i. reflections ii. dilations iii. transformations a a or ... · i. reflections _____ are like...
TRANSCRIPT
I. Reflections
_________________________________
are like mirror images as seen across a
line or a point
1. Reflect the shape over the x-axis.
2. Reflect over the line x = 1
3. Reflect over the line y=-x
II. Dilations
A __________________________________
is a transformation which changes the size
of a figure but not its shape. This reduces
or enlarges the figure to a similar figure.
1. Dilation of 1.5
2. Dilation of 2
3. Write a rule to describe the following
dilation S(1, 4), X(0, 5), F(5, 4), E(1, 2) to
E’(0.5, 1), X’(0, 2.5), S’(0.5, 2), F’(2.5, 2)
III. Transformations
A ________________________________
occurs when a figure is moved by sliding it
up, down, left or right
1. Graph and connect the points (2, 2), (3,
4), (4, 2) and (5, 4), then translate the
points along the vector <2, -2>
2. Translate along the vector: < −4, −3 >
3. Translate the image (x+4, y-1)
IV. Rotations
_____________________________- a point
around which a figure is rotated
_____________________________- which
way a figure is rotated.
a. When a figure is rotated 90°
counterclockwise about the origin,
multiply the y-coordinate by -1 and
switch the x- and y- coordinates.
(x, y) _____________
b. When a figure is rotated 180° about the
origin, multiply both coordinates by -1.
(x, y) _____________
c. When a figure is rotated 270°
counterclockwise (90° clockwise)
about the origin, multiply the x-
coordinate by -1, then switch the x- & y-
coordinates.
(x, y) _____________
1. Rotation 180° about the origin
2. Rotation 180° about the origin 𝑍(1, −3), 𝐾(8, 1), 𝐶(0, −6), 𝑁(10, −4)
________________________________
3. Rotation 90° clockwise about the origin
4. Rotation 90° counterclockwise about
the origin 𝑍(1, −3), 𝐾(8, 1), 𝐶(0, −6), 𝑁(10, −4)
_________________________________
V. Composition of Transformations
The _________________ of
transformations is one, two, or more
transformations.
1. Quadrilateral BGTS has vertices B(–3, 4),
G(–1, 3), T(–1 , 1), and S(–4, 2). Graph
BGTS and its image after a translation
along <5, 0> and a reflection in the x-
axis.
2. ΔTUV has vertices T(2, –1), U(5, –2), and
V(3, –4). Graph ΔTUV and its image after
a translation along <–1 , 5> and a
rotation 180° about the origin.
3. Pre-image: H(2,2), I(-2,2), J(-2,-2),
K(2,-2)
Rotate the figure
180°
𝐻𝐼( , ), 𝐼𝐼( ,
), 𝐽𝐼( , ), 𝐾𝐼( ,
)
Translate the
figure according
to (x,y)→(x+2,y+2)
𝐻𝐼( , ), 𝐼𝐼( ,
), 𝐽𝐼( , ), 𝐾𝐼( ,
)
Reflect the figure
over the line y = x
𝐻𝐼( , ), 𝐼𝐼( ,
), 𝐽𝐼( , ), 𝐾𝐼( ,
)
Practice Test:
1. ∆𝐴𝐵𝐶 is transformed into ∆𝐷𝐸𝐹. What
transformation was performed to make ∆𝐷𝐸𝐹
A. Reflect ∆𝐴𝐵𝐶 across the x-axis, then
translate it 2 units up.
B. Reflect ∆𝐴𝐵𝐶 across the x-axis, then
translate it 2 units down.
C. Rotate ∆𝐴𝐵𝐶 90° clockwise about the
origin, then translate it 2 units up.
D. Rotate ∆𝐴𝐵𝐶 90° clockwise about the
origin, then translate it 2 units down.
2. Triangle ABC has vertices A(1, 2), B(3, 5),
and C(4, 3). Using the origin as a center
of dilation, triangle ABC is dilated by a
scale factor of 3 to create triangle
A’B’C’. Which statement can be used to justify that triangle ABC is similar to
triangle A’B’C’?
