Weebly€¦ · Web viewHonors Math 2 Unit 2 Class PacketSanderson High School Honors Math 2Unit 2 Class PacketSanderson High School 17 1

Honors Math 2 Unit 2 Class Packet Sanderson High School

Day 1: Fred Functions

Warm-up

Use the graph to answer the questions below.

1. List three points that lie on this graph.

2. If x = 0, then y = _____.3. If x = 5, then y = _____.4. If x = 6, then y = _____.5. If x = – 4, then y = _____.6. If y = 3, then x = _____.7. If y = 4, then x = _____ or _____.

_______________

___________________________________________________________________________

To the right is a graph of a “Fred” function. We can use Fred functions to explore transformations in the coordinate plane.

I. Let’s review briefly.

1. a. Explain what a function is in your own words.

b. Using the graph, how do we know that Fred is a function?

2. a. Explain what we mean by the term domain.

b. Using the graph, what is the domain of Fred?

3. a. Explain what we mean by the term range.

b. Using the graph, what is the range of Fred?

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F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)


4. Let’s explore the points on Fred.

a. How many points lie on Fred? Can you list them all?

b. What are the key points that would help us graph Fred?

We are going to call these key points “characteristic” points. It is important when graphing a function that you are able to identify these characteristic points.

c. Use the graph of graph to evaluate the following.

F(1) = _____ F( –1) = _____ F(_____) = –2 F(5) = ______

II. Remember that F(x) is another name for the y-values.

Therefore the equation of Fred is y = F(x).x F(x)

–1

1

2

4

1. Why did we choose those x-values to put in the table?

Now let’s try graphing Freddie Jr.: y = F(x) + 4. Complete the table below for this new function and then graph Freddie Jr. on the coordinate plane above. y = F(x) + 4

x y

–1

1

2


2

4

2. What type of transformation maps Fred, F(x), to Freddie Jr., F(x) + 4? (Be specific.)

3. In y = F(x) + 4, how did the “+4” affect the graph of Fred? Did it affect the domain or the range?

III. Suppose Freddie Jr’s equation is: y = F(x) – 3. Complete the table below for this new function and then graph Freddie Jr. on the coordinate plane above.

y = F(x) – 3x y

–1

1

2

4

1. What type of transformation maps Fred, F(x), to Freddie Jr., F(x) – 3? Be specific.

2. In y = F(x) – 3, how did the “– 3” affect the graph of Fred? Did it affect the domain or the range?

IV. Checkpoint: Using the understanding you have gained so far, describe the affect to Fred for the following functions.

Equation Effect to Fred’s graph

Example: y=F(x) +

18Translate up 18 units

1. y = F(x) – 100

3


2. y = F(x) + 73

3. y = F(x) + 32

4. y = F(x) – 521

V. Suppose Freddie Jr’s equation is: y = F(x + 4).

1. Complete the table.

x x + 4 Y

–5 –1 1

1 –1

2 –1

4 –2

(Hint: Since, x + 4 = –1, subtract 4 from both sides of the equation, and x = –5. Use a similar method to find the missing x values.)

2. What type of transformation maps Fred, F(x), to Freddie Jr., F(x + 4)? (Be specific.)

3. In y = F(x + 4), how did the “+4” affect the graph of Fred? Did it affect the domain or the range?

VI. Suppose Freddie Jr’s equation is: y = F(x – 3). Complete the table below for this new function and then graph Freddie Jr. on the coordinate plane above.

1. Complete the table.

y = F(x – 3)x x – 3 y

–1

4


1

2

4

2. What type of transformation maps Fred, F(x), to Freddie Jr., F(x – 3)? (Be specific.)

3. In y = F(x – 3), how did the “ –3” affect the graph of Fred? Did it affect the domain or the range?

VII. Checkpoint: Using the understanding you have gained so far, describe the effect to Fred for the following functions.


Example: y=F(x + 18) Translate left 18 units

1. y = F(x – 10)

2. y = F(x) + 7

3. y = F(x + 48)

4. y = F(x) – 22

5. y = F(x + 30) + 18

VIII. Checkpoint: Using the understanding you have gained so far, write the equation that would have the following effect on Fred’s graph.


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Example: y=F(x + 8) Translate left 8 units

1. Translate up 29 units

2. Translate right 7

3. Translate left 45

4. Translate left 5 and up 14

5. Translate down 2 and right 6

IX. Now let’s look at a new function. Its notation is H(x), and we will call it Harry.Use Harry to demonstrate what you have learned so far about the transformations of functions.

