Honors Algebra 2/ Trigonometry (Justice) Chapter 3 and 4 Notes

How do you factor quadratic trinomials where a = 1?

(x² + bx + c) Find two factors that multiply to c but add to equal b Ex. x² + 15x + 56 Factors: 7 and 8 (x + 7) (x + 8)

Vertex form of a quadratic function: . vertex . axis of symmetry . domain and range

. (h,k) . x = h . domain and range are the same but the first value of the range changes based on the y value

When imaginary numbers are taken into account, what three types of possible solutions exist?

. 2 real solutions . 1 real solution . 2 complex solutions

How many solutions does a quadratic formula have when: . b² - 4ac > 0 . b² - 4ac = 0 . b² - 4ac < 0

. 2 unique solutions . 1 real solution . 0 real solutions or roots

When is the k value in the range before the infinity symbol, and when is it after?

. Before- when the graph is opening up Ex. (-3, ∞) . After - when the graph is opening down Ex. (-∞, -3)

Find the vertex form of an equation using the given points: VERTEX: (-3,2) POINT: (-5,-6)

. Find the vertex form of the equation y = a(x + 3)² + 2 . Plug in the x and y variables from the point to find the a value and solve -6 = a(-5 +3)² + 2 a = -2 . Finish the equation y = -2(x+3)² + 2

How do you solve systems with three variables?

. Make two pairings with the equations to eliminate 1 of 3 variables (you'll use one twice) . Use the two "new equations" you made in step one and use elimination again to solve . Substitute the values to solve for the other variables

How do you solve matrices of just coefficients and constants on a graphing calculator?

. Press 2nd and x⁻¹ (matrix) . Label the coefficients A and constants B on paper . On the A section of the matrix option, move the cursor to Edit and press enter . Submit the scale (ex. 3 x 3) and insert the values of the A matrix into it . Press 2nd x⁻¹ (matrix) and move to the B option in the Edit section before pressing enter . Set the scale (ex. 3 x 1) and submit the constant values . Return to the main blank screen by pressing 2nd to select Quit . Press 2nd x⁻¹ (matrix), move to the A option, and select enter. [A] should appear, and press x⁻¹ * so it looks like [A]⁻¹ * . Return to the matrix screen and move to the B section, pressing enter so that the equation looks like [A]⁻¹ * [B] and press enter again . A set of values should appear

How do you find the x-intercepts of a graphed quadratic equation?

. Press 2nd, Trace, and 'zero' . Move the cursor to the left of one of the x-intercepts, then to the right, then to the estimated value before pressing 'Enter'. . Repeat with the other x-intercept

How do you find the vertex of a parabola when given the x-intercepts?

. check the 'a' value to see if the graph points up or down . subtract the smaller intercept from the larger intercept and divide by two to find the center . Plug the axis of symmetry into the quadratic equation as x to find y . Graph the parabola using the x-intercepts and the vertex

What are the two ways to solve a quadratic system?

. graphing . substitution

What are the three methods for solving a system?

. graphing . substitution . elimination

In what three ways can a parabola and a line intersect in one point?

. line stretching vertically through center of parabola . tangent . vertex of parabola is on the line

What are other names for the x-intercepts in a quadratic equation?

. roots . zeroes

What are the two methods of solving a system of inequalities?

. tables . graphs

The 'a' value in a quadratic function: . stretch . compression . when negative

. vertical stretch occurs when the absolute value of the 'a' value is greater than 1 (|a| > 1) . vertical compression occurs when the absolute value of the 'a' value is greater than zero but less than one (0 < |a| < 1) . when the a is negative, the graph is flipped upside down so that it stretches into Quadrants III and IV

What two forms of matrices can the following system be represented in? 1) x - 2y + 3z = 12 2) 2x - y - 2z = 5 3) 2x + 2y - z = 4

1 -2 3 * x = 12 2 -1 -2 * y = 5 2 2 -1 * z = 4 And x = 1 -2 3⁻¹ * 12 y = 2 -1 -2 * 5 z = 2 2 -1 * 4 Put it into the latter form [A]⁻¹ * [B] on a graphing calculator

What is an example of solving systems with three variables?

