Algebra II Finals Study Guide - copy
Never say...
"no slope" you should say the slope is 0 if horizontal or undefined if verticle
Adding Complex Numbers
(a+bi) + (c+di) = (a+c) + (b+d)i In order to add a complex number, add their real parts together, then their imaginary parts together, and express the sum as a complex number.
Subtracting Complex Numbers
(a+bi)-(c+di) = (a-c) + (b-d)i Subtract the real parts, subtract imaginary parts, and express the difference as a complex number
A function is increasing when...
- The line rises from left to right - The y-values increase
slope in standard form
-A/B
A function is decreasing when...
-The line falls from left to right -The y-values decrease
4 Conditions to put an equation in standard form
1) A, B, and C must be integers 2) A ≥ 0 3) A+B cannot = 0 at the same time 4) A, B, and C must have no greatest common factor
How to go from general form to vertex form
1) Find a,b, and c 2) Find the vertex by doing (-b/2a, f(-b/2a)) 3) Since the vertex is (h,k) you can plug the vertex and a into vertex form
Steps to graphing a quadratic function in standard form
1) Find and graph the equation of the axis of symmetry (use a dotted line in this class) x=-b/2a **The A.O.S. is always an equation in the form x=? 2) Find and graph the vertex, the vertex is always an ordered pair (-b/2a, ƒ(-b/2a)) 3) Find the y intercept: (0,C) graph (0,C) if it fits on your graph and graph its equidistant/partner point 4) To find other points make a table of values with the vertex in the middle and other x values with their partner point which is equidistant from the axis of symmetry Partner points have the same y values
How to use intercepts to graph the (general) Standard Form of the equation of a line
1) Find the x-int. by setting y equal to 0 and solving for x. Then, plot the x-int. on the x axis. 2) Find the y-int. by setting x= to 0 and solving for y. Then, plot the y-int. on the y axis. 3) Draw a line through the two points using a straightedge and draw arrowheads at either end of the line
Finding x and y intercepts
1) From a graph- look and see 2) From an equation -in standard form: *to find x int. let y=0 *to find y int. let x=0
Steps to solve a system of linear inequalities
1) Graph both inequalities and shade (if < or > it is a dashed line, if ≤ or ≥ it is a solid line) 2) Look and see where they intersect 3) Highlight the boundary lines of the overlapping section
What are the 5 ways to solve a quadratic equation?
1) Graphing 2) Factoring 3) Square roots 4) Completing the square 5) Quadratic Formula
Order of Transformations
1) Horizontal Shifting 2) Vertical stretching or shrinking 3) Reflecting over the x-axis 4) Vertical shifting
Elimination/Addition method steps
1) If necessary, rewrite both the equations in the form Ax+By=C 2) If necessary, multiply either or both of the equations by appropriate non-zero numbers so that the sum of the coefficients of one variable added together is 0. 3) Add the equations from step two together. The sum is an equation in one variable. 4) Solve the resulting equation. 5) Back-substitute the value obtained in step four into either of the given equations and solve for the other variable. 6) Check the solution in both of the original equations
Steps to solving equations containing radicals
1) Isolate the radical 2) Square both sides of the equations (only look at the principal square roots which are the positive square roots) AKA exponenciate 3) Repeat steps one and two if there is another radical 4) Solve the resulting equation 5) CHECK (mandatory) your answers in the original equation (only have to check with real rational answers, no crazy fractions)
How to solve by square roots
1) Isolate the squared term 2) Take the square root of both sides 3) Solve for the variable Remember: Never leave a radicand in the denominator, also when deal with square roots make sure to write ± before the symbol
Steps to solve a system of equation in three variables by elimination: (this can only be done by elimination)
1) Label the equations a, b, and c 2) Choose two of the equations and eliminate one variable 3) Choose another combination of the original equations and eliminate the same variable. Now you have two equations with only two variables. 4) Take the two resulting equations and solve for the two variables 5) Plug in the resulting values of the variables into one of the original equations to solve for the other variable 6) Write solutions as (x,y,z) **Note: in this class you will never get a fraction for one of these variables: will always be a whole number
How to write a quadratic equation from given roots
1) Look at the roots (see what x equals) 2) Rearrange the x= so the whole expression equals zero 3) Write the solutions as factors equal to zero 4) Multiply the factors and combine like terms if necessary 5) Should be in form ax²+bx+c=0
What are the three possible solutions to a system of linear equations and when do they occur?