1. What are Harry’s characteristic points?

__________________________________________

2. Describe the effect on Harry’s graph for each of the following.a. H(x – 2) _______________________________________________

b. H(x) + 7 _______________________________________________

c. H(x+2) – 3 _______________________________________________

3. Use your answers to questions 1 and 2 to help you sketch each graph without using a table.a. y = H(x – 2) b. y = H(x) + 7

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Honors Math 2 Unit 2 Class Packet Sanderson High Schoolc. y = H(x+2) – 3

X. Let’s suppose that Freddie Jr. is y = – F(x) 1. Complete the table.

y = – F(x)x F(x) y

–1 1 –1

1

2

4

2. What type of transformation maps Fred, F(x), to Freddie Jr., –F(x)? (Be specific.)

3. In y = – F(x), how did the negative coefficient of “F(x)” affect the graph of Fred? How does this relate to our study of transformations earlier this semester?

XI. Now let’s suppose that Freddie Jr. is y = F(–x) 1. Complete the table.

y = F(–x)x –x y

–1

1

2

7

F(x)


F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)F(x)


4

2. What type of transformation maps Fred, F(x), to Freddie Jr., F(–x)? (Be specific.)

3. In y = F(–x), how did the negative coefficient of “x” affect the graph of Fred? How does this relate to our study of transformations earlier this semester?

XII. Checkpoint: Harry is H(x) and is shown on each grid. Use Harry’s characteristic points to graph Harry’s children without making a table.

1. y = H(–x) 2. y = – H(x)

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Day 2: Fred Functions

Warm-up

The graph of f ( x )=x2−4 is shown. Answer the following.

1. What is the domain of f(x)?

2. What is the range of f(x)?

3. Evaluate:

a. f(2) = ________

b. f(–1) = ________

c. f(6) = ________

4. Fill in the blank.

a. If f(x) = –4, then x = ________.

b. If f(x) = 0, then x = ________ or ________.

c. If f(x) = –3, then x = ________ or ________.

d. If f(x) = 5, then x = ________ or ________.

__________________________________________________________________________________________

Today we will revisit Fred, our “parent” function, and investigate transformations other than translations and reflections. .

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Honors Math 2 Unit 2 Class Packet Sanderson High SchoolRecall that the equation for Fred is y = F(x).

x F(x)

Let’s suppose that Freddie Jr. is y = 4 F(x) 1. Complete the table.

y = 4 F(x)x F(x) y

–1

1

2

4

2. In y = 4 F(x), the coefficient of “F(x)” is 4. How did that affect the graph of Fred? Is this one of the transformations we studied? If so, which one? If not, explain.

XIII. Now let’s suppose that Freddie Jr. is y = ½ F(x). 1. Complete the table.

y = ½ F(x)x F(x) y

–1

1

2

4

10

F(x)

F(x)



Honors Math 2 Unit 2 Class Packet Sanderson High School2. In y = ½ F(x), the coefficient of “F(x)” is ½. How did that affect the graph of Fred?

XIV. Checkpoint: 1. Complete each chart below. Each chart starts with the characteristic points of Fred.

2. Compare the 2nd and 3rd columns of each chart above. The 2nd column is the y-value for Fred. Can you make a conjecture about how a coefficient changes the parent graph?

XV. Now let’s suppose that Freddie Jr. is y = –3 F(x). 1. Complete the table.

y = –3 F(x)x F(x) y

–1

1

2

4

2. Reread the conjecture you made in #7 on the previous page. Does it hold true or do you need to refine it? If it does need some work, restate it more correctly here.

XVI. Checkpoint: Let’s revisit Harry, H(x).4. Describe the effect on Harry’s graph for each of the following.

Example: –5H(x) Each point is reflected in the x-axis and is 5 times as far from the x-axis.

d. 3H(x) ____________________________________________________________________________

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x F(x) 3 F(x)

–1 1

1 –1

2 –1

4 –2

x F(x) ¼ F(x)

–1 1

1 –1

2 –1

4 –2


e. –2H(x) ____________________________________________________________________________

f.12H(x) ____________________________________________________________________________

5. Use your answers to questions 1 and 2 to help you sketch each graph without using a table.

b. y = 3H(x) b. y = –2H(x) c. y = 12H(x)

The graph of Dipper, D(x), is shown.

