Lesson 6.1: Successive Differences in Polynomials

Learning Goals:

1) How can we determine what type of a function a given set of points is represented by?

2) How can we determine the equation for a given set of points?

Linear Functionsx y=2x+3 First Differences0 3 5−3=21 5 7−5=22 7 9−7=23 9 11−9=24 11 13−11=25 13

What do you notice?

1st difference ¿2 and a=2. y=mx+b y=ax+b

Let’s generalize this…

x ax+b First Differences0 b a+b−b=a1 a+b 2a+b−(a+b )=a2 2a+b 3a−2a=a3 3a+b

y=4 x−5 y=−10x+5

1st Difference a=4 1st Difference a=−10

Quadratic Functions

1

For a linear function the first difference is equivalent to a

Number Square x2 First Differences Second Differences

1 1 4−1=3 3−5=22 4 9−4=5 7−5=23 9 16−9=7 9−7=24 16 25−16=9 11−9=25 25 36−25=116 36

What do you notice? The second differences are all equal

Let’s generalize this…

¿ a x2+bx+c First Differences Second Differences0 c (a+b+c )−c=a+b (3a+b )−(a+b )=2a1 a+b+c (4 a+2b+c )−(a+b+c )=3a+b (5a+b )−(3a+b )=2a2 4 a+2b+c (9a+3b+c )−(4a+2b+c )=5a+b (7a+b )−(5a+b )=2a3 9a+3b+c (16a+4b+c )−(9a+3b+c )=7 a+b4 16a+4b+c

y=2x2 y=−5x2

2nd Difference ¿2a=2 (2 )=4 2nd Difference ¿2a=2 (−5 )=−10

Cubic Functions¿ a x3+b x2+cx+d First Differences Second

DifferencesThird Differences

0 d ¿a+b+c ¿6a+2b 6a1 a+b+c+d ¿7a+3b+c ¿12a+2b 6a

2

For a quadratic function the second difference is equivalent to 2a

What do you think will happen if we have a cubic function?

The third differences will be equal!

2 8a+4b+2c+d ¿19a+5b+c ¿18a+2b3 27a+9b+3c+d ¿37a+7 b+c4 64 a+16b+4c+d

y=3 x3 y=−5x3

3rd Difference ¿6 (3 )=18 3rd Difference ¿6a=6 (−5 )=−30

Example 1:

What type of relationship is indicated by the following set of ordered pairs? Look at the differences in the outputs! Explain how you know. Quadratic relationship because the second differences are the same

x y First Differences Second Differences

0 0 2−0=2 8−2=61 2 10−2=8 14−8=62 10 24−10=14 20−14=63 24 44−24=204 44

Find an equation that all ordered pairs above satisfy. y=3 x2−x+0

2a=6

a=3

Use your calculator! “regression” = relationship

stat→edit→L1 , L2

x−list :L1 y−list :L2

stat→calc→5:QuadReg

Example 2:

a) What type of relationship does the set of ordered pairs (x , y ) satisfy? How do you know?

x y First Difference Second Difference

Third Difference

Fourth Difference

3

For a cubic function the third difference is equivalent to 6a

0 2 1−2=−1 5−(−1)=6 12−6=6 Cubic Relationship

1 1 6−1=5 17−5=12 18−12=6 Because 3rd 2 6 23−6=17 35−17=18 24−18=6 Differences are3 23 58−23=35 59−35=24 All Equal4 58 117−58=595 117

b) What is the equation that satisfies the list of ordered pairs above? Round all values to the nearest tenth.

stat→calc→6 :cubic regression

y=x3+0 x2−2 x+2

y=x3−2x+2

Example 3:

Show that the set of ordered pairs ( x , y ) in the table below satisfied a quadratic relationship. Find the equation of the form y=a x2+bx+c that all of the ordered pairs satisfy.

x 0 1 2 3 4 5y 5 4 −1 −10 −23 −40

stat→calc→5:Quad Regression

y=−2x2+x+5

Homework 6.1: Successive Differences in Polynomials

1. Create a table to find the third differences for the polynomial s3−s2+s for integer values of s from −3 to 3. {Hint: Plug equation into calculator to get the corresponding y-values}

4

2. a) Show that the set of ordered pairs in the table below satisfies a cubic relationship. (Hint: Find the third differences)

b) Find the equation that satisfies the ordered pairs listed above. Round all coefficients to the nearest hundredth.

