apex algebra 2 learning packet - charles county public …algebra 2: weeks 3-4, april 20 – may 1...

Algebra 2: Weeks 3-4, April 20 – May 1

CHARLES COUNTY PUBLIC SCHOOLS

APEX Algebra 2 Learning Packet


Student: _________________________________ School: _____________________________

Teacher: _________________________________ Block/Period: ________________________

Packet Directions for Students Week3

Read through the Instruction and examples on Shifting Functions while completing the corresponding questions on the 4.4.1 Study: Shifting Functions worksheet

Complete 11.4.1 Study: Shifting Functions o Check and revise solutions using the 11.4.1 Study: Shifting Functions Answer Key

Complete Quiz – Shifting Functions

Week 4

Read through the Instruction and examples on Shifting Functions while completing the corresponding questions on the 4.4.1 Study: Shifting Functions worksheet

Complete 11.4.1 Study: Shifting Functions o Check and revise solutions using the 11.4.1 Study: Shifting Functions Answer Key

Complete Quiz – Shifting Functions


Shifting Functions

If you slid your pencil to another spot on your desk, would its shape change? Would it get longer or

shorter? Would it suddenly become an eraser or a highlighter?

The answer to all of these questions, of course, is no. Simply moving an object from one place to

another doesn't change its shape.

The same is true for functions. When you move a function, you actually create another function in the

same family — one that has the same shape as the original.

Shifting Functions Up and Down

There are two ways to shift the graph of a function: vertically (up or down) and horizontally (left or

right). We'll look at vertical shifts first.

Remember, when you shift a graph, you change its location, not its shape.

Shifting Up — Coordinates

Shifting a graph vertically moves it along the y-axis.

To shift a graph up, you must increase the value of the y-coordinate for each of the function's ordered

pairs. In other words, you increase each of the function's output values.

Shifting Up — The Equation

When you shift the graph of a function up, you increase all of the output values by a certain positive

number.

This is the same as adding that number to the right-hand side of the function's equation. That's because

the right-hand side of the equation is the same thing as the function's output.

For example, Suppose F(x) = x2. To shift its graph up by 1 unit, add 1 to the right-hand side of the

equation. The new function is G(x) = x2 + 1.

For Example:

A function transformation takes whatever is the basic function f (x) and then "transforms" it (or

"translates" it), which is a fancy way of saying that you change the formula a bit and thereby move the

graph around.

For instance, the graph for y = x2 + 3 looks like this:

This is three units higher than the basic quadratic, f (x) = x2. That is, x2 + 3 is f (x) + 3. We added a "3"

outside the basic squaring function f (x) = x2 and thereby went from the basic quadratic x2 to the

transformed function x2 + 3.

This is always true: To move a function up, you add outside the function: f (x) + b is f (x) moved

up b units. Moving the function down works the same way; f (x) – b is f (x) moved down b units.


Shifting Down — The Equation

To shift the function down, you need to decrease the output values of a function by a positive number.

So, you need to subtract the positive number from the right-hand side of the function's equation.

Remember, the graph of H(x) = x2 – 1 has the same shape as the graph of F(x) = x2. It's just shifted down

1 unit.

Rule: Shifting Functions Vertically

How does the equation of a function change when it is shifted vertically?

To shift the graph of a function a certain number of units up, you need to add that number to

the function's equation.

To shift the graph of a function a certain number of units down, you need to subtract that

number from the function's equation

Shifting Functions Left and Right

Now that you've shifted functions vertically, it's time to shift them horizontally — along the x-axis.

You might be able to guess how the equation of a function changes when it's shifted horizontally.

Shifting Left

When you shift a function to the left 1 unit, you're actually adding 1 unit to the x-value.

Shifting Left — The Equation

When you shift a function left by a certain number of units, you will add that number to each x in the

equation.

Shifting Right

When you shift a function to the right 1 unit, you're actually subtracting 1 unit to the x-value.

Shifting Right — The Equation

When you shift a function 1 unit to the right, you will subtract 1 from each x in the equation.

This means you can find the equation for the function G(x) by substituting the expression x – 1 for x in

the equation for F(x).

Moving left and right

On the other hand, y = (x + 3)2 looks like this:

In this graph, f (x) has been moved over three units to the left: f (x + 3) = (x + 3)2 is f (x) shifted three

units to the left.


This is always true: To shift a function left, add inside the function's

argument: f (x + b) gives f (x)shifted b units to the left. Shifting to the right works the same way; f (x –

b) is f (x) shiftedb units to the right.

