Week 2 - Trigonometry

Day 6

domain range
2
3 7
4 9
6 10
7

Day 6

1. Opener

domain range
2
3 7
4 9
6 10
7

Day 6

1. Opener
Are these functions?
domain range
2
3 7
4 9
6 10
7

Day 6

1. Opener

a)
domain range b)
2
3 7
4 9
6 10
7

Day 6

1. Opener

a)
domain range b)
2
3 7
4 9
6 10
7
b) What is the range here: {(-2, 3), (5, 1), (-2, 7), (-3, 8)}

Day 6

1. Opener

a)
domain range b)
2
3 7
4 9
6 10
7
b) What is the range here: {(-2, 3), (5, 1), (-2, 7), (-3, 8)}

c) Give a line perpendicular to: 2y - x = 7

2. Exercise
Find the line perpendicular to 2y - x = 7 that goes through (2, 3). Sketch both
lines.

2. Exercise
What is the vertex of the following parabola y = (x + 3)2 + 4.

2. Exercise
What is the vertex of the following parabola y = (x + 3)2 + 4.

V(-3, 4)

2. Exercise
What is the vertex of the following parabola y = -(x - 3)2 + 4.

2. Exercise
What is the vertex of the following parabola y = -(x - 3)2 + 4.

V(3, 4)

2. Exercise
What is the equation of the following parabola?

2. Exercise

y = (x - 1)2 + 1.

2. Exercise

y = -(x - 1)2 + 1.

2. Exercise

y = (x + 2)2 - 3.

2. Exercise
Find the vertex, x and y intercepts and sketch the graph of
y = x2 - 6x + 8.

3. Summarizing
1. Slope 5. Parallel lines
y2 − y1
m= m1 = m2
x2 − x1
2. General equation 6. Perpendicular lines
Ax + By + C = 0 m1m2 = −1
3. Slope-intercept form 7. Standard form
y = mx + b y = ax 2 + bx + c
4. Point-slope equation 8. Vertex form
2
y − y1 = m ( x − x1 ) y = ( x − h) + k

Day 7

1. f (x) = −3x + 5
2. y = 2x − 4
2
3. g(x) = x − 9
2
4. y = ( x − 2 ) + 1

Day 7

Opener

1. f (x) = −3x + 5
2. y = 2x − 4
2
3. g(x) = x − 9
2
4. y = ( x − 2 ) + 1

Day 7

Opener
Sketch the graph of the following functions:

1. f (x) = −3x + 5
2. y = 2x − 4
2
3. g(x) = x − 9
2
4. y = ( x − 2 ) + 1

Graphs of Functions.
Vertically Shifting the graph of y = f(x)


Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0

The graph of f is shifted The graph of f is shifted
Effect on Graph vertically upward a vertically downward a
distance c distance c




y=x2




y=x2+4
y=x2




y=x2+4
y=x2
y=x2-4

Horizontally Shifting the graph of y = f(x)


Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0

Effect on Graph horizontally to the right horizontally to the left a




y=x2




y=(x-4)2
y=x2




y=(x-4)2
y=x2
y=(x+2)2

1. f (x) = x + 4
3
2. y = ( x − 2 )
1
3. g(x) =
x−3

Examples.

1. f (x) = x + 4
3
2. y = ( x − 2 )
1
3. g(x) =
x−3

Examples.
Sketch the graph of the following functions:

1. f (x) = x + 4
3
2. y = ( x − 2 )
1
3. g(x) =
x−3

Inverse Functions
Definition of One-to-One Function.

Inverse Functions

Horizontal Line Test.

Inverse Functions


A function f is one-to-one if and only if every horizontal line

Inverse Functions


A function f is one-to-one if and only if every horizontal line
intersects the graph of f in at most one point.

Example.

Use the horizontal line test to determine if the function is one-to-one.

1. f (x) = 3x + 2
2
2. g(x) = x − 3

Definition of Inverse Function.
Let f be a one-to-one function with domain D and range R. A
function g with domain R and range D is the inverse function of
f, provided the following condition is true for every x in D and
every y in R:

y = f (x) if and only if x = g(y)

Example.
Let f(x) = x2 - 3 for x ≥ 0. Find the inverse function of f.

Theorem on Inverse Functions.
Let f be a one-to-one function with domain D and range R. If g is
a function with domain R and range D, then g is the inverse
function of f if and only if both of the following conditions are true:

(1) g(f(x)) = x for every x in D

(2) f(g(y)) = y for every y in R

Homework 2.
Find the inverse function of f
2
1. f (x) =
3+ x
2
2. f (x) = ( x + 2 ) , x ≥ −2
2x
3. f (x) =
x −1
3x + 4
4. f (x) =
2x − 3
2x + 3
5. f (x) =
x+2
6. f (x) = 2 3 x

Day 9

Domain and Range of f and f-1

domain of f-1 = range of f

range of f-1 = domain of f

Guidelines for Finding f-1 in Simple Cases.
1. Verify that f is a one-to-one function throughout its domain.

2. Solve the equation y = f(x) for x in terms of y, obtaining an
equation of the form x = f-1(y).

3. Verify the following conditions:

(a) f-1(f(x)) = x for every x in the domain of f

(b) f(f-1(x)) = x for every x in the domain of f-1

Example.
4x
Find the inverse function of f (x) =
x−2

Day 10

Exercises.
Find the inverse function of f.

1
1. f (x) =
x+3
3x + 2
2. f (x) =
2x − 5

Quiz 2.
Sketch the following functions:
2
1. y = ( x + 2 )
2. y = x 3 − 1
3. State the domain and range of the following function

Week 2 - Trigonometry

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