1) One solution- occurs when the lines intersect one time 2) No solution- occurs when the lines are parallel 3) Infinitely many solutions- both equation result in the same exact line (does not mean all real numbers)
How to graph a line using slope intercept form
1) Plot point containing the y intercept of the y axis 2) Obtain a second point by using the slope (m). Write m as a fraction and use rise over run, starting at the y intercept to plot this second point 3) Then connect the two points using a straightedge and draw arrowheads on each side of the line
4 Possibilities for a Line's slope
1) Positive slope- line rises from left to right 2) Negative slope- line falls from left to right 3) Zero slope- horizontal 4) Undefined slope- verticle
Journal: Explain how you would solve a Linear Inequality and give an example: include how you know whether to make the boundary a solid line or a dashed line and how you know which side of the boundary to shade. Give an example.
1) Rewrite in slope intercept form the boundary line 2) Graph the boundary line, if it is ≤ or ≥ draw a solid line, if it is < or > draw a dashed line 3) Use a test point in the inequality to see which half-plane to shade, use (0,0) as the test point unless the line passes through the origin 4) Shade the region/half-plane where all the points that make the inequality true lie
Steps to graph an inequality
1) Rewrite in slope intercept form the boundary line 2) Graph the boundary line, if it is ≤ or ≥ draw a solid line, if it is < or > draw a dashed line 3) Use a test point in the inequality to see which half-plane to shade, use (0,0) as the test point unless the line passes through the origin 4) Shade the region/half-plane where all the points that make the inequality true lie
Steps to solve a linear inequality in two variable
1) Rewrite in slope intercept form the boundary line 2) Graph the boundary line, if it is ≤ or ≥ draw a solid line, if it is < or > draw a dashed line 3) Use a test point in the inequality to see which half-plane to shade, use (0,0) as the test point unless the line passes through the origin 4) Shade the region/half-plane where all the points that make the inequality true lie
How to solve polynomial equations by factoring
1) Set the equation equal to 0 2) Factor by grouping 3) Use the ZZP to find the roots (solutions)
Substitution method steps
1) Solve either of the equations for one variable in terms of the other (isolate one variable on one side in one of the equations) 2) Substitute the expression found in step one into the *other* equation 3) Solve the equation containing one variable 4) Back substitute the value found in step three into one of the original equations. Simplify and find the value of the remaining variable 5) Check the proposed solution in both of the systems equations (make sure they both result in a true statement when symplified)
Checklist (going from one equation to another)
1) Start at point slope form 2) Then go to slope intercept 3) Then to standard form
How to graph a quadratic function in vertex form
1)Find a, h, and k 2)Find and graph the axis of symmetry (x=h) 3)Find and graph the vertex (h,k) 4)To find other points make a table of values with the vertex in the middle and other x values with their partner point which is equidistant from the axis of symmetry Partner points have the same y values
The Domain of the Composition of Functions
1)x must be in the domain of g(x) 2)AND g(x) must be in the domain of f(x)
Synthetic Division Steps
1. Arrange polynomial in descending powers, with a 0 coefficient for any missing terms 2. Write c for the divisor, x-c. To the right write the coefficients of the dividend 3. Write the leading coefficient of the dividend on the bottom row 4. Multiply c, times the value just written on the bottom row. 5. Add the values in this new column, writing the sum in the bottom row 6. Repeat the series of multiplications and additions until all columns are filled in 7. Use the numbers in the last row to write the quotient, plus the remainder above the divisor. The degree of the first term of the quotient is one less than the degree of the first term of the dividend. The final value in the row is the remainder.
How to divide a polynomial using long division
1. Arrange the terms of both the dividend and the divisor in descending order of exponents 2.Divide first term in dividend by first term in divisor. The result is the first term of the quotient. Don't forget to line up like terms. 3. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. 4. Subtract the product from the dividend (remember you will have to change signs of all terms being subtracted) 5. Bring down the next term in the original dividend and write it next to the remainder to form a new dividend 6. Use this new expression as the dividend and repeat the process
What is the square root of -16?
