Final exam review Math 2

Unit 1 - Transformations

Key exercises from this unit

Exercise Sample solution

Translations-

* Translate each point individually ● T3,-4(x,y)

Reflections- ● Over x axis ● Over y axis ● Over y=-x ● Over y=x ● Over y=5 ● Over x=5

*reflect each point individually Rules:

- rx axis : (x,y) → ( , ) - r y axis : (x,y) → ( , ) - ry=3/rx=-2: graph each point, count units

from each point to line and continue counting same amount after line.

- ry=x(x,y) → ( , ) - ry=-x(x,y) → ( , )

-

Rotations ● 90o Clockwise ● 180o ● 270o Clockwise ● 90o CC

*rotate each point individually Rules:

- R90 : (x,y) → ( , ) - R180 : (x,y) → ( , ) - R270 : (x,y) → ( , ) - R90 clockwise : (x,y) → ( , )

Dilations ● Dilate with a

scale factor k

*If the scale factor is a fraction, it will be getting smaller *if the scale factor is a number >1, it will be getting bigger *you can’t have a negative scale factor Rule: (x,y) → (kx, ky)

Find a scale factoR given the pre-image/ Image

Symmetry: ● Line symmetry ● Point

symmetry ● Rotational

symmetry

Select practice questions from this unit

Unit 2-Quadratics Key exercises from this unit

***********Also study: Part 1 study guide, notes & homework********** Exercise Sample solution

Simplify imaginary numbers -(i with exponents greater than 1)

i1823 i130005

* Think about the imaginary clock!

Operations with complex numbers -a+bi form

Addition Subtraction Multiplication Conjugates (4-2i) +(5i-2) (4-2i)-(5i+2) (4-21)(5i+2) (2i+3)(2i-3)

Find the original equation given the roots

x=2, x= 2−5

*Use the roots to find the factors, then multiply the factors together

Find the discriminant n2-27n=6

Determine if b2-4ac is: ● a positive perfect square(2 rational) ● positive non-perfect square(2 irrational) ● negative(2 imaginary roots) or ● 0( 1 rational root)

Solve the Quadratic -method: quadratic formula

15n2-27n=6

x = 2a−b±√b −4ac2

Solve The Quadratic -method: square roots

4x2=25

*use when you see 2 perfect square terms *get on opposite sides and take square root

Solve the Quadratic -method: factoring

n2+3n-12=6

*make sure one side is equal to zero, *factor out the GCF first then use ac method *use zero product property (set each factor = 0 and solve)

Solve the quadratic -Method: Completing the Square

5n2+20n-68= 2

*You can always use this method on an equation, even when it does not factor.

1. Get ‘c’ term alone on other side 2. Make sure a=1 (if not, divide both sides

by a) 3. Find magic number and add to both sides

(b/2)2

4. Factor ***remember it will always factor to (x+b/2)

2 5. Square root both sides 6. Solve for both answers

Understanding graphs of quadratic equations:

f(x)=2x2-12x+10

Domain: (- , ) *always the same for quadratics* Range: ( , ) End behavior: As x → , f(x) As x → , f(x) Increasing: Decreasing:

X intercepts / roots / zeros: Opens up or opens down?

Application problems

* Draw a diagram, follow the steps, look for things being multiplied The width of a rectangle is 5 meters less than its length. The area is 84 square meters. Find the dimensions of the rectangle.

Application problems Projectile motion

❖ What is the maximum height?

❖ Where did it land?

*use projectile motion formula: *max height → look for vertex, landing → look at intercepts Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function h(t)= -16t2 +16 +480 , where t is the time in seconds and h is the height in feet. (use calc)

How long did it take for Jason to reach his maximum height? What was the highest point that Jason reached? Jason hit the water after how many seconds?

Quadratic regression What is the curve of best fit? Predict an outcome based on the curve of best fit

The table shows the operating costs of a small business from 2000 to 2005. Assume that t is the number of years since 2000 and C is the cost in thousands of dollars. Use the results from the regression shown to find the best-fitting quadratic model for the data. Round to two decimal places. Then use the model to find the operating cost in 2007. Show your work.

Analyzing a graph on the calculator

1. To graph: Press [Y=], Enter the function, Press [GRAPH]. 2. To find the zeros / vertex: Press [2nd] [CALC] *(vertex is either max or

min) When prompted for "Left Bound?", specify the left bound for x using the arrow keys and press [ENTER].

• When prompted for "Right Bound?", specify the right bound for x using the arrow keys and press [ENTER]. • When prompted for "Guess?", move the cursor close to the zero value or press [ENTER].

Calculator steps for curve of best fit/ quadratic regression

Select Practice Questions from unit 2

Simplify:

2i )(5i )( − 3 + 3

2i ) 2i )( + 1 − ( − 8

2i )(2i )( + 1 − 1 5i 2i 1 1234 + 1 2

Find the discriminant, and state the number/ type of roots:

x x 2 + 2 + 1 = 0

x x 2 2 − 3 − 5 = 0

x x 2 2 + 2 = 3 x 5 9 2 = 2

Find the original equation given the roots

Roots → factors x= 3

2 x=-5

Roots → factors x= 3

−1 x=5

Roots → factors x=3 x=-5

Solve using the Quadratic Formula or completing the square

2p2-20= 12p

4x 2 – x = 5 10x2+9x=7

Interpret the Graph using your graphing calculator:

f(x)= -x2-10x+25 Domain: Range: Increasing: Decreasing: roots: Y intercepts: End behavior:

f(x)= -2x2-6x-8 Domain: Range: Increasing: Decreasing: Roots: Y intercepts: End Behavior:

f(x)= 2x2-6x-8 Domain: Range: Increasing: Decreasing: Roots: Y intercepts: End Behavior:

Answer the questions below using your knowledge of quadratics:

The length of a rectangle is 1 foot more than twice the width. The area is 55 square feet. Find the dimensions of the rectangle.

