Cheryl Corbett, Ivy Cook, and Erin Koch5th Period

http://itech.pjc.edu/falzone/handouts/parent_functions.pdf

http://itech.pjc.edu/falzone/handouts/parent_functions.pdf

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

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

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

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

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

Quadratic Formula

• Example: y=x2+5x+2

2 42

b b acx

a

5 ± √17 2

5 ± √17 2

Trig Identities

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

Unit Circle

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

Logarithmic Rules

• Logarithmic Rule 1:

• Logarithmic Rule 2: • Logarithmic Rule 3:

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)

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

• 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

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

Graph a circle with the following formula:

The vertex is (2,-1)

The radius is √9=3

Ellipses

Graph the ellipse:

Parabolas

Graph the following parabola: x2=8y

Focus

Directrix

Hyperbolas

Graph the following:

Center: (0,0)Foci: (2,0), (-2,0)Asymptotes: y=7/2x, y=-7/2x

Foci

Change of BaseFormula

logb(x) = logd(x)

logd(b)

Change of Base Examples

• Example 1: log3(6) = ln(6) = 1.791759 = 1.63093

ln(3) 1.098612

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

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

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

Limit Example-L’hospital’s Rule

lim 4x⁵ - 7x⁴ + ∏x³ + ex² - 1000 = 0x→∞ 6x¹⁷ - 8x¹² + (1/x)

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

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

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