4i and -4i -16 has two square roots in the complex numbers system 4i is the principal square root
Journal: A. Explain how to use the general (standard) form of a line's equation to find the slope and y-intercept. Give an example. B. Explain how you would write the equation of a line if you know two points through which the line passes. Give an example and all three forms of the linear equation.
A. In order to find the slope do -A/B and to find the y intercept do C/B. 4x+3y=2 slope = -4/3 y-int. = 3/2 B. First plug the points into the slope formula and simplify in order to find the slope. Then, choose one point to plug into the point slope formula with the slope you found. Then, rearrange the equation into slope intercept form. Then rearrange the equation again into standard form making sure that A is positive and that A,B,C are integers.
When writing the interval for increasing, decreasing, and constant areas of a function...
Always use parenthesis, open interval
What is a solution to a system of linear equations in 2 variables?
An ordered pair that satisfies both equations (makes a true statement when plugged into both equations) in the system, the point where the lines intersect
Standard (General Form)
Ax+By=C C must be on right side of equals sign
Why must you begin by expressing all square roots in terms of i when performing operations with square roots of negative numbers?
Because the product rule for radicals only applies to real numbers and a negative radicand is not a real number
x-int. in standard form (ax+by=c)
C/A or (C/A, 0)
y-int. in standard form (ax+by=c)
C/B or (0, C/B)
How to divide a polynomial by a monomial
Divide each term of the numerator by the denominator Remember: You always want to end up with positive exponents
Dividing Complex Numbers
Division uses complex conjugates because the goal is to obtain a real number in the denominator since there should never be a in the denominator and i represents a radical 1. Multiply the numerator and denominator by the complex conjugate of the denominator to get a real number in the denominator 2. Simplify 3. Write answer in the form: a/# + b/# i (put i on the end)
Journal: Explain how you would go about transforming the equation f(x) = |x| into f(x) = -|x - 2| + 3. Be sure to explain all your steps in order.
First I would graph f(x) = |x|. The I would shift the control horizontally to the right to spaces by adding 2 to each x value. There is no stretch or shrink so next I would reflect the previous graph over the x axis by making each y value of the previous step its opposite. Finally, I would shift the previous graph up 3 spaces by adding 3 to each y value.
Journal: Without actually solving the equation, give a general description of how to solve x³ - 5x² = x - 5
First rearrange the equation so it is all equal to 0. Next, factor by grouping and use ZZP to solve for x.
How to solve by graphing
Graph the Quadratic Function: 1) Find and graph the equation of the axis of symmetry (use a dotted line in this class) x=-b/2a **The A.O.S. is always an equation in the form x=? 2) Find and graph the vertex, the vertex is always an ordered pair (-b/2a, ƒ(-b/2a)) 3) Find the y intercept: (0,C) graph (0,C) if it fits on your graph and graph its equidistant/partner point 4) To find other points make a table of values with the vertex in the middle and other x values with their partner point which is equidistant from the axis of symmetry Partner points have the same y values Look and see where function crosses x axis Write the solution set If it does not cross x axis the solution set is no real numbers
What are the three methods for solving a system of linear equations?
Graphing, substitution, and the elimination/addition method
Journal: Suppose you were asked to solve the system: 2x + 3y = -2 3x + y = 39 by the elimination method. To make your work as easy as possible, which variable would you eliminate? Describe how you would do this and then solve.
I would eliminate y because all you have to do is multiply the second equation by negative three, then add the two equations together and solve for x. Then, plug x into one of the beginning equations and solve for y. Then write the solution as an ordered pair. (Do this one for practice: answer is (17,-12)
Journal: If equations for f and g are given, explain how to find f-g.
In order to find f-g subtract the equation for g(x) from the equation for f(x). Then, combine like terms and simplify.
Journal: Describe a procedure for finding (f o g)(x). What is the name of this function?
In order to find the (f o g)(x) plug the whole expression for g(x) into all the x's in the f(x) equation. Then simplify. This is called a composite function.
Journal: Explain how to identify the domain and range of a function from its graph.
