Ch 0 & 1 Notes Algebra 1 Review & Transformations

Name:__________________ Hr: ____

Order of Ops. & Evaluating I can… Simplify numeric expressions using the proper order of operations.

Evaluating expressions

Order of Operations "Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number or variable it is probably an operation. To stop the madness, long long ago people agreed to follow rules when doing calculations,

and they are:

How Do I Remember It All ... ? BEDMAS !

B Brackets first

E Exponents (also Powers and Square Roots, etc.)

DM Division and Multiplication (left-to-right)

AS Addition and Subtraction (left-to-right)

Divide and Multiply rank equally (and go left to right). Add and Subtract rank equally (and go left to right)

Examples: Simplify each expression

1. (7 - √9) • (3 + 1) 2. 30 - |5 − 15|

Evaluating

expressions

Examples: Evaluate each expression if a = 𝟒, b = –5, c = –2, d = 3, & g = 6.

1. 𝑎𝑏2 – d 2. |𝑐 + 𝑏| + a

3. (b – dg) 4. a(b + c) + 𝑑

5. –b(a + (𝑐 − 𝑑)) 6. 𝑎𝑑 − 𝑔2

𝑐

Solving Equations (review) We can solve an equation by using ___________________________________

to ____________ the variable in the equation.

Guidelines:

Simplify both sides first (may include distributing)

Use inverse operations to isolate the variable (get the

variable alone on one side of the equation.)

Undo addition or subtraction, before undoing multiplication

or division. (SADME) Example 1 Solve

1

2 𝑥 – 5 = 10

Check: Substitute in the answer you got.

Example 2 Solve 64 – 12𝑤 = 5𝑤 + 3 Write the original equation. Collect variable

on the ________ side by ________________.

Simplify.

Collect constants on the ________ side by

________________.

___________ each side by _________.

Simplify.

Example 3 Solve

5

2(10x + 15) = 18 – 4(x – 3)

Write the original problem.

Distribute the _______.

Check: Substitute in the answer you got.

Example 4 Solving a Temperature Conversion Formula

Solve K = 5

932 273( )F for F.

Solve Linear Systems by Substitution I can... Solve systems of linear equations by substitution.

Solving Systems using Substitution

Steps:

1) Solve one

equation for one

variable (like x)

2) Substitute the

found expression

into the other

equation

3) Solve for the

other variable (like

y)

4) Substitute that

value into one of

the original

equations to find

the other variable

5) Write solution as

an ordered pair

(like (x,y))

Method 1 – SUBSTITUTION METHOD:

1. Solve 1 equations for one variable

2. SUBSTITUTE that found value into the OTHER equation

3. Solve for the other variable

4. Substitute that value into one of the original equations to find the other

variable

3. Write your answer as an ordered pair (x, y)

**Think about which equation would be the easiest to solve for - BE LAZY**

Use substitution to solve each system of equations.

1) {𝑦 = 2𝑥 − 1

𝑦 = −3𝑥 + 4 _______________ 2) {

−2𝑥 + 𝑦 = −34𝑥 + 2𝑦 = 12

______________

3) {𝑥 + 𝑦 = 3

−2𝑥 + 𝑦 = −6 ____________

Solve Linear Systems by Elimination Solving Systems using Elimination

Steps

Method 2 – Elimination METHOD:

-3y + 3x = -9 0 = y - 6x + 2

1. Arrange equations so like terms are stacked, like this…

2. Create a pair of opposites by multiplying one or both

equations

3. Add the columns together

4. Solve for the remaining variable

5. Substitute to solve for the other variable.

Write answer as ordered pair (____, ____)

1. Stack like terms

2. Create a pair of opposites by multiplying one or both equations

3. Add the columns together

4. Solve for the remaining variable

5. Substitute to solve for the other variable

Use elimination to solve each system of equations.

1) {2𝑥 + 3𝑦 = 11−2𝑥 + 9𝑦 = 1

________

2) {3𝑥 + 4𝑦 = 0𝑥 − 4𝑦 = −8

________

Special Cases: True => All real Numbers False => No Solution

3) {3𝑥 − 5𝑦 = 75𝑥 − 𝑦 = 19

________ 4) {2𝑥 + 3𝑦 = 6

5𝑥 − 5𝑦 = 65 ________

5) {𝑥 + 𝑦 = 6

5𝑥 + 5𝑦 = 30 _________ 6) {

3𝑥 − 𝑦 = 5−6𝑥 + 2𝑦 = 11

________

Think and Discuss What concepts do you need to have extra practice? Be honest.

Reflections – 9.1 I can... If two figures are congruent, is one necessarily a reflection of the other? How do you know when a transformation is a reflection?

An isometry can also be referred to as a ____________ transformation. A __________________ is a transformation with rigid motion where every point is the ___________ ___________________ from the line of reflections. Example 1: Identifying Reflections Tell whether each transformation appears to be a reflection.

A. B.

When given the preimage and the image, how could you locate the line of reflection?

Example 2 – Drawing Reflections Copy the triangle and the line of reflection. Draw the reflection of the triangle across the line.

Type of Reflection Rule (mapping)

Reflection in the 𝑥-axis (𝑥, 𝑦) →

Reflection in the 𝑦-axis (𝑥, 𝑦) →

Reflection in the line 𝑦 = 𝑥

(𝑥, 𝑦) →

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

How is reflecting across the x-axis different than reflecting across the y-axis?

Example 3 – Drawing Reflections in the Coordinate Plane Reflect the figure with the given vertices across the given line.

A. X (2, -1), Y (-4, -3), Z (3, 2); x-axis

B) R (-2, 2), S (5, 0), T (3, -1); y = x

C. Reflect the pre-image across the y-axis. Give the coordinates of the

final image.