List the characteristic points of Dipper.

__________________________________________________

What is different about Dipper from the functions we have used so far?

Since Dipper is our original function, we will refer to him asthe parent function. Using our knowledge of transformational functions, let’s practice finding children of this parent.

Note: In transformational graphing where there are multiple steps, it is important to perform the translations last.

I. Example: Let’s explore the steps to graph Dipper Jr, 2D(x + 3) + 5, without using tables.

Step 1. The transformations represented in this new function are listed below in the order they will be performed. (See note above.)

Vertical stretch by 2 (Each point moves twice as far from the x-axis.) Translate left 3. Translate up 5.

Step 2. On the graph, put your pencil on the left-most characteristic point, (– 5, –1) .

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......................


Vertical stretch by 2 takes it to (– 5, –2). (Note that the originally, the point was 1 unit away from the x-axis. Now, the new point is 2 units away from the x-axis.)

Starting with your pencil at (– 5, –2), translate this point 3 units to the left. Your pencil should now be on (– 8, –2).

Starting with your pencil at (– 8, –2), translate this point up 5 units. Your pencil should now be on (– 8, 3).

Plot the point (– 8, 3). It is recommended that you do this using a different colored pencil.

Step 3. Follow the process used in Step 2 above to perform all the transformations on the other 3 characteristic points.

Step 4. After completing Step 3, you will have all four characteristic points for Dipper Jr. Use these to complete the graph of Dipper Jr. Be sure you use a curve in the appropriate place. Dipper is not made of segments only.

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Honors Math 2 Unit 2 Class Packet Sanderson High SchoolII. Dipper has another child named Little Dip, – D(x) – 4

Using the process in the previous example as a guide, graph Little Dip (without using tables).

1. List the transformations needed to graph Little Dip.(Remember, to be careful with order.)

_____________________________________________

_____________________________________________

2. Apply the transformations listed above to each of the four characteristic points.

3. Complete the graph of Little Dip using your new characteristic points from #2.

III. Dipper has another child named Dipsy, 3 D(– x)

Using the process in the previous example as a guide, graph Dipsy (without using tables).

1. List the transformations needed to graph Dipsy.(Remember, to be careful with order.)

_____________________________________________

_____________________________________________

2. Apply the transformations listed above to each of the four characteristic points.

3. Complete the graph of Dipsy using your new characteristic points from #2.

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.

.

.....................

.....................

Honors Math 2 Unit 2 Class Packet Sanderson High SchoolIV. Now that we have practiced transformational graphing with Dipper and his children, you and your

partner should use the process learned from the previous three problems to complete the following.

1. Given Cardio, C(x), graph: y = 3C(x) + 5

2. Given Garfield, G(x), graph: y = – G(x – 3) – 6

3. Given Horizon, H(x), graph y = – 3H(x)

15


4. Given Batman, B(x), graph: y = B(–x) + 8

5. Given Mickey, M(x), graph: y = −13 M(x)

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Days 3 & 4: Factoring

Warm-up

1. If the base of a triangle has a length of 8x units, and the height is units, write a simplified algebraic expression for the area of the triangle in terms of x.

2. A square has a side length of k. If the length of the square is increased by 6 units, and the width of the square is increased by 4 units to create a new, larger rectangle, write a simplified algebraic expression for the area of the new rectangle in terms of k.

3. Expand each of these expressions to an equivalent standard form quadratic expression.

a. (x+5)(4 x−3) b. (x+3)(x−3)

c. ( s−7 )2 d. (3 s+2 )2

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Factoring PracticeA. Factoring out the GCF:

a. 16 m2 n+12 mn2

b. 14a3 b3 c−21a2 b4 c+7 a2 b3 c

c. 7 xy+2ab

B. Factor by Grouping: for polynomials with 4 or more terms

a. a2 x+b2 x+a2 y+b2 y

b. 3 x3+2 xy−15 x2−10 y

c. 20 ab−35 b−63+36 a

C. Factoring Trinomialsa. When the leading coefficient is 1

i. x2+5 x+4

ii. x2+6 x−16

iii. x2−2x−63

iv. a2−9a+20

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v. x2+5 x+6

b. When the leading coefficient is not 1i. 3 x2−6 x−24

ii. 4 x2+7 x+3

iii. 6 n2+25 n+14

iv. 20 a2−21 a−5

v. 2 x2−24 x+72

vi. 4 x2−16 x+16

vii. 4 y2+36 yz+81 z2

D. Difference of “Two Squares”:a. x2−25

b. 16 x4−z4

19


c.19

x2− 425

y2

20


Extra Factoring Practice

21


Solving by Factoring

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Day 5: Characteristics of Quadratics

Warm-up

For each pair of polynomials, find the sum [f(x)+g(x)] and the difference [f(x)-g(x)].