A. 𝐴𝐵̅̅ ̅̅ = 𝐴′𝐵′̅̅ ̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ = 𝐵′𝐶′̅̅ ̅̅ ̅̅
B. ∠𝐴 ≅ ∠𝐴′ and ∠𝐵 ≅ ∠𝐵′ C. The sum of the angles of each triangle
remains 180 degrees.
D. The area of triangle A’B’C’ is 3 times of triangle ABC.
3. If ∆𝑅𝑆𝑇 is the result of two
transformations on ∆𝑁𝑃𝑄, which two of
these statemtns can be combined to
prove that ∆𝑅𝑆𝑇~∆𝑁𝑃𝑄?
Statement 1: ∠𝑄 ≅ ∠𝑇
Statement 2: ∠𝑁 ≅ ∠𝑅
Statement 3: 𝑆𝑇𝑃𝑄 = 𝑅𝑆𝑁𝑃
Statement 4: 𝑅𝑆 = 34 𝑁𝑃
A. Statement 1 and 2
B. Statement 1 and 4
C. Statement 2 and 3
D. Statement 3 and 4
4. Which of the following transformations
maps Figure A onto Figure B?
A. Translate Figure A 3 units right and 2
units down.
B. Translate Figure A 3 units right, and
then reflect it across the x-axis.
C. Reflect Figure A across the x-axis, and
then translate it 3 units left.
D. Reflect Figure A across the u-axis, and
then translate it 3 units right.
5. If 𝑀𝑁̅̅ ̅̅ ̅ is mapped to 𝑃𝑄̅̅ ̅̅ , which combination
of transformations could NOT have taken
place?
A. Line segment MN was reflected over the x-
axis and then reflected over the y-axis.
B. Line segment MN was translated 10 units to
the right and then reflected over the x-axis.
C. Line segment MN was rotated 180° clockwise about Point N and then
translated 10 units to the right.
D. Line segment MN was reflected over the y-
axis and then rotated 90° counterclockwise
about Point N.
6. Triangle EGF is graphed below.
Triangle EGF will be rotated 90°
counterclockwise around the origin and will
then be reflected across the y-axis, producing
an image triangle. Which additional
transformation will map the image triangle back
onto the original triangle?
A. Rotate 270° counterclockwise around the
origin
B. Rotate 180° counterclockwise around the
origin
C. Reflect across the line y=-x
D. Reflect across the line y=x
7. Which set of transformations will map
Figure STUVW onto Figure MNPQR?
A. A reflection across the y-axis, and then a
reflection across the x-axis
B. A translation of 10 units down, and then
reflection across y-axis
C. A 90° counterclockwise rotation about the
origin, and then a reflection across the y-
axis
D. A 90° counterclockwise rotation about the
origin, and then a reflection across the x-
axis
8. Jose created a tessellation for a carpet
design shown below.
He produced the following pattern.
Which transformations did he use to create the
pattern?
A. Rotations and Reflections
B. Rotations and Translations
C. Reflections and Translations
D. Reflections, Rotations, and Translations
9. What transformations have occurred to
create the function 𝑓(𝑥) = 3𝑥3 − 4 from
the function (𝑔) = 𝑥3
A. The graph of the function has been
stretched horizontally and shifted up four
units.
B. The graph of the function has been
stretched vertically and shifted up four
units.
C. The graph of the function has been
stretched horizontally and shifted down
four units
D. The graph of the function has been
stretched vertically and shifted down four
units.
10. If the graph of 𝑓(𝑥) = √𝑥 + 3 is translated
2 units right and 4 units down, which of
these functions describes the
transformation graph?