3. The distance d feet required to stop a car traveling at 10v mph under dry asphalt conditions is given by the following table.

a) What type of relationship is indicated by the set of ordered pairs?

5

b) Assuming that the relationship continues to hold, find the distance required to stop the car when v=6.

4. Show that the set of ordered pairs in the table below satisfies a quadratic relationship. Find the equation of the form y=a x2+bx+c that all ordered pairs satisfy.

Lesson 6.2: Linear Systems in Three Variable

Learning Goals:

1) What is a linear system of equations?2) How do we solve a linear system of equations in three variables?

6

Review: Solve the following system of equations algebraically for x and y.

1. 5 x+3 y=7

2 x+ y=3

Solutions are the intersection of the graphs!

Do by elimination method:

5 x+3 y=7→5 x+3 y=7 now sub in:5 (2 )+3 y=7

−3(2 x+ y=3)→−6 x−3 y=−9 10+3 y=7

−x=−2→x=2 3 y=−3→ y=−1

x=2∧ y=−1

2. A scientist wants to create 120 ml of a solution that is 30% acidic. To create this solution, she has access to a 20% solution and a 45% solution. How many milliliters of each solution should she combine to create the 30% solution?

Create a system of equations

x=ml of 20%solution x+ y=120

y=mlof 45% solution .20 x+.45 y=.30(120)

Do by elimination method:

−.45 ( x+ y=120 )→−.45 x−.45 y=−54 sub in: 72+ y=120

.20 x+ .45 y=.30(120)→ .20 x+.45 y=36 y=48

−25 x=−18→x=72

How does the process change when we try to solve for three variables?

3. Determine the values for x , y ,∧z in the following system?

2 x+3 y−z=5

4 x− y−z=−1

x+4 y+z=12

1) Create two different systems of two equations7

2) Eliminate the same variable from each new system3) With resulting equations, create a new system and eliminate another

variable4) Substitute answer back into other equations to find other variables

Eliminate the z : Eliminate the z :

2 x+3 y−z=5 4 x− y−z=−1

x+4 y+z=12 x+4 y+z=12

3 x+7 y=17 5 x+3 y=11

Now eliminate the x :

5 (3 x+7 y=17 )→15 x+35 y=85

−3(5 x+3 y=11)→−15 x−9 y=−33

26 y=52→ y=2

3 x+7 (2 )=17

3 x+14=17 2 (1 )+3 (2 )−z=5

3 x=3→x=1 2+6−z=5

8−z=5→z=3

4. 2 x+ y−z=−5

4 x−2 y+z=10

2 x+3 y+2 z=3

Eliminate the z :

2 x+ y−z=−5 2 (2x+ y−z=−5 )→4 x+2 y−2 z=−10

4 x−2 y+z=10 2 x+3 y+2 z=32 x+3 y+2 z=3

6 x− y=5 6 x+5 y=−7

8

Eliminate the x : Sub in for x: Sub in for last variable (z ):

6 x− y=5 6 x−(−2 )=5 2( 12 )+(−2 )−z=−5

6 x+5 y=−7 6 x=3→x=12 1−2−z=−5

−6 y=12→ y=−2 −1−z=−5→z=4

5. x−2 y+3 z=9

−x+3 y=−4

2 x−5 y+5 z=17

Eliminate the z:

5 ( x−2 y+3 z=9 )→5 x−10 y+15 z=45

−3(2 x−5 y+5 z=17)→−6 x+15 y−15 z=−51

−x+5 y=−6

Eliminate the y : Sub in for x: Sub in for last variable ( z ) :

−1 (−x+3 y=−4 ) −x+3 (−1 )=−4 1−2 (−1 )+3 z=9

−x+5 y=−6 −x=−1→x=1 1+2+3 z=9

2 y=−2→y=−1 3 z=6→z=2

Solving a System of Equations in Three Variables:

Pick any two pairs of equations from the system. Eliminate the same variable from each pair using the elimination method.

Remember to look for opposites to eliminate a variable. Solve the system of the two new equations using the elimination method.

Remember to look for opposites to eliminate a variable. Substitute the solution back into one of the original equations and solve for

the third variable.

9

When you solve for a system of 3 linear equations, what does the solution set represent? A point

Homework 6.2: Linear Systems in Three Variable

Solve each of the following systems of equations.