1. f(x) = 3p2 - 2p + 3 and g(x) = p2 - 7p + 7

Sum:________________________ Difference: _________________________

2. f(x) = 7x2 - 8 and g(x) = 3x2 + 1

Sum:________________________ Difference: _________________________

3.

Sum:________________________ Difference: _________________________

4.

Sum:________________________ Difference: _________________________

5.

Sum:________________________ Difference: _________________________

6.

Sum:________________________ Difference: _________________________

Find the key features of the quadratic function represented below.

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Domain:

Range:

Interval of Increase:

Interval of Decrease:

Minimum Point:

Axis of Symmetry:

x-Intercepts:

y-Intercept:

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Introduction:

A __________________ written in _______________ looks like: y=ax2+bx+c The graph of a quadratic function is U-shaped and called a ________________. The highest or lowest point of the graph occurs at the ____________________. Parabolas have a line of symmetry (splits the graph in half) called the

_____________________.

Parabolas Have These Characteristics: If a > 0 then the parabola opens ____, but if a < 0 then the parabola opens

_____. The x-coordinate of the vertex is _________. The maximum or minimum of the graph is at the _________. The axis of symmetry is the vertical line ___________. The y-intercept is __________.

Example 1: Graphing a Quadratic Function in Standard Form

Graph y=2 x2−8 x+6

Steps:1. Find and plot the vertex.2. Look at “a” to see if it opens up or down.3. Draw the axis of symmetry4. Find and plot another point, then use symmetry

to plot another point5. Draw a parabola through those points.

You Try!

Graph y=x2−2 x−3

Day 6: Completing the Square25


Warm-Up

The Galaxy Sport and Outdoor Gear company has a climbing wall in the middle of its store. Before the store opened for business, the owners did some market research and concluded that the daily number of climbing wall customers would be related to the price per climb x by the linear function n ( x )=100−4 x .

a. According to this function, how many daily climbing wall customers will there be if the price per climb is $10? What if the price per climb is $15? What if the climb is offered to customers at no cost?

b. What do the numbers 100 and -4 in the rule of n ( x ) tell about the relationship between climb price and number of customers?

c. What is a reasonable domain for n(x ) tell about the relationship between climb price and number of customers?

Domain: __________________

d. What is the range of n(x ) for the domain you specified in Part c? That is, what are the possible values of n ( x ) corresponding to plausible inputs for the function?

Range: ___________________

e. If the function I (x) tells how daily income from the climbing wall depends on price per climb, why is I ( x )=100 x−4 x2 a suitable rule for that function?

Quadratics in vertex form make graphing parabolas a lot easier! It will also help us a lot as we learn more about parabolas.

But you aren’t usually given a quadratic in vertex form. You are usually given the quadratic in

standard form: ________________________How do you convert from the standard format to the vertex format?

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By using the technique of ________________________________

Steps: 1. Write in Standard Form. 2. “A” must be equal to 1! If it isn’t 1, factor out “A.”

3. Find (b2 )

2

4. Add the # above inside the parenthesis, subtract from the outside. 5. Rewrite the trinomial as a binomial squared. 6. Solve for x!

Examples:

1. y=x2+6 x+2 vertex form:________________min/max:___________________

2. y=x2−10 x−2 vertex form:________________min/max:___________________

3. y=2 x2−8 x+1 vertex form:________________min/max:___________________

4. y=−2 x2+6 x+1 vertex form:________________min/max:___________________

You Try!