Warning: The common temptation is to think that f (x + 3) moves f (x) to the right by three, because "+3"

is to the right. But the left-right shifting is backwards from what you might have expected.

Adding moves you to the left; subtracting moves you to the right.

If you lose track, think about the point on the graph where x = 0. For f (x + 3), what does x now need to

be for 0 to be plugged into f ? In this case, x needs to be –3, so the argument is –3 + 3 = 0, so I need to

shift left by three.

This process will tell you where the x-values, and thus the graph, have shifted. At least, that's how I was

able to keep track of things. Yes, I had trouble with this concept, too.

Given f (x) = –x2 + 5x + 2, find the expression, in terms of f , for a leftward shift of five units.

To shift the graph side to side, I need to add or subtract inside the argument of the function (that is,

inside the parentheses). To move to the left, I need (counter-intuitively) to add inside the parentheses.

To move five units, I'll need to add 5 inside the parentheses.

Then my answer is:

f (x + 5)

Checking Your Work

Were you surprised to find that you add 1 when you shift the graph to the left 1 unit? It is a common

mistake to think you should substitute x – 1 instead.

But remember that you want the y-value to be zero since you are shifting the point (0, 0) along the x-

axis.

Here's a quick example: For F(x + 3), what does x need to be to give you an output of zero?

If you guessed -3, you were right! And when x is -3, the graph has moved to the left 3 units.

Confirm — Shifting Functions Horizontally

Rule: Shifting Functions Horizontally

How does the equation of a function change when it is shifted horizontally?

To shift the graph of a function a certain number of units left, you need to add that number to

each x in the equation (or each input variable).

To shift the graph of a function a certain number of units right, you need to subtract that

number from each x in the equation (or each input variable).

Example:

Graph the function y=−12(x−3)2+2y=−12(x−3)2+2 .

If we start with y=x2y=x2 and replace xx with x−3x−3 , it has the effect of shifting the graph 33 units to

the right.


4.4.1 Study: Shifting Functions

Use the questions below to keep track of key concepts from this lesson's study activity.

1) Fill in the blanks to complete the list.

Shifting Functions

Shifting a function is a function around the xy-plane.

The graph's changes, but its shape does not change.

A function can shift (up or down) or (left or right).

2) Fill in the blanks to complete the chart.

Shift How the graph

changes How the equation changes

Example:

F(x) = x2

Up Moves up along the

-axis

a number to the entire

function F(x) = x2 + 1

Down

Moves down along

the

-axis

a number from the entire

function F(x) = x2 – 1

Left

Moves left along

the

-axis

a number to each x in the

function F(x) = (x + 1)2

Right

Moves right along

the

-axis

a number from each x in the

function F(x) = (x – 1)2

3) Each graph below shows the function F(x) = x2 shifted.

Tell which direction each is shifted and how many units.



Use the graph and its description to write the equation for the shifted function G(x).

Shifted down 1 unit and to the right 2 units

Equation:

Shifted up 3 units and to the left 1 unit

Equation:



ANSWER KEY

1) Fill in the blanks to complete the list. (Pages 1 – 2)

Shifting Functions


moving


location


vertically; horizontally

2) Fill in the blanks to complete the chart. (Pages 3 – 7; 11 – 18)

Shift How the graph


Example:

F(x) = x2

Up

Moves up along the

-axis

y


function

Add

F(x) = x2 + 1

Down

Moves down along

the

-axis

y


function

Subtract

F(x) = x2 – 1

Left

Moves left along the

-axis

x


function

Add

F(x) = (x +

1)2

Right

Moves right along

the

-axis

x

a number from each x in

the function

Subtract

F(x) = (x –

1)2



Tell which direction each is shifted and how many units. (Pages 6 and 17)

Shifted up 3 units

Shifted down 3 units

Shifted left 3 units

Shifted right 3 units



Use the graph and its description to write the equation for the shifted function G(x). (Pages 23 – 25)


Equation:

G(x) = (x – 2)2 – 1


Equation:

G(x) = (x + 1)2 + 3


Quiz: Shifting Functions Vertically and Horizontally Question 1a of 10

The graphs below have the same shape. What is the equation of the blue graph?