1. 2 x+3 y=3

12 x−15 y=−4

10

2. A chemist needs to make 40 ml of a 15% acid solution. He has a 5% acid solution and a 30% acid solution on hand. If he uses the 5% and 30% solutions to create the 15% solution, how many ml of each does he need?

3. x+2 y+3 z=5

3 x+2 y−2 z=−13

5 x+3 y−z=−11

Lesson 6.3: Linear Systems in Three Variable

Learning Goals:

1) What is a linear system of equations?2) How do we solve a linear system of equations in three variables?

1. x+ y+z=2

11

5 x+5 y+5 z=3

4 x+ y−3 z=−6

−5 ( x+ y+ z=2 )−5 x−5 y−5 z=−10

5 x+5 y+5 z=35 x+5 y+5 z=3

0≠−7 FALSE so NO SOLUTIONS

2. x− y+z=−3

x− y−z=−3

5 x−5 y+z=−15

x− y+z=−3 x− y−z=−3 −3(2 x−2 y=−6)

x− y−z=−3 5 x−5 y+z=−15 6 x−6 y=−18

2 x−2 y=−6 6 x−6 y=−18 0=0 (infinite solutions)

12

What is different in this system of equations that you have not seen before? What

does it mean?No Solutions.

Practice: ¿edit →❑x❑ ,then 2nd Quit→rref

3. a+3b+c=3

2a−3b+2c=3

−a+3b−3c=1

a=3b=13c=−1

13

What is different in this system of equations that

you have not seen before? What does it

mean?Infinite Solutions.

4. x+ y+z=−1

4 x+4 y−4 z=−2

3 x+2 y+z=0

x=74∨1.75 y=−5

2∨−2.5 z=−1

4∨−.25

5. p+q+3 r=4

2q+3 r=7

p−q−r=−2

p=25q=5 r=1

6. −2 x−3 y+ z=−6

x+ y−z=5

7 x+8 y−6 z=31

Infinite many solutions!

Homework 6.3: Linear Systems in Three Variable

Solve each of the following systems of equations algebraically.

1. 3 x−5 y+z=9

x−3 y−2 z=−8

5 x−6 y+3 z=15

2. 4 x−3 y+2 z=12

14

x+ y−z=3

−2 x−2 y+2 z=5

3. 4 x+2 y−z=21

−x−2 y+2 z=13

3 x−2 y+5 z=70

4. x+3 y−z=2

x+ y−z=0

3 x+2 y−3 z=−1

Lesson 6.4: Transformations of Functions

Learning Goal: What is a translation and how do we represent that translation of a function in its equation?

Parent Functions

Functions that belong to the same family share key characteristics. The parent function is the most basic function in a family. Functions in the same family are transformations of their parent function. Below are the four parent functions we will work with in this unit. (There are others!)

15

Quadratic Absolute Value Cubic Square Root

y=x2 y=|x| y=x3 y=√x

Using your calculator, sketch the graph of each of the equations below. Explain what happened to the parent function of f ( x )=x2 in each graph.

f ( x )=x2+3 f ( x )=x2−3 f (x )=( x+3 )2 f ( x )= (x−3 )2

Moved up 3 Moved down 3 Moved left 3 Moved right 3

Summary of Translations f ( x )= y

f ( x )+a Vertical Up af ( x )−a Vertical Down af (x+a) Horizontal Left af (x−a) Horizontal Right a

It is a vertical or horizontal movement of a graph

f ( x−a )+b

a is the horizontal (opposite )∧b isthe verticalmovement

16

Example 1: State the transformation that occurs and then sketch the graph of the function without using your calculator.

a) f ( x )=|x−1|+4 Parent: y=|x| b) g ( x )=√ x+2−1 Parent: y=√x

Horizontal: Right 1 Horizontal: Left 2

Vertical: Up 4 Vertical: Down 1

c) h ( x )=( x+3 )3+2 Parent: y=x3 d) f ( x )= (x−5 )2 Parent: y=x2

17

Horizontal: Left 3 Horizontal: Right 5

Vertical: Up 2 Vertical: Stays in same spot

a) b) c) d)

e) g ( x )=|x−2|−3 Parent: y=|x| f) h ( x )=√ x+4 Parent: y=√x

Horizontal: Right 2 Horizontal: Stays in same spot

Vertical: Down 3 Vertical: Up 4

18

Example 2:

a) Given the graph of f (x), graph

g ( x )=f ( x−3 )+1. State the value of g (0 ) .

g ( x )is a translationof f ( x)

Horizontal: Right 3

Vertical: Up 1

g (0 )=2

b) Given the graph of g(x ), graph

h ( x )=g ( x+2 )−3. State the value of h(−2)

h ( x ) is a translationof g(x )

Horizontal: Left 2

Vertical: Down 3

h (−2 )=−3

19

c) Given the graph of f (x), graph

g ( x )=f ( x−2 )−1. State the value of g(3).

g ( x )is a translationof f ( x)

Horizontal: Right 2

Vertical: Down 1

g (3 )=2

Example 3: State the parent function and the translation that is occuring in each of the given functions.

