Pshs 3rd yr_functions

PSHS 3RD YEAR MATH

Session 1

1. Evaluate the function for f(-1) if

21 1 1 1f

2( ) 1f x x x

Diagnostic - Answers

1 1 1

1

2. What is the sum h(x) + g(x) if h(x) = 5x – 1 and g(x) = 3x2 + 6x – 7 ?

25 1 3 6 7h x g x x x x23 11 8x x

3. If F(x)= 2x + 3 and G(x) = 5x – 4, what is the composite function

G(F(x))?

5 4G F x F x

5 2 3 4x

10 15 4

10 11

x

x

4. What is the domain of the square root function ?

• DOMAIN

The set of all possible values for x, OR

The x-coordinates of the points of the graph of the function

( )f x x 5

2 ways to analyze:1.) Knowing the graph of the function2.) Using the given equation of the function

The graph of y x

What are the x-coordinates of the points on the graph? 0 to DOMAIN

The graph of ( )f x x 5

• Courtesy of Wolfram Alpha: www/wolframalpha.com

• GENERAL EQUATION: y a x h k

at x=5

h=horizontal shift

DOMAIN of

is

5x

( )f x x 5

5. Function or not?Relation Function Not a

Functiona.{(5,6), (-2,3), (3,1), (5,2), (8, -4)

}

b. y = 9 – x2

c.

d. x = (y + 6) (y – 3)

-1 1

-2

6. Find the inverse of the function . • HOW TO GET

• 1.) Use y for “f(x)”.

• 2.) Switch x and y.

• 3.) Isolate y.

• 4.) This new y will be your

1f x2 3f x x

1f x

1f x

2 3y x

2 3x y

3

2

xy

13

2

xf x

7. Which of the following is NOT divisible by (x – 1) ?

• 3 Ways to answer:

1.) Using LONG division.

2.) Using SYNTHETIC division.

3.) Using the REMAINDER THEOREM.

Remainder Theorem:

3 2

3 2

3 4

1 3 1 4 0

x x

Therefore,

is divisible by(x - 1).

3 23 4x x

8. What are the roots of the function ?

• ROOTS – also known as the x-coordinates of the x-intercepts, OR the solutions to the equation if h( x ) were equal 0.

• HOW DO WE GET THE x-intercepts?

• Let y = 0, meaning let h( x ) = 0.

3 22 15h x x x x

3 20 2 15x x x

3 2

2

0 2 15

0 2 15

0 5 3

x x x

x x x

x x x

Which is the correct set of roots of f( x )?

x = -3, 0 and 5OR

x = -5, 0 and 3

9. What is the equation of the line that passes through (5, -6) and is perpendicular to the line whose equation is ?• Point (5, -6) is a point on the line we are

looking for.

• Perpendicular means that the slopes of

• and the unknown line are NEGATIVE RECIPROCALS of each other.

4 16 0x y

4 16 0x y

10. Given the function :

• Find the intercepts.

• To get the x-intercept:

• Therefore,

• x-intercept = (1/2, 0).

1 2f x x

0 1 2

2 1

1

2

x

x

x

1 2 0

1

y

y

• To get the y-intercept:

Therefore, y-intercept = (0, 1).

10.b. Graph the function.

11. Given the function :

• Find the vertex of the parabola.

• 2 ways:

• 1.) Vertex-form of the quadratic equation.

• 2.) Use the vertex formula.

24 1 2f x x

2y a x h k

24: ,

2 4

b ac bvertex

a a

11.b. Find the domain and range.

• Domain – set of all real numbers:

• Range – based on 2 things:

• 1.) the y-coordinate of the VERTEX of the parabola.

• 2.) the direction where the parabola is opening leading coefficient.

• Therefore, the range is 2y

x

11.c. Graph the function.

Vertex: (1, 2)

Trigonometric functions

• Sine, Cosine, and Tangent

• These are RATIOS of the sides of the right triangle

sinopposite

hypotenuse

cosadjacent

hypotenusetan

opposite

adjacent

SOH CAH TOA

RECALL: Pythagorean Theorem

a

b

c

If side a = 5 cm. and side c = 13 cm., what isthe length of side b?

2 2 2a b c

where a and b are the LEGS, and c is the hypotenuse.

RECALL: The 30-60-90 TRIANGLE THEOREMS

• The side opposite 30 degrees will have a length of ½ of the length of the longest side (hypotenuse).

• The side opposite 60 degrees will be times the length of the longest side.

30

60

3

2

a

b

c

RECALL: THE 45-45-90 TRIANGLE

• In terms of the sides, what kind of triangle is the 45-45-90 triangle?

• The length of the longest side is times the length of a leg.

a

c

a

2

Trigonometric identities

sintan

cos

1csc

sin

1sec

cos

1cot

tan

2 2sin cos 1

THE UNIT CIRCLE

For angle measuresgreater than 90 deg,we use REFERENCE ANGLES.

Reference anglesAngle rotation starts from the positive x-axis, then moving counter-clockwise.The reference angle is measured from the x-axis.

What is the reference Angle of the ff?1.) 120 deg =2.) 225 deg =3.) 330 deg =

UNITS OF ANGLES: DEGREES & RADIANS

360 2

180

902

454

603

306

What IF YOU FORGOT THE CONVERSIONS?

• What is 270 degrees in radian measure?

• Just memorize one thing:

360 270

2 ?

360 2

What are the trigonometric function values of the ff. angles?

Angle Sin x Cos x Tan x Csc x Sec x Cot x

240 deg

-225 deg

-150 deg

Using the periodicity of the sine and cosine functions

171. sin

47

2. cos6

53. tan

3

GRAPHS OF THE TRIGONOMETRIC FUNCTIONS

FEATURES of y = sin x

1. Amplitude2. Period3. Phase shift (horizontal)4. Vertical shift

siny x

siny x2siny x

siny x

sin 3y x

siny x sin2

y x

siny x sin2y x

Transformations of the graph

sin( )y a bx c k

a - amplitude

k – Vertical shiftc

c/b – Horizontal (phase) shift

b/ (2pi) - Period