In order to find the domain of a function from a graph, look to see which x values the graph crosses and how far the graph extends out to the left or right. In order to find the range, look to see the y values that the graph crosses and how far the graph extends up or down.
Inverse Functions
Let f and g equal two functions such that f(g(x))=x and the g(f(x))= x for every x in the domain
Steps to find the domain:
Look to see what x cannot equal Pay attention to fractions and square roots
Note
Make sure to say which shifts you do not do
What is the key to completing the elimination/ addition method?
Obtain, for one variable, coefficients that differ only in sign
Journal: What must be done to a functions equation so that it graph is stretched vertically?
One must multiply the functions equation by a positive number greater than one in order for the graph of the function to be stretched vertically
Reflection
Reflection of f(x) over the x axis is shown by a negative sign in front of the f(x) When reflecting each y value of the previous step becomes its opposite and the x values stay the same
How to solve by completing the square
Steps for Finding C 1) Make sure a=1 2) Find b/2 3) Take b/2 and square it Solving by Completing the Square 1) Make sure a=1 2) Add or subtract c from both sides (want to get ax²+bx =c) 3) Find b/2 and square it 4) Add (b/2)² to both sides 5) Factor the left and simplify the right 6) Solve by square roots
Complex conjugate definition
The complex conjugate of a+bi is a-bi and the complex conjugate of a-bi is a+bi. When complex conjugates are multiplied you get a real number
Journal: Explain the two ways to find x and y intercepts from the general (standard) form of a line's equation and how to use these intercepts to graph the equation. Give an example.
The first way is to set y to 0 and solve for x to find the x intercept. Then, set x equal to 0 and solve for y to find the y intercept. Then, plot both these points on the graph and connect them using a straightedge. The second way is to do C/B for the y intercept and C/A for the x intercept. Then plot these points and connect them using a straightedge.
State the Remainder Theorem and how it can be used to find f(-6) if f(x) = x⁴ + 7x³ + 8x² + 11x + 5. What advantage is there to using the Remainder Theorem in this situation rather than evaluating f(-6) directly? Show the comparison.
The remainder theorem states that the remainder in synthetic division by "c" is the value of the function at the number "c". f(c) In the example treat -6 as c in synthetic division and the x⁴ + 7x³ + 8x² + 11x + 5 as the dividend. Then, do synthetic division. The advantage of using the remainder theorem is it takes less time than evaluating f(x) like normal and there is less room for error as well because there are less steps to take.
What do you need in order to write the equation of a line?
The slope and an ordered pair on the line
What is a conjugate?
The sum and difference of the same two terms
When do you have a system with no solution? How would you write the solution set?
The system has no solution when the lines are parallel. The solution set is ∅, the empty set. If the variables cancel when doing the addition or substitution method and you are left with a false statement it shows that there is no solution.
When are two complex numbers equal?
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal
Journal: Without graphing, what is the mathematical relationship between 2 lines that are parallel? between two lines that are perpendicular? Be sure to give examples.
Two lines that are parallel have the same slope. Two lines that are perpendicular have negative reciprocal slopes of each other.
Multiplying Complex Numbers
Us the distributive property and FOIL, then replace any occurrences of i² with -1 and simplify
How to find an inverse function
Use switch and swap 1)Switch x and y 2)Solve for y
If you are given on 2 points on the line, how would you write an equation?
Use the two points to find the slope by plugging in to the slope formula, then plug one of the points and the slope into point slope formula
When can synthetic division be performed?
When the divisor is in the form x-c
When do you obtain a real number from multiplying complex numbers?
When you multiply: (a+bi)(a-bi)
What is the solution to a quadratic equation?
Where the parabola intersects the x axis, in the form (#,0)
What can you do if you are solving a system by elimination and get an ugly fraction as the first variable?
You can return to the original system and use the addition method and find the value of the other variable
When do you have a system with Infinitely Many Solutions? Does this mean All Real Numbers? Why or Why Not?
You have a system with infinitely many solutions when the two lines are identical. If you attempt to solve a system like this by the addition or substitution method, both variables will be eliminated and a true statement will result. Infinitely many solutions does not mean all real numbers, it means that the ordered pairs that satisfy the system are infinite, referring to an infinite number of ordered pairs that are on the line.
