3-2,3-3 Graphs and Properties of Functions TEAGUE

Example 1)

Find y when x is -14

Find y when x is -4

At x = 4 is the y value + or -

At x = -4 is the y value + or -

The intervals above the x-axis

(-12,-2)(8, 12]

Most left point and most right point

Lowest and highest point

Where the graph touches the x-axis and same as f(x) = 0 in part e

-6

6

6

-3

-12<x<-2, 8<x12

-12, -2, 8

{x|-14x12}

{y|-6y9}

-12, -2, 8,

times

Where the graph touches the y-axis

If you draw a horizontal line at 1

If you draw a vertical line at

Give x where y is -6

Give x where y is 9

Example 2)

-3

3 times

1 times

x =-14, 4

- times

x = -6

times

Example 3)

from left to right from bottom to top

(-,5)

times

1

(0,3)

times

Example 4)

Example 5)

Is the point (2,11) on the graph?

Set the equation = -3 and get 4x2 – x=0

factor x(4x-1)=0

Factor the original to get x-intercepts (slide and divide)

(-,10]

times

(-,)

times

(10,0)(-15,0),(0,10)

15

(-2,15)

0,¼

(0,-3),( ¼,-3),

(-,)

times 1,¾ -3

Example 6) 𝒇(𝒙) =𝒙−𝟓

𝒙+𝟐 * domain set bottom = 0

a) Give the domain: {x|x≠-2} * x-intercepts set top = 0

b) Is the point (1,𝟑 𝟐⁄ ) on the graph of f? 𝟏−𝟓

𝟏+𝟐=

−𝟒

𝟑 NO, doesn’t = 3 2⁄

c) If x=2, what is f(x)? 𝟐−𝟓

𝟐+𝟐=

−𝟑

𝟒 List the point (2,

−3

4)

d) If f(x) = 2, what is x? x−5

x+2= 2

2x+4=x-5 x=-9 List the point (-9,2)

e) What are the x-intercepts? Set y=0 x−5

x+2= 0 x-5=0 x=5

don’t use the bottom bc it makes it undefined List the points (-5,0)

* to get the y-intercepts – just cover up the x values = −𝟓

𝟐

Example 7) 𝒇(𝒙) =𝟖𝒙𝟐

𝒙𝟒+𝟏𝟔

a) Is (2,1) on the graph? 𝟖(𝟏𝟐)

(𝟏𝟒)+𝟏𝟔= 1

b) If x = 3, then f(x)=____? 𝟖(𝟑𝟐)

(𝟑𝟒)+𝟏𝟔=

𝟕𝟐

𝟗𝟕 List the point (3,

𝟕𝟐

𝟗𝟕)

c) If f(x)=1, then x______? 𝟖𝒙𝟐

(𝒙𝟒)+𝟏𝟔= 1

cross multiply 8x2 = x4+16

0= x4-8x2 + 16

(x2-4)(x2-4)

(x-2)(x+2) x = -2,2 List the points (-2,1),(2,1)

d) If f(x) = 2, what is x? 𝟖𝒙𝟐

(𝒙𝟒)+𝟏𝟔= 2 0= 2x4-8x2 + 32 0= x4-4x2 + 16 doesn’t factor

List the point none

e) What are the x=intercepts? Set y=0 𝟖𝒙𝟐

(𝒙𝟒)+𝟏𝟔= 0 8x2 = 0 x=0

* to get the y-intercepts – set zero for x values = 𝟎 List the points (0 ,0)

D: (-,) because bottom doesn’t factor

Example 8):

−44(7)2

(27)2 + (7) + 6= 10.04 ft

−44(10)2

(27)2 + (10) + 6= 9.96 ft

The ball won’t go through the hoop

10 = −44(15)2

𝑣2 + (15) + 6

-11v2=9900

-11 -11

v2=900 v = 30

Example 9):

Even or Odd Functions:

EX a:

EX b:

EX c:

Example 10):

Example 11):

Example 12):

The x values for the two top points

The y value for the two top points

a) What values of x is a local maximum?

At x = -3 max is 2

b) What values of x is a local minimum?

At x = 1 min is 2

c) Find Intervals of increasing and decreasing

INCREASING ON: DECREASING ON:

Example 13):

a) (-1,0),(0,1),(1,0)

b) Domain [-3,3] Range [0,3]

c) IncreasinG: (-1,0), (1,3) Decreasing: (-3,-1),(0,1) Constant: none

d) even

EVEN FUNCTION STARTS AND ENDS ODD FUNCTION STARTS ONE SIDE

ON THE SAME SIDE – SYMMETRICAL AND ENDS ON THE OTHER