Alg 3 1

Functions and Coordinate Geometry

Functions

I Vocabulary

a) Function: A 1-1 correspondence between 2 sets such that for each value in the domain set there is only 1

value in the range set. (Each “x” can only have one corresponding “y” value)

b) domain: “x” values

c) range: “y” values

II Functions ?

A. a) = + +2y x 5x 6 _____

b) y = 2x + 3 _____

c) y = |x| _____

d) 2y = 4 _____

e) ( )2y = - x + 2 _____

f) 2y = x + 2 _____

B. Function y = x - 11

Another way to write it f(x) = x - 11

Substitute numbers in for x f(23) = 12 when x = 23 then y = 12 (23,12)

f(-5) =

f(99) =

ex. If f(x) = |x| f(2) = ______ Function? ______

C. Piecewise-defined functions

y = 2 1

2 3 1

x x

x x

− <

− ≥ Function? _______

y = 2

3 1

2 1

x x

x x x

+ ≤ −

+ + ≥ − Function? _______

y = 2

3 1 2

3 2

x x

x x x

− ≤

− + ≥ Function? _______

f(x) = y

(domain, range)

Alg 3 2

Functions and Coordinate Geometry

III Restrictions

1) Denominator ≠ 0 5+ x

x,

2 -3x

x -5,

2

x

x +5x - 6

______ ______ ______________

2) negative number 4− , 4 16−

3 8 2− = −

IV Examples

1)

f(-1) =

f(0) =

2)

Find f (x + 2)

Functions

a) � � √� domain = __________ range = _____________

b) � ��

√� domain = __________

c � ���

�� domain = __________

even roots

ok!

≠x ≠x ≠x

= + +2f(x) x 2x 3

1f =

2

= − +2f(x) x 2x 3

Alg 3 3

Functions and Coordinate Geometry

V Examples

1. If ( ) 32 1f x x= − 2. If ( ) 2 2f x x= + then find ( ) ( )3

3

f x f

x

−

−

find f(-3)

3. If f(x) = 2 3 1− +x x

a) find f(x + 2) b) find 3f( x)

3f(2)

Alg 3 4

Functions and Coordinate Geometry

Homework Assignment #1

(1) ( ) 4 3 2f x = 3x 16x 7x 28x 13− − − − , ( ) 2

g x = x + 2x 4− . Find each of the following.

(a) ( )g 4 (c) ( )( )f g 3−

(b) ( )f 6 (d) 2g(x 3)−

(2) Determine whether each of the following defines y as a function of x. If it is a function, find the domain

please.

(a) y 5x 4= − (f) y 5 4x = −

(b) 2

y x + 2x 3= − (g) 2

y x 5x 6 = − −

(c) 2 2

x + y = 1 (h) 2

xy

x + 1=

(d) 3x + 5

y 2x 3

=−

(i) 5 2x

y 3x + 4

−=

(e)

23

2

3x + 2x 5y

x 5

−=

−

(3) Let the function f be defined by ( )f x = x + 1. If ( )2 f = 20ζ , find ( )f 3ζ .

(4) Let the function f be defined by ( )f x = x + c , where c is constant. If ( )f 2 = 10 , find the value of the

constant c.

Alg 3 5

Functions and Coordinate Geometry

Answers

(1) (a) 20 (c) 27

(b) –1 (d)

22x 8x 2− −

(2) (a) all real number (f) 54

x ≤

(b) Not a function (g) x 1 or x 6≤ − ≥

(c) Not a function (h) all real numbers

(d) 32

x ≠ (i) 543 4 < x − ≤

(e) x 5≠ ±

(3) 28

(4) 8

Alg 3 6

Functions and Coordinate Geometry

Linear Equations

I: An equation whose graph is a line is called a linear equation.

Linear:

3 2 6

4

2

3 1

x y

x

y

y x

+ =

=

= −

= +

Not Linear

23 2

1 14

x y

x y

+ =

+ =

II DEFINITIONS:

Slope (m)

x-intercept

y-intercept

parallel lines

perpendicular lines

vertical lines

horizontal lines

III STANDARD FORM SLOPE INTERCEPT FORM

Ax +By = C y = mx + b

2x -3y = -2

write in standard form write in slope intercept form

y = x + 3 2x + 3y = 6

POINT-SLOPE FORM

� � �� � �� � ���

Alg 3 7

Functions and Coordinate Geometry

IV EXAMPLES

1. Find the x and y intercepts for

a) 3x + y = 6 b) 1

4x +

1

3y = 6

2. Find the slope for the lines through the following points:

a) (3,-5) (-3,-3) b) (-2,-3) (-1,1)

c) ( 1

2 , 1

2) (

1

3 ,1

4)

d) Find “t” if the line through ( -1,1) and (3,2) is parallel to the line through (0,6) and (-8, t).

e) Show the figure with the following vertices is a parallelogram.

