ln(x²-9)ex = y²-9ex-9 = y²±√(ex-9) = y f¯¹(x) = √(ex-9)
ANSWER
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Citations
• http://itech.pjc.edu/falzone/handouts/parent_functions.pdf
• http://www.mathwords.com/c/conic_sections.htm
• ©Cheryl Corbett, Erin Koch, and Ivy Cook. 02/18/10