1) x + y + 2z = 3 2) 2x + y + 3z = 7 3) -x - 2y + z = 10 . Eliminate x from 2 and 3 by multiplying 3 by 2 2x + y + 3z = 7 -2x - 4y + 2z = 10 becomes -3y + 5z = 27 . Isolate another y and z equation by eliminating x from 1 and 3, then multiply it be -y + 3z = 13 . Isolate z by eliminating y from each equation, by multiplying the above equation by -3 . -4z = -12, so z = 3 . Substitute the value of z into each equation and solve normally to find the remaining two variables

Solve the following equation

3x² - 1 = 53 3x² = 54 x² = 18 √x² = √18 x = 3√2

What kind of function is a parabola?

A quadratic function

What is the intersection in systems of inequalities?

A region, not a point

Elimination

Eliminate one of the variables by adding or subtracting the equations (must be in standard form to do so; Ax + By = C) Ex. x - 3y = 5 -2x - 4y = -5 Multiply one value to subtract one variable out 2(x - 3y = 5) = 2x - 6y = 10 Combine -10y = 5 y = -1/2 Plug the variable into one of the equations -2x - 4(-1/2) = y x = 7/2) (7/2, -1/2)

What are two methods of making a = 1 in an equation so you can complete the square?

Ex. 2x² - 8x ___ = 11 1. Divide the entire equation by the a value and solve as necessary (can be messy with large fractions) x² - 4x + 4 = 19/2 (once solved, becomes 2 ±√38/2 2. Factor the a value out, values on opposite side of equation affected 2 (x² - 4x + __ ) = 11 2( x² - 4x + 4) = 11 + 8 (4 * 2 = 8) 2(x−2)² = 19

What are qualities of the difference between two squared binomials?

Ex. 4x² - 25 Both a and c are perfect squares, and they form identical binomials except that one is positive and one is negative The above equation forms (2x + 5) and (2x - 5)

How can you convert the vertex form a quadratic function into standard form?

Ex. f(x) = (x-1)² + 2 Separate the squared value into two binomials: (x-1) (x-1) + 2 FOIL: x² - 2x + 3

What does the GCF do?

Factors/divides out the greatest common factor Ex. 6x² + 4x + 2 becomes 2(3x² + 2x + 1)

How can an answer for a system of equations be represented if only x and y variables are present?

In an ordered pair Ex. x = -2, y = 5 (-2,5)

What is the formula for finding the y value of the vertex in standard form?

Insert the y value into the equation and solve

Convert the following system to a matrix: 1) 2x + y = 8 2) -6x + 5 + 1/2z = 3 3) x + 2y - 10z = 7

Matrix: 2 1 0 8 -6 5 1/2 3 1 2 -10 7

When are the maximum and minimum values used?

Maximum: when the 'a' value is negative Minimum: when the 'a' value is positive

When you clear fractions in sets of equations, do you need to have the same least common denominator for both equations?

No. Each can have a different equation tailored to its values

If only the vertex shifts in a quadratic equation, does the position of the points change?

No; (1,1), (2,4), (3,9), and (4,16) still apply, just from the vertex and not the origin

How do you find the equation of a parabola using plotted points?

Press STAT, Calc, and 5; the a, b, and c values will be provided

Why do quadratic equations have two answers and how can you solve them?

Quadratic equations form x², which once isolated the square root of both sides of the equation must be found to find x Every time the square root is taken out of a value, a positive and a negative version of it exist (√x² = |x| = ±x

What would horizontal shifts to the left and right look like in a quadratic formula (h)?