The product of two consecutive even integers is 6 more than three times their sum. Find the integers.

The product of two consecutive integers is 56. Find the integers.

Answer the following questions using the projectile motion formula:

An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s (t ) = –4.9t 2 + 19.6t + 58.8, wheres is in meters. When does the object strike the ground?

An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be the object's maximum height? When will it attain this height? s (t ) = –16t 2 + 64t + 80

After the semester is over, you discover that the math department has changed textbooks (again) so the bookstore won't buy back your nearly-new book. You and your friend Herman decide to get creative. You go to the roof of a twelve-story building and look over the edge to the reflecting pool 160 feet below. You drop your book over the edge at the same instant that Herman chucks his book straight down at 48 feet per second. By how many seconds does his book beat yours into the water?

Quadratic regression

The table shows how wind affects a runner’s performance in the 200 meter dash, where s represents the speed of the wind in meters per second and t represents the change in finishing time. Negative wind speed means the runner is running with the wind while positive wind speed means the runner is running with the wind. a. Write an equation for the curve of best fit. b. Predict the change in finishing time when the wind is -8 meters per second

11. A study compared the speed, x, in miles per hour and the average fuel economy y(in miles per gallon) for cars. The results are shown in the table.

a. Find a quadratic model in standard form for the data.. b. Predict your fuel economy if you were going 42 mph.

Unit 3- Square root and Inverse functions Key exercises from this unit

Exercise Sample solution

Simplify- Radicals 2√500

*find the biggest perfect square that fits in the number to break it up

Add/ Subtract -Radical expressions

( +4)+( +3) √2 2√2 ( +4)-( +3) √2 2√2

*only combine(add/ subtract) like radicals

Multiply- Radical expressions

+4)( +3) (√2 2√2

Divide/ rationalize the denominator - Radicals

+2√35−2√3

* multiply top and bottom of the fraction by the conjugate of the denominator to get rid of it

Simplify: Using exponent rules

20x y3 22(x y z)3 −2

Product rule: Power rule: Quotient rule: Negative exponent: Fractional exponent:

Graph Square root functions:

f(x) = − √x + 2 − 3

* identify how the square root is being transformed from the parent function Domain: Range: ( , ) Starting point: Opens up or opens down?

Direct variation:

x y = k y varies directly with x. If y = -4 when x = 2, find y when x = -6

Inverse variation:

y = xk

y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

Simple interest:

You get a summer job at a bakery. Suppose you save $1400 of your pay and deposit it into an account that earns simple annual interest. After 9 months, the balance is

$1421. Find the annual interest rate. * I=Prt → I is the simple interest earned → P is the principal, → t is the time in years. → r is the annual interest rate (written as a decimal)

Compound interest: You deposit $1500 into an account that earns 2.4% interest compounded annually. Find the balance after 6 years.

* When an account earns interest compounded annually, the balance A is given by the formula A=P(1+r)t where → A is the balance/ ending amount → P is the principal, → t is the time in years → r is the annual interest rate (written as a decimal)

Unit 4-Triangle congruence Key exercises from this unit

Topic Sample exercise

distance

Find the perimeter of a triangle with endpoints (3,3), (8,0) & (4,6)

midpoint

Find the midpoint of a segment with endpoints (2,-8) and (-7, 5)

Midsegment

Angle bisectors

Perpendicular bisectors:

Parallel line angle relationships

Triangle interior angle sum

.

Exterior angles

Classifying triangles

SSS What additional information is needed to prove the triangles congruent using SSS?

SAS What givens are needed to prove the 2 triangles using SAS?

ASA What givens are needed to prove the two triangles congruent using ASA?

AAS What givens are needed to prove the triangles congruent using AAS?

HL What givens are needed to prove the triangles congruent using HL?

POTENTIAL REASONS bank

If... Possible reasons:

2 angles are congruent

Vertical angles are congruent Alternate exterior angles are congruent Alternate interior angles are congruent Corresponding angles are congruent Reflexive property Transitive property Definition of angle bisector Substitution property Base angles theorem (isosceles triangle- base angles are congruent) CPCTC

2 sides are congruent Definition of segment bisector Definition of perpendicular bisector Definition of midpoint Reflexive property Transitive property Substitution property Corollary to base angles theorem (isosceles triangle-sides are congruent) CPCTC

2 Triangles are congruent

SSS SAS ASA AAS HL

2 values are equal (algebraic proofs)

Substitution property Transitive property Addition property of equality Subtraction property of equality Multiplication property of equality Division property of equality Distribution property

Unit 5-Trigonometry Key exercises from this unit

Exercise Sample solution

45-45-90- triangle ratios

30-60-90--triangle ratios

Pythagorean theorem-to find a missing side

SohCahToa trig to find a missing side

SohCahToa trig to find a missing angle

Find angle c

Angles of elevation and depression

From a point 115 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 64.3o . Find the height of the tree to the nearest foot.

Unit 6-Probability