Download IV Examples

Alg 3(11)
Ch. 2.1- 2.3
1
2.1 Functions
POWERPOINT
I Vocabulary
a) Function: A 1-1 correspondence between 2 sets such that for each value in the domains set there is only 1
value in the range set. (No “x” repeat)
b) domain: “x” values
c) range: “y” values
II Functions ?
A.
a)
b)
c)
d)
y  x 2  5x  6
y = 2x + 3
y = |x|
y2 = 4
e) y = -  x + 2 
2
_____
2
f) y = x + 2
B.
_____
Function
Another way to write it
Substitute numbers in for x
ex.
C.
_____
_____
_____
_____
If f(x) = |x|
y = x - 11
f(x) = x - 11
f(x) = y
f(23) = 12
f(-5) =
f(99) =
when x = 23 then y = 12
(domain, range)
f(2) = ______
Function? ______
Piecewise-defined functions
2 x
y= 
2 x  3
x  3
y=  2
x  x  2
3x  1
y=  2
x  x  3
x 1
x 1
x  1
x  1
x2
x2
(23,12)
Function? _______
Function? _______
Function? _______
Alg 3(11)
Ch. 2.1- 2.3
2
III Restrictions
1) Denominator  0
5+ x
,
x
2 - 3x
,
x -5
x
x + 5x - 6
2
x ______ x  ______ x  ______________
even roots
2)
negative number
4 ,
16
3
8  2
ok!
IV Examples
1)
4
f(x)  x2  2x  3
f(-1) =
f(0) =
 1
f  =
2
2)
f(x)  x2  2x  3
Find f (x + 2)
Functions
a) y  x
y
b)
y
c)
domain = __________
2x
x 4
domain = __________
1
x
domain = __________
2
range = _____________
Alg 3(11)
Ch. 2.1- 2.3
3
V Examples
1. If f  x   2 x3 1
2. If f  x   x2  2
find f(-3)
3. If f(x) = x 2  3x  1
a) find f(x + 2)
b) find 3f( x)
3f(2)
then find
f  x   f  3
x 3
Alg 3(11)
Ch. 2.1- 2.3
Ch 2.2 Graphing
4
POWERPOINT
I Def: An equation whose graph is a line is a linear equation.
3x  2 y  6
Linear:
x4
y  2
y  3x  1
3x  y 2  2
Not Linear
1 1
 4
x y
II STANDARD FORM
SLOPE INTERCEPT FORM
Ax +By = C
2x -3y = -2
y = mx + b
write in standard form
y=x+3
write in slope intercept form
2x + 3y = 6
3x = 2y + ½
-5y + 2x = 10
III GRAPHING
x-intercept
slope = m =
y-intercept
parallel lines
perpendicular lines
vertical lines
horizontal lines
IV EXAMPLES
1. Find the x and y intercepts for
a) 3x + y = 6
b)
1
1
x + y=6
4
3
Alg 3(11)
Ch. 2.1- 2.3
2. Find the slope for the lines through the following points:
a) (3,-5) (-3,-3)
c) (
1 1
, )
2 2
(
b) (-2,-3) (-1,1)
1 1
, )
3 4
d) graph a line with slope - ¼ through the point (2,3)
e) Find “t” if the line through ( -1,1) and (3,2) is parallel to the line through (0,6) and (-8, t).
f) Show the figure with the following vertices is a parallelogram.
A (1,2) B (4,-1) C (2,-2) D (-1,1)
5
Alg 3(11)
Ch. 2.1- 2.3
g) Graph y = x + 6
h) Graph y = -2x + 1
i) Graph
1
1
x  y 1
3
2
6
Alg 3(11)
Ch. 2.1- 2.3
2.3 Line Equations
7
I Slope – Intercept Form
y = mx + b
m = _________
b = __________
To write the equation of a line you need the slope and a point on the line.
1. slope = 2, y – intercept = 4
(0,4)
2. slope = 2, point (2,3)
a) sub in point and slope
b) find “b”
y = mx + b
3 = 2(2) + b
b = -1
3. 2 points
(1,2) (3,4)
y = 2x + 4
y = 2x -1
a) find “m”
b) use one point with slope to find “b”
____________
4. 2 intercepts
x-int = -2
y-int = 3
a) write intercepts as points and find “m”
b) use y – intercept for “b”
_____________
5. point and a || line
(2,3) y = 2x + 3
a) pull of slope from || line
b) use point to find “b”
( same as #2)
_____________
6. point and  line
(2,3) y = 2x + 3
a) pull of slope ( _______________________ )
b) use pt. to find “b”
______________
Alg 3(11)
Ch. 2.1- 2.3
8
7. vertical line and point
(2,3)
a) x = “x” value of point
8. horizontal line and point
(2,3)
a) y = “y” value of point
____________
____________
Ex.
1. Write the equation of the line through (-1,3) and (3, 6)
2. Write the equation of the line through (-2, 6) and || to the line 3x – 2y = 4
Alg 3(11)
Ch. 2.1- 2.3
9
Algebra Review Worksheet 2.1-2.3
(1) Find the domain of each of the following.