RIGHT: (x-h)² LEFT: (x+h)²

Substitution

Replacing one variable with an expression of the other Ex. y = 2x - 1 y = -2x + 5 Set the two values equal to each other; 2x - 1 = -2x + 5 x = 1.5 Plug the value you found into the equation 2(1.5) - 1 = y y = 2 (1.5, 2)

What kind of solutions are present when b² - 4ac is a perfect square?

The solutions will be rational and you can factor

What would the vertex, axis of symmetry, domain, and range be for a function with the equation y = x²?

Vertex: (0,0) Axis of symmetry: x = 0 Domain: (-∞,∞) Range: [0,∞)

When do systems of equations have no solutions?

When the end equation sets two different values equal to each other Ex. 0 = -6

When would there be no constant in a projectile motion formula?

When the object starts on the ground

What variable are imaginary numbers based off of?

i (italicized)

What is the imaginary unit 'i' equal to?

i = √-1 Ex. √-25 = √-1 * √25 = 5i

How many answers should come out of a quadratic system?

two x values and their corresponding y values placed in ordered pairs

When are systems of equations used?

when solving a situation that involves more than one unknown value/variable

Explicit

when x is isolated in an equation (or other variables) Ex. x = 2y + 3

What is the vertex form of a quadratic function?

y = a(x - h)² + k

What is the formula for a quadratic function in its most essential form?

y = x²

Simplify the following value: √37/4

√37/4 is technically √37/√4, so it can be simplified to √37/2

What is the formula for finding the vertex in standard form?

(-b/2a, f(-b/2a))

How should the y intercept be written?

(0,y)

Solve the following equation with a squared binomial:

(x - 7)² = 50 √(x-7)2 = √50 7 ± 5√2

What would the binomial look like for the following equations: . a² + 2ab + b . a² - 2ab + b

. (a + b)² . (a - b)²

Simplify the following square roots using i: . √-72 . √-108 . √-216

. 6i√2 . 6i√3 . 6i√6

How do you complete the square when the c value for the standard form of a quadratic equation can't be used to make a binomial?

. Transfer the c value to the other side of the equation x² + 8x -6 becomes x² + 8x = 6 . Find the c value and complete the square using (b/2)² and add to both sides (b/2)² = 16, so x² + 8x + 16 = 22 . Create a binomial (x + 4)² = 22 solve accordingly by square rooting both sides √(x+4)² = √22 becomes x = -4 ±√22

Solve the following problem: 2x² + 16 = 76

2x² = 60 x² = 30 x = ±√30 (can't be further simplified)

Solve the following quadratic system: y = (x-3)² + 5 y = 8

8 = (x-3)² + 5 √3 = √(x-3)² x = 3±√3 'Y' is already given, so put each possibility in an ordered pair (3+√3, 8) (3-√3, 8)

What does each column of a matrix represent?

A variable

What is each number in a matrix called?

An element

What can the factored form of quadratic equations do?

Build a graph from the zeroes

What shouldn't you do when converting vertex form to standard form?

Distribute the squared value Ex. (x-1)² should not be interpreted as (x²-1²)

How do you simplify the square roots of numbers which contain squares?

Draw a tree to find the square and the number it multiplies by Ex. √20 Two factors = 4, 5 √4, √5 The square root of 4 divides even but 5 doesn't so the answer is 2√5 **If the factors it is divided into can be divided further multiply the remaining factors together and place them inside of the square root

Graphing in systems of equations

Find the point of intersection . Zoom for systems of equations on calculator: 6 . Press 2nd, Trace, and 5 . Move the cursor to the intersection . Hit Enter three times

How do you solve quadratic systems of inequalities?

Graph both parabolas and their corresponding shaded regions

What does v₀t stand for in projectile motion formulas?

Initial velocity

How many answers exist for linear vs. quadratic equations?

LINEAR: 1 QUADRATIC: 2

How do you solve quadratic trinomials where a ≠ 1?