A (1,2) B (4,-1) C (2,-2) D (-1,1)

Alg 3 8

Functions and Coordinate Geometry

Writing Equations

To write the equation of a line you need the slope and a point on the line.

1. 2 points

(1,2) (3,5)

2. 2 intercepts

x-int = -2

y-int = 3

3. point and a || line

(2,3) y = 2x + 3

4. point and ⊥ line

(2,3) y = 2x + 3

5. vertical line and point

(2,3)

6. horizontal line and point

(2,3)

Alg 3 9

Functions and Coordinate Geometry

Homework Assignment #2 – Linear Equations

(1) Sketch a graph of each of the following lines please.

(a) 4x 2y = 6− (b) 2x + y = y 4−

(c) y + 5 = 5

(2) Find all values of the constant k if the line connecting the points (−4 , 1) and (k , 5) is to be: (a) parallel, (b) perpendicular to the line connecting the points (3 , k) and (−3 , 6).

(3) Write an equation of a line which satisfies each of the following please.

(a) Passes through (−2 , 5) with slope 23− .

(b) Passes through (4 , −7) with slope 52 .

(c) Passes through (3 , 6) and (−6 , 0).

(d) Passes through (−5 , 3) and (2 , 3).

(e) Passes through (8 , −1) and is parallel to 3x − 4y = 2.

(f) Passes through (8 , −1) and is perpendicular to 3x − 4y = 2.

(4) ABC∆ has vertices A (−1 , −1) , B (1 , 3) , C (4 , 2). Write the equation of the altitude to BC����

please.

Alg 3 10

Functions and Coordinate Geometry

Answers

(1) (a) y = 2x 3− (b) x = 2−

(c) y = 0

(2) (a) k = −6 or k = 8 (b) k = 0

(3) (a) 32

y 5 (x 2)− = − +

(b) 25

y 7 (x 4)+ = − (a lot easier than) 543

52 x y −=

(c) 23

y (x 6)= + or 23

y 6 (x 3)− = −

(d) y = 3

(e) 34

y 1 (x 8)+ = −

(f) 43

y 1 (x 8)+ = − −

(4) y 1 = 3(x 1)+ +

Alg 3 11

Functions and Coordinate Geometry

Coordinate Geometry

1. Given a right triangle with two sides 3 and 5 as shown, what is the length of the hypotenuse? 2. Using the same methodology, what is the distance between two points (-1, 1) and (3, 4)? 3. What about the distance between points (-3,- 2) and (5, 4)? 4a. Can you give a formula for the distance between (3, 2) and ( ,x y )?

4b. What about the distance between ( 1 1,x y ) and ( 2 2,x y )?

3

5

(-1,1)

(3,4)

Alg 3 12

Functions and Coordinate Geometry

DISTANCE FORMULA If you label the two points ( 1 1,x y ) ,( 2 2,x y )

5. Keeping this formula in mind, can we find some points that have a distance of 5 from the origin? Hint : ( 1 1,x y ) = (0, 0) for the origin

6. What is an equation for the set of all these points? 7. How would these points change if we centered the circle around (1, 2) instead of the origin? 8. How would the equation change???

( )1 1x , y

( )2 2x , y

2 1| x x |−

2 1| y y |− d 2 2 2

2 1 2 1

2 2

2 1 2 1

d = (x x ) + (y y )

d = (x x ) + (y y )

− −

− −

Alg 3 13

Functions and Coordinate Geometry

CIRCLES A circle is the set of all points in the plane, each of which is at a fixed distance r from a given point called the center ( h ,k ) of the circle; r is the radius of the circle.

r = 2 2( ) ( )− + −x h y k or

2 2 2( ) ( )= − + −r x h y k

Center Radius

9. Find the center and the radius 2 24 ( 6) ( 3)x y= − + + ________ ________

10. 2 2 10x y+ = ________ ________

11. Find the center and radius 2 26 10 2x x y y− + − = ________ ________

12. 2 216 16 64 32 64x y x y+ − + = ________ ________

13. Write the equation of the circle with center (-5,2) and radius 3 .

P (x,y)

(h,k)

Alg 3 14

Functions and Coordinate Geometry

Midpoint is the point directly between two points Midpoint is just an averaging of the x and y coordinates

1 2 1 2,2 2

x x y y+ +

So for the two points (-3,- 2) and (5, 4)… midpoint is 3 5 2 4

,2 2

− + − +

=(1,1)

6. Find midpoint of AB if A1 1,

2 4

and B 1 2,

3 3

Alg 3 15

Functions and Coordinate Geometry

Homework Assignment #3 – Distance and Midpoint

(1) Find the distance between each of the following pairs of points please.