G(x) = _____

# Choice

A. G(x) = (x + 2)2 - 1

B. G(x) = (x - 2)2 + 1

C. G(x) = (x - 2)2 - 1

D. G(x) = (x + 2)2 + 1


Question 2a of 10


G(x) = _____

# Choice

A. G(x) = (x + 2)2 - 1

B. G(x) = (x + 2)2 + 1

C. G(x) = (x - 2)2 - 1

D. G(x) = (x - 2)2 + 1


Question 3a of 10


G(x) = _____

# Choice

A. G(x) = (x + 3)2 + 2

B. G(x) = (x - 3)2 - 2

C. G(x) = (x + 3)2 - 2

D. G(x) = (x - 3)2 + 2


Question 4a of 10


G(x) = _____

# Choice

A. G(x) = (x - 1)2 - 3

B. G(x) = (x + 1)2 - 3

C. G(x) = (x - 1)2 + 3

D. G(x) = (x + 1)2 + 3


Question 5a of 10

The graph of F(x), shown below, has the same shape as the graph of G(x) = x2, but it is shifted down 3

units and to the left 2 units. What is its equation?

F(x) = _____

# Choice

A. F(x) = (x - 2)2 + 3

B. F(x) = (x + 2)2 + 3

C. F(x) = (x + 2)2 - 3

D. F(x) = (x - 2)2 - 3


Question 6a of 10

The graph of F(x), shown below, has the same shape as the graph of G(x) = x2, but it is shifted up 3 units

and to the right 1 unit. What is its equation?

F(x) = _____

# Choice

A. F(x) = (x + 1)2 + 3

B. F(x) = (x - 1)2 - 3

C. F(x) = (x + 1)2 - 3

D. F(x) = (x - 1)2 + 3


Question 7a of 10

If H(x) = 4x2 - 16 were shifted 9 units to the right and 1 down, what would the new equation be?

# Choice

A. H(x) = 4(x - 9)2 - 17

B. H(x) = 4(x - 17)2 - 9

C. H(x) = 4(x + 9)2 - 17

D. H(x) = 4(x - 7)2 + 16

Question 8a of 10

If G(x) = were shifted 3 units to the left and 3 units up, what would the new equation be?

# Choice

A.

B.

C.

D.


Question 9a of 10

Suppose that G(x) = F(x + 1) - 3. Which statement best compares the graph of G(x) with the graph

of F(x)?

# Choice

A. The graph of G(x) is the graph of F(x) shifted 1 unit to the right and 3

units down.

B. The graph of G(x) is the graph of F(x) shifted 1 unit to the right and 3

units up.

C. The graph of G(x) is the graph of F(x) shifted 1 unit to the left and 3

units up.

D. The graph of G(x) is the graph of F(x) shifted 1 unit to the left and 3

units down.

Question 10a of 10

Suppose that G(x) = F(x - 2) + 7. Which statement best compares the graph of G(x) with the graph

of F(x)?

# Choice

A. The graph of G(x) is the graph of F(x) shifted 2 units to the left and 7

units down.

B. The graph of G(x) is the graph of F(x) shifted 2 units to the left and 7

units up.

C. The graph of G(x) is the graph of F(x) shifted 2 units to the right and 7

units up.

D. The graph of G(x) is the graph of F(x) shifted 2 units to the right and 7

units down.


Shifting Functions

If you slid your pencil to another spot on your desk, would its shape change? Would it get longer or

shorter? Would it suddenly become an eraser or a highlighter?

The answer to all of these questions, of course, is no. Simply moving an object from one place to

another doesn't change its shape.

The same is true for functions. When you move a function, you actually create another function in the

same family — one that has the same shape as the original.

Shifting Functions Up and Down

There are two ways to shift the graph of a function: vertically (up or down) and horizontally (left or

right). We'll look at vertical shifts first.

Remember, when you shift a graph, you change its location, not its shape.

Shifting Up — Coordinates

Shifting a graph vertically moves it along the y-axis.

To shift a graph up, you must increase the value of the y-coordinate for each of the function's ordered

pairs. In other words, you increase each of the function's output values.

Shifting Up — The Equation

When you shift the graph of a function up, you increase all of the output values by a certain positive

number.

This is the same as adding that number to the right-hand side of the function's equation. That's because

the right-hand side of the equation is the same thing as the function's output.

For example, Suppose F(x) = x2. To shift its graph up by 1 unit, add 1 to the right-hand side of the

equation. The new function is G(x) = x2 + 1.

For Example:

A function transformation takes whatever is the basic function f (x) and then "transforms" it (or

"translates" it), which is a fancy way of saying that you change the formula a bit and thereby move the

graph around.

For instance, the graph for y = x2 + 3 looks like this:

This is three units higher than the basic quadratic, f (x) = x2. That is, x2 + 3 is f (x) + 3. We added a "3"

outside the basic squaring function f (x) = x2 and thereby went from the basic quadratic x2 to the

transformed function x2 + 3.

