warm-up please fill out the chart to the best of your ability
TRANSCRIPT
Warm-UpPlease fill out the chart to the best of your
ability
Assignmentp. 168# 1, 4, 6
Section 1.5 Shifting, Reflecting, and Stretching Graphs
Objectives: Students will know how to identify and graph shifts, reflections, and nonrigid
transformations of functions.
Extra Practicehttp://www.khanacademy.org/exercise/shifting_and_reflecting_functions
Extra Exampleshttp://www.khanacademy.org/video/algebra-ii--shifting-quadratic-graphs?topic=california-standards-test-algebra-2
The Original SixConstant Function f(x) = c
Identity Function f(x) = x
y
x
y
x
X Y
-1 -2
0 -2
1 -2
X Y
-1 -1
0 0
1 1
Absolute Value Function f(x) = |x| Square Root Function
y
x
X Y
-1
1
0 0
1 1
y
x
X Y
-1
0 0
4 2
f ( x) x
Quadratic Function f(x) = x2
Cubic Function f(x) = x3
y
x
y
x
X Y
-1
-1
0 0
2 8
X Y
-2
4
0 0
2 4
Vertical and Horizontal ShiftsUse a graphing utility to graph:
Y1 = f(x) = x2. Then, on the same viewing screen, graph Y2 = (x – 4)2. How did we change the equation? How did the graph change?
Y3 = (x + 4)2, Y4 = x2 – 4, and Y5 = x2 + 4. How did we change the equations? How did the graphs change?
Let c be a positive real number. The following changes in the function y = f(x) will produce the stated shifts in the graph of y = f(x).
h(x) =f(x - c) Horizontal shift c units to the rightY2 = (x – 4)2
h(x) =f(x + c) Horizontal shift c units to the leftY3 = (x + 4)2
h(x) =f(x) - c Vertical shift c units downwardY4 = x2 – 4
h(x) =f(x) + c Vertical shift c units upwardY5 = x2 + 4
Example 1. Given f(x) = x3 + x, describe and graph the shifts in the graph of f generated by the following functions.
a) g(x) = (x + 1)3 + x + 1.
b) h(x) = (x - 4)3 + x.
Let . Write the equation for the function resulting from a vertical shift of 3 units downward and a horizontal shift of 2 units to the right of the graph of
xxf )(
f(x) = | x 2 | 3
Warm UpWrite about what it means to reflect over the
y-axis and x-axis without using the word symmetry?
Assignmenthttp://www.khanacademy.org/exercise/shiftin
g_and_reflecting_functionsRegister me as your coach and do 10
problems
Reflecting Graphs
Reflecting Graphs Use a graphing utility to graph:
Y1 = f(x) = (x – 2)3. Then, on the same viewing screen, graph Y2 = -(x – 2)3.
Y3 = (-x - 2)3.
The following changes in the function y = f(x) will produce the stated reflections in the graph of y = f(x).
h(x) =f(-x) Reflection in the y-axis
h(x) = -f(x) Reflection in the x-axis
Example 2. Given f(x) = x3 + 3, describe the reflections in the graph of f generated by the following functions.
a) g(x) = -x3 + 3. Reflected in the ???-axis.
b) h(x) = -(x3 + 3) = -x3 - 3. Reflected in the ???-axis.
Example 3. Below is the graph ofa) y = b) Graph y = -f(x). c) Graph y
= f(-x) + 1
2x
1. 2. 3.
Widening and NarrowingDistort the shape of the graphIs not shifting or reflecting it. Come from equations of the form y = cf(x).
If c > 1, then there is a vertical stretch of the graph of y = f(x). If 0 < c < 1, then there is a vertical shrink.
Example 4. Given f(x) = 1- x2, describe the graph of g(x) = 3 – 3x2.
Because 3 – 3x2 = 3(1- x2), the graph of g is a vertical stretch (each y-value is multiplied by 3) of the graph of f.
X f(x)=1- x2 g(x) = 3 – 3x2
-1 0 0
0 1 3
1 0 0
2 -3 -9
X Y
-2 (8/3)
-1 (2/3)
0 0
1 (2/3)
2 (8/3)
X Y
-2 8
-1 2
0 0
1 2
2 8
X Y
-2 4
-1 1
0 0
1 1
2 4
2xy 2
3
2xy 22xy 2xy
X Y
-2 4
-1 1
0 0
1 1
2 4
Please describe the following function
g(x) = -2f(x)Reflection? Wider or Narrower?
h(x) = Reflection? Wider or Narrower
)(2
1xf
Warm UpIn the mail, you receive a coupon for $5 off of a
pair of jeans. When you arrive at the store, you find that all jeans are 25% off. You find a pair of jeans for $36.
1. If you use the $5 off coupon first, and then you use the 25% off on the remaining amount, how much will the jeans cost?
2. If you use the 25% off first, and then you use the $5 off on the remaining amount, how much will the jeans cost?
Jean fiend Let the cost of the jeans be represented by a
variable x. Write a function f(x) that represents the cost of the jeans after the $5 off coupon.
Write a function g(x) that represents the cost of the jeans after the 25% discount.
Function CompositionWrite a new function r(x) that represents the
cost of the jeans if the 25% discount is applied first and the $5 off coupon is applied second.
Write a new function s(x) that represents the cost of the jeans if the $5 off coupon is applied first and the 25% discount is applied second.
Compositions of FunctionsThe composition of the function f with the
function g is (f g)(x) = f(g(x)).f(x) = x – 5, g(x) = .75x (f g)(x) = f(g(x)) = [.75x] - 5
The composition of the function g with the function f is
(g f )(x) = g(f(x)).
g(x) = .75x, f(x) = x – 5 (g f )(x) = .75(x - 5)
Welcome to my domainThe domain of (f g) is the set of all x in the
domain of g in the domain of f.
Domain of f
Domain of g and domain of f g
Example 2. f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.
Find (f g)(x) Find (g f )(x)
f g x f g x f 2x 1
2x 1 2 2 2x 1 4x 2 4x 1 4x 2
4x 2 8x 3
g f x g f x g x2 2x 2 x 2 2x 12x2 4x 1
1.5 Combinations of Functions
Objectives: Students will know how to find arithmetic combinations and compositions of
functions.
Arithmetic Combinations of FunctionsLet f and g be functions with overlapping
domains. Then for all x common to both domains:
(f g)(x) = f(x) g(x)(fg)(x) = f(x) • g(x)
provided g(x) 0.
f
g
x f (x)
g(x),
p
Example 1. f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.a)
b)
c)
d)
f g x f (x) g(x)
x 2 2x 2x 1
x2 4x 1
f g x f (x) g(x)
x 2 2x 2x 1
x2 1
fg x f x g(x)
x2 2x 2x 1
2x3 5x 2 2x
f
g
x
f x g x
x2 2x2x 1