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A math student is pestered by a classmate who wants to copy his homework. The student hesitates, because he doesn't want to be punished for cheating.
His classmate calms him down: "Nobody will be able to trace my homework to you: I'll be changing the names of all the constants and variables: a to b, x to y, and so on." Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying.
After the deadline, the student asks: "Did you really change the names of all the variables?" "Sure!" the classmate replies. "When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x+1, I called it the timber of x+1..."
TODAY: Ch 13: Graph Reflections, Shifts, Dilations
OFFICE HOURS: today: 1:30-2 today here,
tomorrow: 10-11am in PDL C-326
HOMEWORK: Ch. 11&12 are due Tuesday & Ch. 13 is due Thursday
Quiz #4: Thursday, from Exponential & Logarithmic Fcts. (Ch. 10-12). It will test your basic skills: properties, standard form, graphs, solving equations.Note: Use of a graphing calculator (or any other unauthorized resource) during a quiz or exam constitutes cheating and will be treated accordingly.
AnnouncementsMonday, November 05, 2012
Lecture 19 Page 1
Solve the equations:
a)
b)
Ch 12 Practice Skills Problems:
Lecture 19 Page 2
Graph REFLECTION
CHANGE IN FORMULA
CHANGE IN DOMAIN? CHANGE IN RANGE?
About x-axis no Multiply by -1(& reverse inequalities)
Composing a function with another, linear function results in the graph of being reflected, shifted, or dilated.
VERTICAL operations:
(if is the "inside" function, i.e. the operations are on the "outside" of )
•
HORIZONTAL operations:
(if is the "outside" function, i.e. the operations are on the "inside" of )
•
REFLECTIONS:I.
About the x-axis:
To reflect the graph of about the x-axis, we need to:
New Domain:
a)
New Range:
About the y-axis:
To reflect the graph of about the y-axis, we need to:
New Domain:
b)
New Range:
Summary:
Ch 13: Graph Operations: Reflections, Shifts, and Dilations
Lecture 19 Page 3
(& reverse inequalities)
About y-axis Multiply by -1(& reverse inequalities)
no
Extra example: Sketch the graphs of
Lecture 19 Page 4
GRAPH SHIFTby 2 units
CHANGE IN FORMULA
CHANGE IN DOMAIN?
CHANGE IN RANGE?
UP no add 2
DOWN no subtract 2
RIGHT add 2 no
LEFT subtract 2 no
II. SHIFTS
UP by c units: a.
DOWN by c units: b.
VERTICAL shifts. To move a graph:1.
Restricted to domain: •
Range: •
Our main (graphical) example below is:
An important principle to remember is:
LEFT by c units: a.
HORIZONTAL shifts. To move a graph:2.
RIGHT by c units: a.
Summary:
Extra example: Sketch the graph of
Lecture 19 Page 5
Extra example: Sketch the graph of
Lecture 19 Page 6
GRAPH DILATIONby 2
CHANGE IN FORMULA
CHANGE IN DOMAIN?
CHANGE IN RANGE?
no Multiply by 2
no Divide by 2
Divide by 2 no
Multiply by 2 no
Multiplying/dividing the entire function f(x) by a POSITIVE constant c results in a VERTICAL dilation of its graph.
a)
Multiplying/dividing on the inside of a function f(x) results in a HORIZONTAL dilation of its graph
b)
Summary:
DILATIONSIII.
Lecture 19 Page 7
Suppose we have
Vertically: First: _______________________, then _________________
Horizontally: First: _______________________, then ___________________
Basic rule:
Suppose we have
Vertically: same as above
Horizontally: First: _______________________, then ___________________
When we have more than one vertical (or horizontal) operation to perform, the ORDER MATTERS!
Order of operations:
Lecture 19 Page 8
Ex 1: Let's start with the function What curve is this? Sketch its graph.
Graph
a)
Graph
b)
So far, the order did not matter because we haven't had to do any 2 vertical or any two horizontal operations.
Ex 2: Sketch
We start with the graph of the function y=?? and we do:
Vertically: First ____________________ then _______________________•
Horizontally: First __________________ then _______________________•
Examples:
Lecture 19 Page 9