5. y=x2−12 x+5 vertex form:______________min/max:_________________

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6. y=x2+4 x−4 vertex form:______________min/max:_________________

7. y=5 x2−10 x−8 vertex form:______________min/max:_________________

8. y=−x2+8 x+36 vertex form:______________min/max:_________________

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Day 7 – Completing the Square and Transformational Graphing

Quadratic Functions

The parent function for quadratic functions is ____________________

The _____________ is NOT a transformation

Example 1: Identify the transformations y= (x−3 )2+5

You Try 1: Identify the transformations

You Try 2: Identify the transformations

Domain:____________________

Range:______________________

Example 2: Find and graph the new characteristic points for the equation y= (x−3 )2+5

a. Determine the transformations that affect the x-values

b. Determine the transformations that affect the y-values

a. Graph the new points

New Domain:_______________

New Range:________________

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x y

x y


Example 3: Find and graph the new characteristic points for the equation y=3 x2+6 x



a. Graph the new points

New Domain:____________

New Range:______________

You Try 3: Find and graph the new characteristic points for the equation y=−2x2+4



30

x y

x y


b. Graph the new points

New Domain:_______________

New Range:________________

You Try 4: Find and graph the new characteristic points for the equation y=−2 x2+16 x−35

c. Determine the transformations that affect the x-values

d. Determine the transformations that affect the y-values

c. Graph the new points

New Domain:_______________

New Range:________________

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x y


Example 4: Write an equation for a quadratic function that is shifted right 6 and up 4

You Try 5: Write an equation for an quadratic function that is reflected over the x-axis, a vertical stretch of 2, and shifted up 6

32


Day 8: Graphing QuadraticsWarm-up:1. Farmer Bob is planting a garden this spring. He wants to plant squash,

pumpkins, corn, beans, and potatoes. His plan for the field layout in feet is shown in the figure below. Use the figure and your knowledge of polynomials, perimeter, and area to solve the following:

a. Write an expression that represents the length of the south side of the field.

b. Simplify the polynomial expression that represents the south side of the field.

c. Write a polynomial expression that represents the perimeter of the pumpkin field.

d. Simplify the polynomial expression that represents the perimeter of the pumpkin field. State one reason why the perimeter would be useful to Farmer Bob.

e. Write a polynomial expression that represents the area of the potato field.

f. Simplify the polynomial expression that represents the area of the potato field. State one reason why the calculated area would be useful to Farmer Bob.

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Making Graphs by HandIf you’re asked to graph of a quadratic function by hand, here’s a set of steps that will produce a reasonably accurate graph:

Use the a value (from any of the three forms) to get the shape: if a > 0, if a < 0. Find the zeros using the easiest method (factoring, square roots, completing the square, quadratic

formula). Provided that there are two zeros, this will give you two points to draw. Find the vertex (using vertex form or –b/2a or averaging the zeros). This usually gives a third point

to draw. Find the y-intercept (evaluate f(0), or just take the c value from ax2 + bx + c form). This usually gives

a fourth point to draw. Draw a dotted line for the axis of symmetry (vertical line through the vertex). Then you can often get

a fifth point by drawing the reflected image of the y-intercept point. Make sure that all the points you’ve found fit with the shape that you anticipated ( or ).

If everything looks OK, draw a parabola shape passing through the points you have.

Example: Make a graph of f(x) = x2 + 6x + 8.

The a value is 1, so expect the shape. Factoring gives the zeros: f(x) = (x + 2)(x + 4) so there are points (–2, 0) and (–4, 0). Completing-the-square leads to vertex form: f(x) = (x + 3)2 – 1 so the vertex is (–3, –1). f(0) = 8 so the y-intercept is at (0, 8). The axis of symmetry x = –3 shows that there’s a reflected image point at (–6, 8). Draw a graph using the five known points:

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2. Do all of the following for the function f(x) = 2x2 – 8x + 6, without using a calculator.

a. Using factoring, find the zeros.

b. Using completing-the-square, find the vertex.

c. Find the y-intercept.

d. Draw the 4 points found so far, use the axis of symmetry to find a 5th point, then sketch the graph of f(x).

35


3. Using the same sequence of steps as in problem 2, find five points then draw the graph for f(x) = –x2 + 6x – 8, without using a calculator.

36


Day 9: Graphing Quadratic Inequalities Warm-Up:

37


Graphing Quadratic Inequalities

Use the same techniques as we used to graph linear inequalities and quadratic equations.

Steps:1. Graph the boundary. Determine if it should be solid (___ or ___) or dashed

(___ or ___)2. Test a point in each region. 3. Shade the region whose ordered pair results in a true inequality.

Example 2: Graph y≤x2−6 x+2 Example 3: Graph y<−x2+3 x−4

You Try!

Graph y>2 x2+4 x−3 Graph y≤−x2+2 x+8

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Weebly€¦ · Web viewHonors Math 2 Unit 2 Class PacketSanderson High School Honors Math 2Unit 2 Class PacketSanderson High School 17 1

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