(a) f x  
3x  4
5x  7
(b) f x  
(c) f x   7x  8
(2) Given the function f x    2x
(d) f x  
2
x2  1
x 2  x  12
 5x  3 , find each of the following.
(a) f 0 
(c) f  x  2
3x  2
(b) f  2

(d) 3 f 5
(3) Sketch a graph of each of the following.
(a) y = 3x + 4
(b) 4x  3y = 12
(4) Write the equation of the line which satisfies each of the following.
(a) Passes through (3 , 5) with slope 2
(b) Passes through (7 , 4) with slope 3
5
(c) Passes through (4 , 1) and (0 , 0)
(d) Passes through (3 , 5) and (2 , 5)
(e) Passes through (3 , 6), and is parallel to the line 4x  2y = 11
(f) Passes through (1 , 9), and is perpendicular to the line 5x + 3y = 2
(5) Triangle ABC has vertices A(2 , 1) , B(1 , 5) , and C(6 , 1).
(a) Is ABC a right triangle?
(b) Write the equation of BC
(c) Write the equation of the altitude to BC
Alg 3(11)
Ch. 2.1- 2.3
10
Answers 2.1-2.3
(1)
(2)
(4)
(a) x 
(b) x   23
7
5
(c) All Real Numbers
(d) x  4 , x   3
(a) 3
(b) 21
(c)  2x 2  3x  1
(d) 84
(a) y = 2x + 11
(b) y 
3x  1
5
5
(c) y   14 x
(d) y = 5
(e) y = 2x  12
(f) y 
(5)
3 x  42
5
5
(a) yes
(b) y 
4 x  31
7
7
(c) y   74 x 
9
2
Alg 3(11)
Ch. 2.1- 2.3
11
More Review of Slopes and Intercepts
slope =
change
---------------------------------
Find the slope of the line between the two points given.
1. (3, -8) and (-5, 2)
2. (-10, -3) and (7,2)
3. (-7, -6) and (3, -6)
4. (8, 2) and (8, -1)
Graph.
5. (1, -3) and m = 3
6. (2, 1) and m = -3/4
Find the intercepts.
7. y = 7x + 5
8. y = -9x + 15
Parallel, Perpendicular, or Neither?
9. 2x + 3y = 4
3x + 2y = 6
10. 0.5x + 2y = 1
4x - y = 3
Alg 3(11)
Ch. 2.1- 2.3
11. 6x - 9y = 4
2
x- y = 11
3
Graph.
13. (0,0) and parallel to y =2x + 1
12
12. y - 7 = 0
3x = 5
14. (1,4) and parallel to x + y = 1
15. (-4, 1) and perpendicular to a line whose slope is m = -5/3.
Alg 3(11)
Ch. 2.1- 2.3
13
Answers for More Review Slopes and Intercepts
1) -4/5
2) 5/17
3) 0
4) undefined
5)
6)
7) (0,5), (-5/7, 0)
8) (0, 15) (5/3, 0)
9) neither
13)
15)
10) perpendicular
11) parallel
14)
12) perpendicular
Alg 3(11)
Ch. 2.1- 2.3
14
2.1-2.3 Additional Review
Find each value if f(x)=
5
.
x2
6. f(3)
7. f(-4)
8. f(1/2)
9. f(-2)
10. f(0)
11. f(m-2)
Write each equation in standard form
2
3
12. x   y 
7
4
13. 3y - 5 = 0
Determine the slope of the line passing through each pair of points.
14) (3, 4) and (-2, 1)
15) (6, 0) and (6, 3)
Find the slope-intercept form of each equation.
16) 4x + 7y = 12
17) 3x - 2y = 4
Find the x- and y-intercepts
18) 5x - 4y = 8
19) 3x - y = -11
20)
2
4
x y 1
3
7
21) 3y = 7
Alg 3(11)
15
Ch. 2.1- 2.3
Write an equation for the line that satifies each of the given conditions in slope-intercept form. Graph the lines
on a separate sheet of graph paper. Label at least two points!
22) slope= -5, passes through (-3, -8)
23) slope= 4/5, passes through (10,-3)
24) passes through (4,3) and (7,-2)
25) passes through (3,11) and (-6,5)
26) passes through (7,2) and (3,-5)
27) x-intercept = 3, y-intercept = 2
28) x-intercept = -5, y-intercept=-5
29) vertical line passing through (1,5)
Alg 3(11)
Ch. 2.1- 2.3
16
Answers for 2.1-2.3 Review
6) f(3)  1
7) f(-4) = -5/2
8) f(1/2) = 2
9) undef
10) f(0) = 5/2
11) f(m-2) = 5/m
12) 28x + 8y = 21
13) 3y = 5
14) 3/5
15) undef
16) y = -4/7x + 12/7
17) y = 3/2x- 2
18) (0,-2), (8/5, 0)
19) (0,11) (-11/3, 0)
20) (0, 7/4), (3/2, 0)
21) no x-int, (0, 7/3)
22) y = -5x-23
23) y = 4/5 x -11
24) y = -5/3 x + 29/5
25) y = 2/3 x + 9
26) y = 7/4 x – 41/4
27) y = -2/3 x + 2
28) y = -x – 5
29) x = 1