Function Parent Function Translationsf ( x )=x2−10 y=x2 Down 10g ( x )=|x+2| y=|x| Left 2

h ( x )=√ x−1+6 y=√x Right 1, Up 6f ( x )= (x−4 )3−5 y=x3 Right 4, Down 5

Example 4: If the parabola is f (x) and the vertex is (−2 ,5), state the vertex after the given translations:

a) f ( x−2 )+2 (−2 ,5) b) f ( x )−5 (−2 ,5)

Right 2, Up 2 +2 ,+2 Down 5 −5

(0 ,7) (−2 ,0)

c) f ( x+1 )−3 (−2 ,5) d) f (x−6) (−2 ,5)

Left 1, Down 3 −1 ,−3 Right 6 +6 ,

(−3 ,2) (4 ,5)

20

Homework 6.4: Transformations of Functions

1. Sketch the graph of 2. Sketch the graph of

f ( x )=√x−1+4 f ( x )=|x+3|−2

3. Given the function f (x) shown below, 4. If a quadratic function f (x) has

create a graph for h ( x )= f (x )+2. a turning point at (−3 ,7) then

Find the value of h (2 ) . where does the quadratic function g defined by

g ( x )=f ( x+4 )+5 have a

turning point?

(1) (−7 ,12)(2) (−4 ,5)(3) (1 ,12 )(4) (4 ,5)

21

5. State the parent function and transformation that is occurring in each function below in the order it is occurring.

Function Parent Function Translationsf ( x )=x2+3

g ( x )=|x−2|−1h ( x )=√ x−1+6f ( x )= (x+4 )3

Lesson 6.5: Reflections of Graphs

Learning Goal: What are reflections and how do we represent the reflection of a graph in its equation?

Warm-Up: State the transformation that is occurring, then sketch the graph of the function without using your calculator.

a) f ( x )= (x−3 )2+2 b) g ( x )=√ x+1−3

y=x2 y=√x

Right 3, Up 2 Left 1, Down 3

22

f ( x−a )+bwhere a is thehorizontal translaton∧b is the vertical translation

r x−axis=negate y

r y−axis=negate x

Example 1

a) Given the graph of f (x) below, sketch the graph of f (−x).

23

REFLECTING FUNCTIONS IN THE X AND Y AXESThe function −f (x) is a reflection of f (x) in the x-axis.The function f (−x) is a reflection of f (x) in the y-axis.

b) Given the graph of g(x ) below, sketch the graph of −g(x ).

c) Given the graph of f (x) below, sketch the graph of −f (x).

24

d) Given the graph of g(x ) below, sketch the graph of g(−x ).

Example 2: After a reflection in the y-axis, the quadratic function

g ( x )=4 x2−7 x+2 would have the equation

(1) y=−4 x2+7 x+2 r y−axis=negate x or x→−x

(2) y=−4 x2+7 x−2 y=4 x2−7 x+2

(3) y=4 x2+7 x+2 y=4 (−x )2−7 (−x )+2

(4) y=4 x2+7 x−2 y=4 x2+7 x+2

When doing a combination of translations and reflections, the following order should be used:

1) Horizontal translation

2) Reflection

3) Vertical translation

Example 3: State the transformations in the order in which they occur.

Functions Transformation H – R - Vf ( x )=−g ( x−2 )+3 Right 2

Reflect x- axisUp 3

f ( x )=g (−x )−5 Reflect y- axis25

Down 5f ( x )=g (−x+2) Left 2

Reflect y-axisf ( x )=−g ( x−4 )+1 Right 4

Reflect x-axisUp 1

f ( x )=−g (−x+1 )−4 Left 1Reflect x-axisReflect y-axisDown 4

Example 4: Write the equation of the following functions, given the parent function and the transformation performed.

a) f ( x )=x2, reflected in the y-axis, translated 5 units to the right and 3 units down.