A: A’:

B:

B’:

C:

C’:

X’:

Y’:

Z’:

A

B

C

R’:

S’:

T’:

Translations – 9.2 I can...

What is the difference between a translation and a reflection?

A translation is a transformation where all the points of a figure are moved the same distance in the same direction. Example 1 – Identifying Translations Tell whether each transformation appears to be a translation. A.

B.

How do you know that the distance between each point and its image is the same for each pair of corresponding points? How do you know that the direction each point has moved is the same for each pair of corresponding points?

Example 2 – Drawing Translations Copy the quadrilateral and the translation vector. Draw the translation along the vector.

Type of Translation Rule (mapping)

Horizontal along vector ⟨𝑎, 0⟩ (𝑥, 𝑦) →

Vertical along vector ⟨0, 𝑏⟩ (𝑥, 𝑦) →

Both along vector ⟨𝑎, 𝑏⟩ (𝑥, 𝑦) →

Notation: A transformation maps every point of a figure onto its image and

may be desribed with arrow notation .. Prime notation ( ‘ ) is sometimes used to identify image points.

How is the translation vector related to

𝐷𝐷′̅̅ ̅̅ ̅?

Example 3 – Drawing Translations in the Coordinate Plane Translate the triangle with vertices D (−3, −1), E (5, −3), and F (−4, −4) along the vector -4, 7.

D’ ( , ) E’( , ) F’( , ) Equation: (x,y) -> ( , )

How do you find the vector that moves an object directly from its starting position to its final position?

Example 4 – Real World Application A sailboat has coordinates 100° west and 5° south. The boat sails 50° due west. Then the boat sails 10° due south. What is the boat’s final position? What single translation vector moves it from its first position to its final position?

Example 5 What is the rule that describes the translation that maps PQRS onto P’Q’R’S’?

Create an equation in terms of x and y for this translation.

Example 6

Translation T <3, -2> (ABCDE) X moves __________ 3 Y moves __________ 2

Think and Discuss

Rotations – 9.3 I can...

A ________________ is a transformation in which a figure is ________ about a __________ point. The __________ point is the _________ ___ _________. Rotations can be clockwise or counterclockwise.

How is a rotation different from a reflection?

Example 1 – Identifying Rotations Tell whether each transformation appears to be a rotation. A. B.

A

B

C

D E

A’ (___, ____) B’ (___, ____) C’ (___, ____) D’ (___, ____) E’ (___, ____)

How can you verify your answer?

Example 2 – Drawing Rotations Copy the figure and the angle of rotation. Draw the rotation of the triangle about point Q by A.

Type of Translation Rule (mapping)

90° about the origin (𝑥, 𝑦) →

180° about the origin (𝑥, 𝑦) →

270° about the origin (𝑥, 𝑦) →

If all three vertices of a triangle are in Quadrant II and the triangle is rotated 90° about the origin, in which quadrant will the vertices of the rotated triangle lie?

Example 3 – Drawing Rotations in the Coordinate Plane

A. Rotate JKL with vertices J (2, 2), K (4, −5), and L (−1, 6) by 180° about the origin.

B. Rotate JKL with vertices J (2, 2), K (4, −5), and L (−1, 6) by 90° about the origin.

C. Rotate JKL with vertices J (2, 2), K (4, −5), and L (−1, 6) by 270° about the origin.

Think and Discuss

Composition of Transformation – 9.4 I can...

Composition of Transformations -> Glide Reflection -> A composition of two isometries is an isometry.

Example 1 – Drawing Compositions of Isometries

KLM has vertices K (4, -1), L (5, -2), and M (1, -4). Rotate KLM 180° about the origin and then reflect it across the y-axis.

Rotate Reflect

K’ K’’

L’ L’’

M’ M’’

The composition of two reflections across two parallel lines is equivalent to a translation.

• The translation vector is _______________________ to the lines. • The length of the translation vector is _______ the distance between the lines.

The composition of two reflections across two intersecting lines is equivalent to a rotation.

• The center of rotation is the ________________ of the lines. • The angle of rotation is ______________ the measure of the angle formed by the lines.

Is the answer the same if the order of transformations is reversed?

Example 2 – Real World Application Sean reflects a design across line p and then reflects the image across line q. Describe a single transformation that moves the design from the original position to the final position.

Any translation or rotation is equivalent to a composition of ________

reflections.

How is the orientation of an image related to its preimage after a glide reflection?

Symmetry – 9.5 I can...

Symmetry -> Line Symmetry -> Line of Symmetry ->

How do you locate a figures line of symmetry?

Example 1 – Identifying Line Symmetry Tell whether each figure has line symmetry. If so, copy the shape and draw all lines of symmetry. A. B. C.

A figure has rotational symmetry (or radial symmetry) if it can be rotated about a point by an angle greater than 0° and less than 360° so that the image coincides with the pre-image.

How do you determine a figure’s angle of rotational symmetry?

Example 2 – Identifying Rotational Symmetry Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order. A. B. C.

Example 3 – Real World Application Describe the symmetry of each icon. Copy each shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and the order. A. B.

A three-dimensional figure has plane symmetry if a plane can divide the figure into two congruent reflected halves. A three-dimensional figure has symmetry about an axis if there is a line about which the figure can be rotated (by an angle greater than 0° and less than 360°) so that the image coincides with the pre-image.

How can you tell if a 3-d figure has plane symmetry?

Example 4 – Identifying Symmetry in Three Dimensions Tell whether each figure has plane symmetry, symmetry about an axis, or neither. A. square pyramid B.

Think and Discuss