This is always true: To move a function up, you add outside the function: f (x) + b is f (x) moved

up b units. Moving the function down works the same way; f (x) – b is f (x) moved down b units.


Shifting Down — The Equation

To shift the function down, you need to decrease the output values of a function by a positive number.

So, you need to subtract the positive number from the right-hand side of the function's equation.

Remember, the graph of H(x) = x2 – 1 has the same shape as the graph of F(x) = x2. It's just shifted down

1 unit.

Rule: Shifting Functions Vertically

How does the equation of a function change when it is shifted vertically?

To shift the graph of a function a certain number of units up, you need to add that number to

the function's equation.

To shift the graph of a function a certain number of units down, you need to subtract that

number from the function's equation

Shifting Functions Left and Right

Now that you've shifted functions vertically, it's time to shift them horizontally — along the x-axis.

You might be able to guess how the equation of a function changes when it's shifted horizontally.

Shifting Left

When you shift a function to the left 1 unit, you're actually adding 1 unit to the x-value.

Shifting Left — The Equation

When you shift a function left by a certain number of units, you will add that number to each x in the

equation.

Shifting Right

When you shift a function to the right 1 unit, you're actually subtracting 1 unit to the x-value.

Shifting Right — The Equation

When you shift a function 1 unit to the right, you will subtract 1 from each x in the equation.

This means you can find the equation for the function G(x) by substituting the expression x – 1 for x in

the equation for F(x).

Moving left and right

On the other hand, y = (x + 3)2 looks like this:

In this graph, f (x) has been moved over three units to the left: f (x + 3) = (x + 3)2 is f (x) shifted three

units to the left.


This is always true: To shift a function left, add inside the function's

argument: f (x + b) gives f (x)shifted b units to the left. Shifting to the right works the same way; f (x –

b) is f (x) shiftedb units to the right.

Warning: The common temptation is to think that f (x + 3) moves f (x) to the right by three, because "+3"

is to the right. But the left-right shifting is backwards from what you might have expected.

Adding moves you to the left; subtracting moves you to the right.

If you lose track, think about the point on the graph where x = 0. For f (x + 3), what does x now need to

be for 0 to be plugged into f ? In this case, x needs to be –3, so the argument is –3 + 3 = 0, so I need to

shift left by three.

This process will tell you where the x-values, and thus the graph, have shifted. At least, that's how I was

able to keep track of things. Yes, I had trouble with this concept, too.

Given f (x) = –x2 + 5x + 2, find the expression, in terms of f , for a leftward shift of five units.

To shift the graph side to side, I need to add or subtract inside the argument of the function (that is,

inside the parentheses). To move to the left, I need (counter-intuitively) to add inside the parentheses.

To move five units, I'll need to add 5 inside the parentheses.

Then my answer is:

f (x + 5)

Checking Your Work

Were you surprised to find that you add 1 when you shift the graph to the left 1 unit? It is a common

mistake to think you should substitute x – 1 instead.

But remember that you want the y-value to be zero since you are shifting the point (0, 0) along the x-

axis.

Here's a quick example: For F(x + 3), what does x need to be to give you an output of zero?

If you guessed -3, you were right! And when x is -3, the graph has moved to the left 3 units.

Confirm — Shifting Functions Horizontally

Rule: Shifting Functions Horizontally

How does the equation of a function change when it is shifted horizontally?

To shift the graph of a function a certain number of units left, you need to add that number to

each x in the equation (or each input variable).

To shift the graph of a function a certain number of units right, you need to subtract that

number from each x in the equation (or each input variable).

Example:

Graph the function y=−12(x−3)2+2y=−12(x−3)2+2 .

If we start with y=x2y=x2 and replace xx with x−3x−3 , it has the effect of shifting the graph 33 units to

the right.



Use the questions below to keep track of key concepts from this lesson's study activity.

1) Fill in the blanks to complete the list.

Shifting Functions




2) Fill in the blanks to complete the chart.

Shift How the graph


Example:

F(x) = x2

Up Moves up along the

-axis


function F(x) = x2 + 1

Down

Moves down along

the

-axis


function F(x) = x2 – 1

Left

Moves left along

the

-axis


function F(x) = (x + 1)2

Right

Moves right along

the

-axis

a number from each x in the

function F(x) = (x – 1)2


Tell which direction each is shifted and how many units.



Use the graph and its description to write the equation for the shifted function G(x).


