Cochino’s Math

By Jocelyn Hernandez, Leslie Zamudio, Horacio Sevilla & Yesenia

Gonzalez

Functions and Their Graphs

• Properties of Lines• Slope m=y2-y1/x2-x1,

• Point slope y-y1=m(x-x1), Slope Intercept y=mx+b

• Vertical X=Y Horizontal Y=X

Basic Functions, Functions and Graphs

• Function: A function has a value in the domain with exactly one value in the range

• Domain: The input or (x)• Range: The output or (y)• Vertical line test: if it passes it’s a function• Solving for a function: Algebraic, Numeric or Graphically• Continuity of a graph: Contiguous• Increasing and decreasing graphs.• Asymptotes: a line that a graph approaches but never

reaches

Transformations (Shifts, Stretches, and Reflections)

• Horizontal Translations: • Y=f(X-C) Translation to the right by C units • Y=f(X+C) Translation to the left by C units • Vertical Translations: • Y=F(X)+C Translation up by C units • Y=F(X)-C Translation down by C units• Reflections across the X-axis: y=-f(x)• Reflections across the Y-axis= f(-x)

Transformations (Shifts, Stretches, and Reflections)

• Stretches and Shrinks: • Horizontal: Y=F(X/C) {A stretch by a factor of C

if C>1}• {a shrink by a factor of C if

C<1}• Vertical: Y=C*F(X) {A stretch by a factor of C if

C>1}• {A shrink by a factor of C if

C<1}

Combination of Functions

• Sum: (F+G)=F(X)+G(X)• Difference: (F-G)(X)=F(X)-G(X)• Product: (FG)(X)=F(X)*G(X)• Quotient: (F/G)(X)=F(X)/G(X), Provided G(X)

cannot equal 0

Polynomials and Rational Functions

• Ways to solve a Quadratic Equation: Factoring, Using the Quadratic Formula, Completing the Square ax2 + bx = c

• Polynomial Function• One-To-One Functions• Horizontal Line Test -If some horizontal line intersects the graph of the function more than once,then the function is not one-to-one. -If no horizontal line intersects the graph of the function more than once,then the function is one-to-one.

Polynomials and Rational Functions

• Synthetic Division: • Real zeros and complex numbers

-complex numbers are numbers such as : 4+3i , 5i+i etc...

• real zeros are the intercepts of a quadratic equation

Polynomials and Rational Functions

• Rational Functions :To graph a rational function, you find the asymptotes and the intercepts, plot a few points, and then sketch in the graph example equation:

Polynomials and Rational Functions

• Fundamental Theorem of Algebra• Any polynomial of degree n ... has n roots, but

you may need to use complex numbers example of a polynomial

• this one has 3 terms

• The Degree of a Polynomial with one variable is the largest exponent of that variable.

Synthetic Division

Analytical Geometry• Parabolas•

F(x)=x^2

Exponential and Logarithmic Functions

• Exponential Function : f(x)=a.bx

• Logarithmic Function: y= Logbx• Properties of Logs:• Product: logbMN=logbM+logbN

• Quotient: LogbM\N=logbM-LogbN

• Power: LogbNy=ylogbN

Exponential and Logarithmic Functions

• Basic Common Logarithms Functions:• -Log101=0 because 100=1

• -Log1010=1 because 101=10

• -Log1010y=y because 10y=10y

• -Loglogx=x because Logx=logx• Log always finds the exponent!!

Ellipses with center(0,0)

• standard equation (x2/a2)+ (y2/b2)=1 • (y2/a2)+ (x2/b2)=1 • focal axis x-aixs y-axis• foci (c,0) (0,c)• vertices (a,0) (0,a)• semimajor axis a a• semiminor axis b b• pythagorean relation a2=b2+c2 a2=b2+c2

Ellipses with center (h,k)

• standard equation(x-h)2/a2 +(y-k)2/b2=1 • (y-k)2/a2 +(x-h)2/b2=1• focal axis y=k x=h• foci (hc,k) (h,kc)• vertices (ha,k) (h,ka)• semimajor axis a a• semiminor axis b b• Pythagorean relation a2=b2+c2

• a2=b2+c2

Hyperbolas

Hyperbolas center (0,0)

• standard equation (x2/a2)+ (y2/b2)=1 • (y2/a2)+ (x2/b2)=1 • focal axis x-aixs y-axis• foci (c,0) (0,c)• vertices (a,0) (0,a)• semimajor axis a a• semiminor axis b b• pythagorean relation c2=a2+b2 c2=a2+b2

• asymptotes y=b/a x y=a/b x

Hyperbolas center (h,k)

• standard equation (x-h)2/a2 +(y-k)2/b2=1 • (y-k)2/a2 +(x-h)2/b2=1• focal axis y=k x=h• foci (hc,k) (h,kc)• vertices (ha,k) (h,ka)• semimajor axis a a• semiminor axis b b• pythagorean relation c2=a2+b2 c2=a2+b2

• asymptotes y=b/a (x-h)+k y=a/b (x-h)+k