A. 𝑔(𝑥) = √𝑥 − 2 − 1
B. 𝑔(𝑥) = √𝑥 + 2 − 1
C. 𝑔(𝑥) = √𝑥 − 2 + 7
D. 𝑔(𝑥) = √𝑥 + 2 + 7
Unit 2-Review- (parallel lines and transversals, congruent triangles, midsegents)
1. Identify each set of angles below as corresponding, vertical, alternate interior, alternate exterior, consecutive or
linear pair. (use the figure to the right to answer a-h)
a. _______________________________
b. _______________________________
c. _______________________________
d. _______________________________
e. _______________________________
f. _______________________________
g. _______________________________
h. _______________________________
Solve for the missing angle value given: (use the figure to the right o answer a-f)
a. b. c. d. e. f.
Solve for the missing variable and the missing angle values that are indicated (#4-7)
2. 5.
Equation:________________________ Equation:_____________________________
X=_____ X=_______
Fill in the following proof (#8-9)
8. Given the diagram below, prove that x=5
Statement Reason
1. 23x-5=21x+5
2. Subtraction Property
3. 2x-5=5
4. Addition Property
5. 2x=10
6.
7. x=5
Solve for the indicated values (#10-14)
10. 11. 12.
x=__________ x=__________ 13. 14.
a=_______ b=______ c=______
d=______ f=______
Determine the distance or midpoint of the following line segments (#15-17)
15. (4, 6) (1,5) Distance:___________________ Midpoint:___________________
16. (7, -5) (9, -1) Distance:___________________ Midpoint:___________________
17. AB=___________
BC=___________
AC=___________
Perimeter of =_________________
Determine if the following figures are congruent, if they are give a congruence statement and why the two shapes are
congruent. a. Are the triangles congruent b. Give a congruence statement c. Why are the triangles
congruent
18. 19. 20. 21.
a. __________________ a. __________________ a. __________________ a.__________________
b. __________________ b. __________________ b. __________________ b. _________________
c. __________________ c. __________________ c. __________________ c. _________________
B
A C
D
B
D C A
E
L
O
N
M
Z
Y
X
W
D
X
Unit 3 - Trigonometry Review Name:
*For trig functions make sure your calculator is in _______________________________ mode.
Trig Function Sides
Sin
Cos
Tan
• The __________________________ should always be with the trig function
o In some cases you won’t know the angle, so the variable, usually _________________________, will be
with the trig function
Find the value of each trigonometric ratio
__________________________ ______________________________ ______________________________
1. Find the length of the missing side. Round your answer to the nearest tenth.
• Trying to find the ________________
• Know the _________________________
• Use:
2. Find the length of the missing side. Round your answer to the nearest tenth.
• Trying to find the ________________
• Know the _________________________
• Use:
3. Find the length of the missing side. Round your answer to the nearest tenth.
• Trying to find the ________________
• Know the _________________________
• Use:
4. Find the length of the missing side. Round your answer to the nearest tenth.
• Trying to find the ________________
• Know the _________________________
• Use:
5. Find the missing angle. Round your answer to the nearest degree.
Know the _______________________ and the _________________________
Use:
6. A flagpole casts a 100 foot shadow. From the ground to the top of the flagpole you measure an angle
of How high is the flagpole?
Trying to find the ________________
Know the _________________________
Use:
Pythagorean Theorem Review
The Pythagorean Theorem can be used in any _____________________________ triangle where you know two
_____________________, and are trying to find another side
o Unlike the trig functions above, you use the Pythagorean Theorem when you don’t know or aren’t trying to find angles
o Pythagorean Theorem ___________________________________________________
a and b represents the _____________________________________
c represents the _____________________________________________
Find the missing side in each right triangle below:
4. The size of a television screen is given by the length of the diagonal of the screen. What size is a television
screen that is 21.6 inches wide and 16.2 inches high?
5. The bottom of a 13-foot straight ladder is set into the ground 5 feet away from a wall. When the top of the
ladder is leaned against the wall, what is the distance above the ground it will reach?
30-60-90 Right Triangles:
● The ____________________________ of all right triangles are opposite of the
____________________________angle.
● Short leg is opposite of the ___________________________ angle
● Longest leg is opposite of the __________________________side.