METHOD 1: Guess and Check Best used if a and c are prime numbers Find the coefficients for x and fill in possible values that multiply to c until one works METHOD 2: The AC Method . Multiply the a value by the c value, and find a b value that is the sum of two numbers whose product is the ac value . Then, rewrite the formula but substitute the two numbers in the place of bx to simulate the OI values of foiling . Divide into two binomials and take out the GCF for each . The final result is the two GCFs combined into a binomial and the common binomial formed without the GCF

Solve the following word problem:

New customers at a piano store get a discount with their first purchases. It costs $300 for 6 lessons and $480 for 12 lessons. Each package includes a one time registration fee. How much does the registration fee cost and how much does each lesson cost? 6x + y = 300 12x + y = 480 Cancel out 12x + y - 480 -12x -2y = -600 -y = -120 The registration fee is $120 Solve to find the remaining variables Each lesson costs $30

What does a matrix contain?

Rows and columns

What can all quadratic trinomials be written as?

The product of two linear binomials

What is the quadratic formula and when is it used ?

The quadratic formula is used to find the x-intercepts of a quadratic equation when completing the square or factoring can't be used easily

What method is used to solve systems of inequalities?

The same method used to solve systems of equations by graphing . graph the equation . find the intersection

What are the solutions to a quadratic equation when y = 0?

The x-intercepts

In what way can a line and a parabola intersect so that there are no solutions?

They don't intersect

What would vertical shifts up and down look like in a quadratic formula (k)?

UP: x² + k DOWN: x² - k

If (x + 4)² = 22 is changed so that it becomes y = (x+4)² -22, what form is it in?

Vertex form

When do systems of equations have infinitely many solutions?

When the end equation sets a value equal to itself Ex. 0 = 0

When is the only time you can complete the square, and what must you do if this isn't the case?

You can only complete the square when a = 1 and you need to change the equation as necessary to create that condition

How do you complete the square in algebra?

You create a squared binomial and find the c value of a quadratic equation Ex. x² + 4x + ___ Find the c value using the formula (b/2)² C = 4 , so x² + 4x + 4 Find the binomial using the formula (x + b/2)² (x + 2)²

What three equations can you get from the following word problem?

You manage a clothing store and budget $6000 to restock 200 shirts. T-shirts cost $12 each, polo shirts cost $24 each, and rugby shirts cost $36 each. There are twice as many polo shirts as rugby shirts SYSTEM: 12t + 24p + 36r = 6000 t + p + r = 200 2p = r

What does each variable in standard form represent?

a = stretch or compression c = y intercept

Matrix

a rectangular array of numbers

system of equations

a set of two or more equations

Tangent line

any line that will touch the function in one point and no others

Discriminant

b² - 4ac in a quadratic formula; tells you what kind of solutions you'll get

What is the standard form of a quadratic function?

f(x) = ax² + bx + c

What is the projectile motion formula for feet?

h(t) = -16t² + v₀t + h₀

What is the projectile motion formula in meters?

h(t) = -4.9t² + v₀t + h₀

What are the dimensions of matrices?

the number of elements in a column x the number of elements in a row Ex. 3 x 4

What is a solution to a system of equations?

the set of values for the variables that make all equations true

Implicit

when x isn't isolated in an equation (or other variables) Ex. x - 2y = 3

Solve the following quadratic system: y = -x² - x + 6 y = x + 3

x + 3 = -x² - x + 6 0 = -x² - 2x + 3 2±√4 -4 (-3)/2 x = -3, 1 y = 0, 4 (-3,0) (1,4)

What is the formula for finding the x value of the vertex in standard form?

x = -b/2a

What is the highest powered term in a quadratic formula?

x²

Convert the following equation from standard form to vertex form

y = -2x² + 4x -5 . separate the 'a' value when solving . find the vertex using -b/2a -4/-4; x = 1 so y = -3 . Combine y = -2(x-1)² -3