Fill in the blanks of parent graph: f ( x )=¿¿¿

f ( x )=(−x−5)2−3

b) f ( x )=|x|, reflected in both the x-axis and y-axis, translated 11 units to the left and 4 units up.

Fill in the blanks of parent graph: f ( x )=¿¿

f ( x )=−|−x+11|+4

c) g ( x )=√ x, reflected in the x-axis, translated 21 units to the right.

Fill in the blanks of parent graph: f ( x )=¿√¿¿ x¿¿

f ( x )=−√x−21

Example 5: For each of the following functions, state the transformation and sketch the graph without using your calculator.

26

a) f ( x )=−|x+3|−4 b) g ( x )=√−x−2+3

c) h ( x )=−(−x−1 )3

27

d) f ( x )=−( x+2 )2+1

Homework 6.5: Reflections of Graphs

1. Write the equation of the following function, given the parent function and the transformation performed.

f ( x )=x3, translated right 2 units, reflected in the x-axis, and translated down 4 units.

2. Write the equation of the parent function f ( x )=|x| that has been translated 9 units to the left and reflected in the y-axis.

28

State the order of the transformations occurring and sketch the graph of

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

5. If f ( x )=−2x2+5 x−3 and g(x ) is a reflection of f (x) across the y-axis, then an equation of g is which of the following?

(1) g ( x )=−2 x2−5x−3

(2) g ( x )=−2 x2+5 x+3

(3) g ( x )=2 x2+5 x−3

(4) g ( x )=2 x2+5 x+3

6. State the order of the transformations that are occurring in each of the given equations.

Function Transformationsf ( x )= (−x+2 )2−1g ( x )=−|x−5|+1h ( x )=−√−x+3+2

29

f ( x )=−( x−4 )3

Lesson 6.6: Dilations of Functions

Learning Goals:

1) What are dilations and how do we represent a dilation of a function in its equation?

2) How can we state the transformation of a graph given its equation in the correct order?

3) How can we graph a function given its equation using our knowledge of transformations?

4) How can we write the equation of a function given the parent function and transformations performed?

Warm-Up:

a) The graph of f (x) is given below. b) The graph of f (x) is given below.

Sketch the graph of g ( x )=f ( x+1 )−4 Sketch the graph of h ( x )=−f ( x)+2.

30

Really means multiply y by scale factor.

31

VERTICAL SCALINGThe function h ( x )=k ∙ f (x) represents the vertical scaling of a function by a

factor of k.{Compression when 0<k<1 and stretch when k>1}

HORIZONTAL SCALING

The function f (kx ) represents a horizontal scaling of f ( x ) by a factor of 1k . {compression}

The function f ( 1k x ) represents a horizontal scaling of f (x) by a factor of k.

Really means multiply x by scale factor.

Scale factor is reciprocal of what you see in the question.

Example 1:

a) Consider the graph below. b) Consider the graph below.

Sketch the graph of g ( x )=12f ( x). Sketch the graph of g ( x )=f (12 x) .

32

Example 2:

a) The graph of g(x ) is shown on the grid b) The graph of f (x) is shown on

below. Sketch the graph of 2 f ( x ). the grid below. Sketch a graph of f (2 x ) .

33

Summary of Transformations

Function Transformationf ( x )+a Vertical translation, up af ( x )−a Vertical translation, down af (x+a) Horizontal translation, left af (x−a) Horizontal translation, right a−f (x) Reflection, negate y, r x−axisf (−x) Reflection, negate x, r y−axisk ∙ f ( x )∨1

k∙ f (x) Vertical, Dilation, multiply y by either k∨1k

f ( k ∙ x )∨f ( 1k ∙ x) Horizontal, Dilation, multiply x by either k∨1k

Performing multiple transformations:

If a function has multiple transformations, they are applied in the following order:

H orizontal translation

D ilation

R eflection

V ertical translation

For example: f ( x )=−( x−2 )3−4; this function will be translated RIGHT 2, then REFLECTED over the x-axis, and then translated DOWN 4.

34

Example 4:

State the parent function and the order in which the transformations are occurring in each function.