● Always start with short leg:
Long Leg=____________________________________
Hypotenuse=__________________________________
Examples: Solve for the missing side values
1. 2.
x=________ m=________
y=________ n=________
3. 4.
x=________ x=________
y=________ y=________
45-45-90 Right Triangles:
● The ____________________________ of all right triangles are opposite of the
____________________________angle.
Because we are working with 45-45-90 Triangle, we know that because of the two congruent angles, the legs will
be _______________ creating a right isosceles triangle.
Leg:____________________________________
Hypotenuse:_____________________________
Examples: Solve for the missing side values
1. 2.
x=________ x=________
y=________ y=________
3. 4.
m=________ a=________
n=________ b=________
Unit 4 - Graphing Quadratics Name:
For each quadratic equation below, describe the transformation and identify the vertex, range, and
intervals of increase/decrease:
1. f(x) = 4(x – 2)2 – 9
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
2. y = ½(x +4)2 – 3
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
3. f(x) = –3(x – 3)2 + 8
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
4. f(x) = –1/4(x – 1)2
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
5. y = x2 + 4
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
6. y = 2(x + 7)2
____________________________________________________
____________________________________________________
Vertex = __________________________Max or Min
Range = _________________________________________
Increase = _______________________________________
Decrease = ______________________________________
Classwork/Homework – Quadratic Review Name:
Find all the key features of the quadratic functions listed below. Remember, if your equation is in standard
form ( you will need to use the equation to find your axis of symmetry/vertex. If the
equation is in vertex form your vertex is the point .
1.
Axis of Symmetry: _________________________________
Vertex: ______________________________________________
Y-Intercept: ________________________________________
X-Intercept(s); _____________________________________
Domain: ____________________________________________
Range: ______________________________________________
Increase: ___________________________________________
Decrease: __________________________________________
2.
Transformation: ___________________________________
______________________________________________________
Axis of Symmetry: _________________________________
Vertex: ______________________________________________
Y-Intercept: ________________________________________
X-Intercept(s); _____________________________________
Domain: ____________________________________________
Range: ______________________________________________
Increase: ___________________________________________
Decrease: __________________________________________
Unit 5 - Radicals
Simplify the following expressions under the radical.
1) √ 2) √ 3) √ 4) √
5) √ 6) √ 7) √ 8) 2√
Changing Forms – Radical to Exponent/Exponent to Radical
Rewrite each of the following in exponential form.
1. (√ ) 2. (√ ) 3. ( √ )
Rewrite each of the following in radical form.
4. 5.( ) 6. ( )
Unit 6 – Solving Quadratics Name:
Recall A quadratic function/equation is a function were the highest exponent is ______________________
There are two main things that you will be asked to do with quadratic functions
1. ______________________________________ 2. ______________________________________
For both factoring and solving, make sure your equation is in the form ____________________________________
Factoring Quadratic Functions
Basic Factoring Equations in the form
Where there is no number in front of the leading
coefficient (x2), factoring by finding two
numbers that multiply to _____ that also add up to
_______
Before, we drew a factoring triangle to help
1.
2. 3.
Factoring with a GCF Equations in the form
the leading coefficient or a is a common factor
Divide each term by a, and leave a in front of
your factored form
Once you’ve divide by a, follow the basic factoring steps
4.
5. 6.
Factoring with no GCF Equations in the form
the leading coefficient or a is not a GCF
Multiply a and c then rewrite your equation
Follow the steps from the basic factoring, but to
undo multiplying by a, divide both factoring by a
o If the fraction simplifies, simplify it. If it doesn’t slide the denominator in front of x in your factor.
7.
8. 9.
Solving Quadratic Equations using the Quadratic Formula
When solving quadratic equations, the quadratic formula works for all quadratics
o Always make sure your equation is in the form ___________________________________________________
Recall the quadratic formula:
1.
2.
3.
4.
5.