Equation:


Equation:



ANSWER KEY

1) Fill in the blanks to complete the list. (Pages 1 – 2)

Shifting Functions


moving


location


vertically; horizontally

2) Fill in the blanks to complete the chart. (Pages 3 – 7; 11 – 18)

Shift How the graph


Example:

F(x) = x2

Up

Moves up along the

-axis

y


function

Add

F(x) = x2 + 1

Down

Moves down along

the

-axis

y


function

Subtract

F(x) = x2 – 1

Left

Moves left along the

-axis

x


function

Add

F(x) = (x +

1)2

Right

Moves right along

the

-axis

x

a number from each x in

the function

Subtract

F(x) = (x –

1)2



Tell which direction each is shifted and how many units. (Pages 6 and 17)

Shifted up 3 units

Shifted down 3 units

Shifted left 3 units

Shifted right 3 units



Use the graph and its description to write the equation for the shifted function G(x). (Pages 23 – 25)


Equation:

G(x) = (x – 2)2 – 1


Equation:

G(x) = (x + 1)2 + 3


Quiz: Shifting Functions Vertically and Horizontally Question 1a of 10


G(x) = _____

# Choice

A. G(x) = (x + 2)2 - 1

B. G(x) = (x - 2)2 + 1

C. G(x) = (x - 2)2 - 1

D. G(x) = (x + 2)2 + 1


Question 2a of 10


G(x) = _____

# Choice

A. G(x) = (x + 2)2 - 1

B. G(x) = (x + 2)2 + 1

C. G(x) = (x - 2)2 - 1

D. G(x) = (x - 2)2 + 1


Question 3a of 10


G(x) = _____

# Choice

A. G(x) = (x + 3)2 + 2

B. G(x) = (x - 3)2 - 2

C. G(x) = (x + 3)2 - 2

D. G(x) = (x - 3)2 + 2


Question 4a of 10


G(x) = _____

# Choice

A. G(x) = (x - 1)2 - 3

B. G(x) = (x + 1)2 - 3

C. G(x) = (x - 1)2 + 3

D. G(x) = (x + 1)2 + 3


Question 5a of 10

The graph of F(x), shown below, has the same shape as the graph of G(x) = x2, but it is shifted down 3

units and to the left 2 units. What is its equation?

F(x) = _____

# Choice

A. F(x) = (x - 2)2 + 3

B. F(x) = (x + 2)2 + 3

C. F(x) = (x + 2)2 - 3

D. F(x) = (x - 2)2 - 3


Question 6a of 10

The graph of F(x), shown below, has the same shape as the graph of G(x) = x2, but it is shifted up 3 units

and to the right 1 unit. What is its equation?

F(x) = _____

# Choice

A. F(x) = (x + 1)2 + 3

B. F(x) = (x - 1)2 - 3

C. F(x) = (x + 1)2 - 3

D. F(x) = (x - 1)2 + 3


Question 7a of 10

If H(x) = 4x2 - 16 were shifted 9 units to the right and 1 down, what would the new equation be?

# Choice

A. H(x) = 4(x - 9)2 - 17

B. H(x) = 4(x - 17)2 - 9

C. H(x) = 4(x + 9)2 - 17

D. H(x) = 4(x - 7)2 + 16

Question 8a of 10

If G(x) = were shifted 3 units to the left and 3 units up, what would the new equation be?

# Choice

A.

B.

C.

D.


Question 9a of 10

Suppose that G(x) = F(x + 1) - 3. Which statement best compares the graph of G(x) with the graph

of F(x)?

# Choice

A. The graph of G(x) is the graph of F(x) shifted 1 unit to the right and 3

units down.

B. The graph of G(x) is the graph of F(x) shifted 1 unit to the right and 3

units up.

C. The graph of G(x) is the graph of F(x) shifted 1 unit to the left and 3

units up.

D. The graph of G(x) is the graph of F(x) shifted 1 unit to the left and 3

units down.

Question 10a of 10

Suppose that G(x) = F(x - 2) + 7. Which statement best compares the graph of G(x) with the graph

of F(x)?

# Choice

A. The graph of G(x) is the graph of F(x) shifted 2 units to the left and 7

units down.

B. The graph of G(x) is the graph of F(x) shifted 2 units to the left and 7

units up.

C. The graph of G(x) is the graph of F(x) shifted 2 units to the right and 7

units up.

D. The graph of G(x) is the graph of F(x) shifted 2 units to the right and 7

units down.

apex algebra 2 learning packet - charles county public …algebra 2: weeks 3-4, april 20 – may 1...

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