Function Transformation H D R Vf ( x )= (2x+1 )2−4 H: Left 1,

D: Multiply x by 12,R:V: Down 4

g ( x )=14|−x|+2 H:

D: Multiply y by 14 ,R: r y−axis,V: Up 2

h ( x )=−3√ x−2−1 H: Right 2,D: Multiply y by 3R: r x−axisV: Down 1

Example 5:

a) Write the equation of the function whose parent function is f ( x )=x2 and has been translated 3 units to the right, reflected over the x-axis, horizontally scaled by a factor of 2, and translated 4 units up.

? g ( x )=−( 12 x−3)2

+4

b) Write the equation of the function whose parent function is f ( x )=√x and has been vertically scaled by a factor of 2, horizontally scaled by a factor of 5, and translated 10 units to the left.

? g ( x )=2√ 15 x−10c) Write an equation of the function whose parent function is f ( x )=|x| and has been vertically scaled by a factor of 7, reflected in the x-axis, reflected in the y-axis, translated 5 units to the right and translated 2 units up.

? g ( x )=−7|−x−5|+2

35

Homework 6.6: Dilations of Functions

1. The graph of f ( x )=10−x2 represents the graph of f ( x )=x2 after

(1) a vertical shift upwards of 10 units followed by a reflection in the x-axis.

(2) a reflection in the x-axis followed by a vertical shift of 10 units upwards.

(3) a leftward shift of 10 units followed by a reflection in the y-axis.

(4) a reflection across the x-axis followed by a rightward shift of 10 units.

2. Which of the following equations represents the graph shown below?

(1) y= (x+3 )2+4

(2) y=−( x+3 )2+4

(3) y=−( x−3 )2+4

(4) y= (−x−3 )2−4

3. Consider the graph of f (x) below. Sketch the graph of 2 f (x ) on the same set of axes.

36

4. Write the equation of the function whose parent function is f ( x )=x2 and has been vertically stretched by a factor of 3, translated 10 units to the left, and reflected in the y-axis.

5. Write the function whose parent function is f ( x )=|x| and has been horizontally scaled by a factor of 5, translated 6 units to the right, reflected in the x-axis, and translated 2 units down.

6. Consider the function f (x) graphed on the grid.

a) Sketch the graph of g ( x )=f (12 x) on the

same set of axes.

b) How would you describe its graph compared to the graph of f ( x ) .

7. State the order of the transformations being performed in the given equations.

Function Translationsf ( x )=−x2+3

g ( x )=2|−x−2|−1

h ( x )=−√ 12 x−1+6Lesson 6.7: The Definition of a Parabola

37

Learning Goals:

1) What is the definition and properties of a parabola?2) What are the equations of a parabola?3) How can we sketch the graph of a parabola given its equation?

Warm-Up: State the parent function and the translation that is occurring in each of the given functions.

Function Parent Function Translationsf ( x )= (x−4 )2−5

Vertex ¿ (4 ,−5 )y=x2

Vertex ¿(0 ,0)Right 4Down 5

On the following diagram, draw and label the vertex and axis of symmetry.

Discuss with a partner: Suppose you are viewing the cross-section of a flat mirror and a parabola-shaped mirror. Where would the incoming light be reflected in each type of design? Sketch your ideas below.

38

Parabola

Geometry Review: When given a point in a plane, what is the set of all points that are equidistant from that point? Demonstrate by drawing a picture.

Definition of a Parabola: When given a line and a point not on the line, the set of all points in the plane that are equidistant from the point and the line is called a parabola.

39

FOCUS: is the point located p units away from the vertex inside the parabola on the axis of symmetry.

DIRECTRIX: is the line perpendicular to the axis of symmetry located p units from the vertex.

VERTEX FORM EQUATIONS OF PARABOLAS

Parabolas can be horizontal or vertical. In all parabolas, the coordinate of the vertex is located at (h , k ). The standard form for the equation of a parabola is y=a x2+bx+c. In this

form, it is difficult to graph the parabola without the use of a graphing calculator.

The vertex form for the equation of a parabola is y=a(x−h)2+k , where a= 14 p

. In this form, it is eacy to graph the parabola by hand. In all parabolas, the variable p represents the distance from the vertex to

the focus OR the distance from the vertex to the directrix.

40

SUMMARY

Before you can determine the properties of a parabola, how must you express its equation? Vertex Form = non-squared variable (x∨ y ) is isolated.

When given the equation, how can you tell which direction a parabola will open?

Horizontal x=¿ and vertical y=¿

look at sign (+ or -) of coefficient to determine specific direction!

When given the equation, what are the coordinates of the vertex? Vertex ¿(h , k)

What does the distance p represent on your parabola?

