Functions – Transformations, Classification, Combination Calculus P-3b

Functions - Graphing Vertical Line Test Identify 8 Basic Functions f(x)=x f(x)=x2 f(x)=x3 f(x)=x ½ f(x)=abs(x) f(x)=1/x f(x)=sin x f(x)=cosx Functions - Graphing

The squaring function

The square root function

The absolute value function

The cubing function

The cube root function

Rational Function

Sine Function

Cosine Function

Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; the graph of y = f(x)  b is the graph of y = f(x) shifted down b units. Vertical Translation

Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted right d units; the graph of y = f(x + d) is the graph of y = f(x) shifted left d units. Horizontal Translation

Shifts Vertical shifts Horizontal shifts Moves the graph up or down Impacts only the “y” values of the function No changes are made to the “x” values Horizontal shifts Moves the graph left or right Impacts only the “x” values of the function No changes are made to the “y” values Shifts

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function. Combining a vertical & horizontal shift

The graph of f(x) is the reflection of the graph of f(x) across the x-axis. The graph of f(x) is the reflection of the graph of f(x) across the y-axis. If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and (x, y) is on the graph of f(x). Reflections

Reflecting Across x-axis (y becomes negative, -f(x)) Across y-axis (x becomes negative, f(-x)) Reflecting

Sequence of transformations Follow order of operations. Select two points (or more) from the original function and move that point one step at a time. f(x) contains (-1,-1), (0,0), (1,1) 1st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (-1,1) 2nd transformation would be 3 times all the y’s, pts. are now (-3,-3), (-2,0), (-1,3) 3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-4), (-2,-1), (-1,2) Sequence of transformations

Transformations

Polynomial Degree – Highest degree of any term in polynomial Coefficient and degree of leading coefficient determines polynomial graph at end points Polynomial Functions

Even degrees – End points both go same direction Even degrees – End points both go same direction. Up if coefficient is positive, down if coefficient is negative Odd degrees – End points go opposite directions. Up on right if coefficient is positive, down on right if coefficient is negative Polynomial Functions

Discontinuity exists when denominators are zero Rational Functions

Algebraic Functions – Functions that can be expressed as a finite number of operations.

Algebraic Functions

Algebraic Functions - Practice Find: Algebraic Functions - Practice

Solve this problem, from the inside parenthesis out. Composite Functions

Practice

Practice

Composite Function - Practice Find: f(g(x)) and g(f(x)) Composite Function - Practice

Even and Odd Functions

Even and Odd Functions Even Functions f(-x)=f(x) Graph is symmetric wrt y-axis Odd Functions f(-x)=-f(x) Graph is symmetric wrt origin Even and Odd Functions

Example

Even and Odd Functions Odd (0,0), (-1,0),(1,0) Determine if the following are even or odd, and then find the zeros of the functions Odd (0,0), (-1,0),(1,0) Even and Odd Functions

Even and Odd Functions Odd (0,0) Determine if the following are even or odd, and then find the zeros of the functions Odd (0,0) Even and Odd Functions

Right 3 Left 3 Down 3 Practice Problems

Practice Problems Up 3 Reflect across x-axis Reflect across y-axis Stretch by 3 in y direction Practice Problems

Practice Problems

Exploration

pgs 27 – 29 31, 33, 37, 39-42, 49, 51, 52, 53, 55, 59, 60, 72 Homework