(a) (6 , 3) and (−2 , 3) (c) (9 , −17) and (−3 , 7)

(b) (−3 , 6) and (2 , −6) (d) ( ) , 65

31

and ( ) , 43

21

(2) Triangle ABC∆ has vertices A(−2 , 2) , B(6 , 8) , C(4 , −1). Find each of the following.

(a) The equation of AC����

.

(b) The length of AC .

(c) The length of the median to AB .

(d) The equation of the altitude to AC����

.

(e) The length of the altitude to AC����

.

(3) Triangle ABC∆ has vertices A(−2 , −1) , B(0 , 1) , C(6 , −5). Find each of the following.

(a) The perimeter of ABC∆ .

(b) The length of the median to the longest side of ABC∆ .

(c) The equation of the perpendicular bisector of the shortest side of ABC∆ .

(d) The coordinates of the point where the three medians intersect (the centroid).

(e) The coordinates of the point where the three perpendicular bisectors intersect (the circumcenter of

the triangle).

Alg 3 16

Functions and Coordinate Geometry

Answers

(1) (a) 8 (c) 512

(b) 13 (d) 12

5

(2) (a) 1 x y21 +−=

(b) 53

(c) 102

(d) y = 2x − 4

(e) 54

(3) (a) 8 2 + 4 5

(b) 2 5

(c) y = x 1− −

(d) ( )543 3 , −

(e) ( )2 , 3−

Alg 3 17

Functions and Coordinate Geometry

TANGENTS Tangents are ⊥ to radii at the point of tangency. Draw the given circle and the tangent line at the indicated point for each of the following. Write the equation of the tangent line.

1. 2 2 80x y+ = (8,4)

2. 2 214 18 39x x y y+ + + = (5,-4)

3. Triangle ∆ABC has vertices A(2 , 1) , B(−1 , −5) , and C(6 , −1).

(a) Write the equation of BC

(b) Write the equation of the median to BC

(c) Find the length of the median to BC

(h,k)

(8, 4 )

Alg 3 18

Functions and Coordinate Geometry

Assignment #4 - Circles

(1) Find the center and radius of each of the following circles please.

(a) ( ) ( ) 49 5 y 2 x 22 =−+− (b) ( ) 100 y 6 x 22 =++

(2) Sketch a graph of each of the following circles please.

(a) 0 9 4y 6x y x 22 =+−−+

(b) 0 5 8y 4x y x 22 =−−++

(c) 0 20 4y 10x y x 22 =++−+

(d) 0 2y y x 22 =++

(3) Write the equation of the circle if the endpoints of a diameter are (−5 , 6) and (1 , 4).

(4) Write the equation of the line which is tangent to the circle 13 y x 22 =+ at the point (2 , 3).

(5) Write the equation of the line which is tangent to the graph of the circle

0 17 8y 2x y x 22 =−+−+ at the point (6 , −1).

Alg 3 19

Functions and Coordinate Geometry

Answers

(1) (a) center (2 , 5) , radius = 7 (b) center (−6 , 0) , radius = 10

(2) (a) ( ) ( ) 4 2 y 3 x 22 =−+−

(b) ( ) ( ) 25 4 y 2 x 22 =−++

(c) ( ) ( ) 9 2 y 5 x 22 =++−

(d) ( ) ( ) 1 1 y 0 x 22 =++−

(3) ( ) ( ) 10 5 y 2 x 22 =−++

(4) 313

32 x y +−=

(5) 9 x y35 +−=

Alg 3 20

Functions and Coordinate Geometry

Graphing Absolute Value

y = |x| x y

0 0 -1 1 1 1 -2 2 2 2 -3 3 3 3

y = |x| + 3

y = | x – 4 |

y = 4 | x | - 6

x y

x y

x y

Alg 3 21

Functions and Coordinate Geometry

= −1

y x 64

y = | x + 5 | - 1 website: http://www.purplemath.com/modules/graphabs.htm Summary y = a |x - h| + k SHIFTS

x y

x y

Alg 3 22

Functions and Coordinate Geometry

Piecewise Graphing Graph f(x) each time with restricted domain

f(x) = 2x + 1 when x < 0 f(x) = x – 4 when 0 < x < 3 f(x) = 3x when x ≥ 3

f(x) = –2x + 3 when x ≤ –2 f(x) = x + 1 when –2 < x < 2 f(x) = 2x – 2 when x ≥ 2

Alg 3 23

Functions and Coordinate Geometry

Homework Assignment #5– Graphing Absolute Value/Piecewise Functions

Sketch a graph of each of the following please.