P is the distance from vertex to focus. P is the distance from vertex to directrix.

41

Directions: Identify the vertex and the distance of p for each parabola. Also identify which direction the parabola opens.

Vertical Parabolasy=± 1

4 p(x−h)2+k

Horizontal Hyperbolasx=± 1

4 p( y−k)2+h

a) x−3=−18y2 b) −16 y=(x−1)2

x=−18y2+3 −16 y

−16=(x−1)2

−16

x=−18

( y+0 )2+3 y=−116

( x−1 )2+0

Opens left V=(3 ,0) Opens down V=(1 ,0)

4 p=8 4 p=16

p=2 p=4

Based on the lesson so far, how would you graph the parabolas below?

Vertical Parabolasy=± 1

4 p(x−h)2+k

Horizontal Hyperbolasx=± 1

4 p( y−k)2+h

1. Sketch 12 ( x+2 )=( y+3)2 on the grid below and label the focus and directrix.

12(x+2)12

=( y+3)2

12

x+2= 112

( y+3 )2 Opens right with a vertex at V= (−2 ,−3 )

x= 112

( y+3)2−2

4 p=12 2 p=2 (3 )=6

p=3

42

How do you find the width of the parabola?

Vertical Parabolasy=± 1

4 p(x−h)2+k

Horizontal Hyperbolasx=± 1

4 p( y−k)2+h

2. Sketch −8 ( y+1 )=(x−2)2 on the grid below and label the focus and directrix.

−8( y+1)−8

=−18

(x−2)2

Opens Down with a Vertex at V=(2 ,−1)

y+1=−18

( x−2 )2

y=−18

( x−2 )2−1

2 p=2 (2 )=4

43

4 p=8

p=2

Summary of How to Graph Parabolas

1. Express parabola in vertex form.

Vertical Parabolasy=± 1

4 p(x−h)2+k

Horizontal Hyperbolasx=± 1

4 p( y−k)2+h

2. Locate the vertex (h , k ) on your graph.

3. Decide whether the parabola opens up, down, left, or right.

4. To locate the axis of symmetry, draw a line through the vertex going in the same direction the parabola opens.

5. To locate the focus, count p units from the vertex on the axis of symmetry in the same direction the parabola opens {see picture below}.

6. To locate the directrix, count p units from the vertex on the axis of symmetry in the opposite direction of the focus {see picture below}.

44

7. Plot two points located 2 p units from the focus in a direction that is parallel to the directrix {see picture below}.

8. To draw the parabola start at the vertex and draw to each endpoint of the width.

Homework 6.7: The Definition of a Parabola

1. Sketch the parabolas below, labeling its focus and directrix.

a) y=−14x2+1 b) x= 18 ( y−2)2

45

c) y−1=−14x2 d) −8 ( x+1 )=( y−2)2

e) Focus: (2 ,4) Directrix: y=−2 f) Focus: (0 ,5) Directrix: x=−1

46

Lesson 6.8: Equations of Parabolas and Congruent Parabolas

Learning Goals:

1) How can we write the equation of a parabola given the focus and directrix?2) How can we determine the equation of a congruent parabola?

Do now: Answer the following questions to help prepare for today’s lesson.

1. What is the definition of a parabola? (Use the diagram to help with your explanation)

A parabola is the set of all points in a plane that are equidistant from a point (the focus) and a line (the directrix).

47

2. What is the vertex form of the equation of a vertical parabola?

y=± 14 p

(x−h)2+k

3. What is the vertex form of the equation of a horizontal parabola?

x=± 14 p

( y−k)2+h

4. What do the values of (h , k ) and p represent about a parabola?

(h , k )=vertex

p=distance¿vertex ¿ focus∨vertex ¿directrix

Practice Identifying the Vertex

y= 124x2 x=−1

8y2+3 x= 1

20( y−1)2

y= 124

¿ x=−18

( y−0)2+3 x= 120

( y−1)2+0

V=(0 ,0) V=(3 ,0) V=(0 ,1)

y=−14

( x−2 )2−2 y= 112x2+5 x=−1

16( y+6 )2−3

V=(2 ,−2) V=(0 ,5) V=(−3 ,−6)

Vertical Parabolasy=± 1

4 p(x−h)2+k

Horizontal Hyperbolasx=± 1

4 p( y−k)2+h

48

How To Write the Equation of a Parabola

1. Draw a picture2. Find the vertex (h , k )3. Determine the value of p4. Determine which way parabola opens5. Write the equation

Directions: Write the equation of the following parabolas in vertex form.

1. x=± 14 p

( y−k)2+h

opens right! V= (−6 ,0 ) p=2

x=18( y−0)2−6

x=18y2−6

2. x=± 14 p

( y−k)2+h

opens left! V= (0 ,0 ) p=3

x=−112y2

3. Vertex: (2 ,4) Focus: (2 ,0) 4. Vertex: (0 ,0) Directrix: x=3

Draw a rough sketch Draw a rough sketch

opens down! Opens left!

y=± 14 p

(x−h)2+k x=± 14 p

( y−k)2+h

V= (2,4 ) p=4 V= (0 ,0 ) p=3

y=−116

( x−2 )2+4 x=−112

( y−0 )2+0∨x=−112

y2

49

3. 4. 5. 6.

5. Focus: (0 ,5) Directrix: y=−1 6. Focus: (4 ,−1) Directrix: x=−4

Draw a rough sketch Draw a rough sketch

opens up! opens right!

y=± 14 p

(x−h)2+k x=± 14 p

( y−k)2+h

V= (0 ,2 ) p=3 V= (0 ,−1 ) p=4

y= 112

(x−0)2+2∨ y= 112x2+2 x= 1

16( y+1 )2+0∨x= 1

16( y+1)2

7. Find the equation of the set of points which are equidistant from (4 ,−2) and the line y=4.

They do not use the words focus, directrix, vertex or even parabola!!

We know to write the equation of a parabola because the question is the definition of a parabola.

y=−112

(x−4 )2+1

V= (4 ,1 ) p=3

Geometry Review: When are two graphs considered congruent to each other?

When they are the same size and shape

50

1. Determine which equation represent the graph of a parabola that is congruent to the parabola shown to the right.

a. y= 120x2

b. y= 110x2+3

c. y=−120x2+8

d. y=15 x2+5

e. x= 110y2

f. x=15 ( y−3)2

g. x= 120y2+1

Look at the p distance. p=5. 14 p

= 14 (5)

= 120 ANSWERS: A, C, G

51

Congruent Parabolas All parabolas with the same distance between the vertex and focus ( p)

will be congruent. For example, the following parabolas are congruent:y= 14x2 y=−1

4(x+3)2 y= 14 x

2+3 y=−14

( x−1 )2+2

x= 14y2 x=−1

4( y+3)2 x=14 y

2+3 x=−14

( y−1 )2+2

These all have p=1 Why would having the same distance ( p) make any two parabolas

congruent?the width of the parabolas will be equal, making parabolas congruent

How to Write the Equation of Congruent Parabolas

When are two parabolas congruent to each other?

Same p distance.

2. Write the equation for two different parabolas that are congruent to the parabola with focus point (0 ,3) and directrix line y=−3.

p=3

y= 112x2+1

x= 112y2+1

x=−112y2+6

3. Parabola P is given below.

Which equation below is congruent to parabola P?

(1) x=−13y2 (2) y= 1

12(x+1)2 (3) x+3=16 y

2 (4) y=−13x2

52

4. Parabola A is given below. Write an equation of a parabola that is congruent to parabola A, but with a vertex at (2 ,5).

y= 14(x−2)2+5

y=−14

(x−2)2+5

Equal p-values!

p=1

V=(2 ,5)

Homework 6.8: Equations of Parabolas and Congruent Parabolas

1. Sketch the parabolas below, labeling its focus and directrix.

a) y=18 x2 b) x=−1

12( y−2)2

53

c) −16 ( y+2 )=x2 d) Vertex: (0 ,0) Focus: (0 ,−3)

2. Write the equation of the following parabolas in vertex form.

a)

54

b) Vertex: (5 ,0 ) Focus: (8 ,0 ) c) Vertex: (−1 ,−3) Directrix: x=4

3. Let P be the parabola with focus (0 ,0) and directrix y=−2.

a) Write an equation whose graph is a parabola congruent to P with the focus (0 ,4 ) .

b) Write an equation whose graph is a parabola congruent to P with the same directrix y=−2, but different focus.

4. The vertex of a parabola, f (x), has coordinates (−3 ,5). Determine the coordinates of the vertex of the parabola defined by f (x+3)

(1) (−3 ,8 ) (2) (−6 ,2 ) (3) (0 ,5 ) (4) (−6 ,5)

55