(1) 3 x y −=

(2) 1 x 2 y +=

(3) 2 x y +−=

(4) 3 2 x y −−=

(5) ( )

≥+

<−−=

0 x if 1 x

0 x if 1 x xf

(6) ( )x 1 if x 1

f x x if 1 x 1

x 2 if x 1

− − ≤ −

= − − < < − ≥

Alg 3 24

Functions and Coordinate Geometry

Algebra Review Worksheet

(1) Find the domain of each of the following.

(a) ( )7 5x

4 3x xf

−+

= (b) ( ) 2 3x xf +=

(c) ( ) 8 7x xf += (d) ( )12 x x

1 x xf

2

2

−−

−=

(2) Given the function ( ) 3 5x x2 xf 2 −+−= , find each of the following.

(a) ( )0f (b) ( )2f −

(c) ( ) 2 x f + (d) ( )5f 3

(3) Write the equation of the line which satisfies each of the following.

(a) Passes through (−7 , −4) with slope 53

(b) Passes through (4 , −1) and (0 , 0)

(c) Passes through (3 , 5) and (−2 , 5)

(d) Passes through (3 , −6), and is parallel to the line 4x − 2y = 11

(e) Passes through (1 , 9), and is perpendicular to the line 5x + 3y = 2

(4) Find the distance between each of the following pairs of points. Find the midpoint of each segment.

(a) (5 , 8) and (1 , 2) (b) (−3 , 9) and (7 , −1)

(c) (8 , 6) and (3 , −6) (d) ( ) ( ) , and , 45

25

43

21 −

Alg 3 25

Functions and Coordinate Geometry

Answers

(1) (a) 57 x ≠ (b)

32 x −≥

(c) All Real Numbers (d) 3 x , 4 x −≠≠

(2) (a) −3 (b) −21

(c) 1 x3 x2 2 −−− (d) −84

(3) (a) 51

53 x y +=

(b) x y41−=

(c) y = 5

(d) y = 2x − 12

(e) 542

53 x y +=

(4) (a) distance = 132 , midpoint (3 , 5) (b) distance = 210 , midpoint (2 , 4)

(c) distance = 13 , midpoint ( )0 , 211 (d) distance = 22 , midpoint ( ) ,

41

23 −

Alg 3 26

Functions and Coordinate Geometry

(5) Triangle ∆ABC has vertices A(−2 , 9) , B(4 , 1) , C(2 , −3).

(a) Find the perimeter of ∆ABC.

(b) Write the equation of the longest side of ∆ABC

(c) Find the length of the median to the shortest side of ∆ABC.

(d) Write the equation of the perpendicular bisector of AC.

(6) Write the equation of the circle with center (2 , −5) and radius 4.

(7) Write the equation of the circle if the endpoints of a diameter are the points (5 , −2) and (−3 , 6).

(8) Write the equation of the circle whose center is (4 , 6) , if the graph is tangent to the

y-axis.

(9) Write the equation of the line which is tangent to the graph of the circle 20 y x 22 =+ at the point (4 ,

2).

(10) Sketch a graph of each of the following.

(a) 0 4 4y 2x y x 22 =−−++

(b) 0 3 4y 6x y x 22 =−+−+

(c) 0 4 4x y x 22 =+−+

(d) 0 7 8y y x 22 =+++

(e) y 2 x 1 5= + −

(f)

2x 1 if x 2

y 1x 1 if x 2

2

− <=

+ ≥

Alg 3 27

Functions and Coordinate Geometry

Answers

(5) (a) perimeter = 10 + 52 + 104

(b) y = −3x + 3

(c) 55

(d) 3 x y31 +=

(6) ( ) ( ) 16 5 y 2 x 22 =++−

(7) ( ) ( ) 32 2 y 1 x22 =−+−

(8) ( ) ( ) 16 6 y 4 x 22 =−+−

(9) 10 x2 y +−=

(10 (a) ( ) ( ) 9 2 y 1 x 22 =−++

(b) ( ) ( ) 16 2 y 3 x 22 =++−

(c) ( ) ( ) 0 0 y 2 x 22 =−+−

(d) ( ) ( ) 9 4 y 0 x 22 =++−