How to solve by factoring
Zero Product Property -If a*b=0 then either a=0 or b=0 or both equal 0 -If you are multiplying two factors and the answer is 0 then at least one of the factors has to be equal to 0 -To solve by factoring the equation must be in the form: ax²+bx+c=0
Real part of complex number
a in a+bi
Function
a relation in which no member of the domain is listed with a different range, no x is repeated with a different y
complex number definition
a system of numbers based on adding multiples of i, such as 5i, to real numbers, complex numbers include real and imaginary numbers
Journal: a) How would you go about changing the Standard Form of a Quadratic Equation into the Vertex Form and vice versa. Use examples. b) Describe the relationship between the solution of ax2 + bx + c = 0 and the graph of f(x) = ax2 + bx + c?
a) First, find a,b, and c. Then, find the vertex by doing x=-b/2a and y is f(-b/2a). The vertex= (h,k). Now plug the vertex and a into vertex form. b) The solution to a quadratic equation are the x intercepts of the graph of the function
Journal: a) What is the relationship between the roots, zeros and x-intercepts of a quadratic equation? Journal: b) What is the Zero Product Property and how is it used in solving quadratic equations. Give an example.
a) Roots, zeros, and x-intercepts all refer to the solution of a quadratic equation. They are all the same thing b) The ZZP says that when you are multiplying 2 factors and the answer is 0, at least one or both of the factors have to equal 0. a*b=0 a or b or both= 0 You use the ZZP to solve quadratic equations by factoring.
Journal: a) What is meant by the slope of a line? When is the slope undefined? When is the slope equal to zero? Write a linear equation with each of these slopes and indicate which is which. b) Explain how to use slope and y-intercept to graph a linear function. Give an example.
a) The slope of a line is the steepness of a line and is the change in y over the change in x. The slope is undefined when the line is vertical. The slope is 0 when the line is horizontal. y= 3 is 0 slope x=3 is undefined slope b) In order to used slope and y-int. to graph a linear function: 1) Plot point containing the y intercept of the y axis 2) Obtain a second point by using the slope (m). Write m as a fraction and use rise over run, starting at the y intercept to plot this second point 3) Then connect the two points using a straightedge and draw arrowheads on each side of the line
standard form of complex and imaginary numbers
a+bi complex and imaginary numbers are the set of all numbers in this form
How to add rational expressions:
a/b ± c/d = ad± bc all over bd
Pure imaginary number
an imaginary number in the form bi
relation
any set of ordered pairs
The slopes of perpendicular lines...
are the negative reciprocals
The slopes of parallel lines...
are the same
Quadratic equation must be in the form
ax²+bx+c=0
Imaginary part of complex number
b in a+bi
When performing operations with square roots of negative numbers...
begin by expressing all square roots in terms of i
Discriminant
b²-4ac (without the radical) tells you how many and what type of solution you will get
x int in standard form (ax+by=c)
c/a
y int in standard form (ax+by=c)
c/b
Not everything that can be divided by long division...
can be divided by synthetic division
Don't forget to
color code the steps and put them in order (for a step like no _____________ just write in pencil and highlight the last step
The graph of imaginary solutions...
does not touch the x axis
f(x)
f of x, the value of the function at the number x, equal to y
(f o g)(x)=
f(g(x))
(f+g)(x)=
f(x) + g(x)
(f*g)(x)=
f(x)*g(x)
(f-g)(x)=
f(x)-g(x)
(f/g)(x)=
f(x)/g(x) and g(x)≠0
Journal: Write an equation for a parabola that has been shifted horizontally, shrunk, and reflected over the x-axis. Explain each step as you go along.
f(x)= -1/2 (x+2)²
Odd Function
f(x)=-f(x) equals same number but has different signs f(x)=x³-2x f(-2)=(-2)³-2(-2) =-8+4 =-4 f(2)=2³-2(2) = 8-4 =4 Symmetric over the origin
Even functions
f(x)=f(-x) f(2)=2²-5=-1 f(-2)=(-2)²-5=-1 so f(x)=x²-5 is an even function Even functions are also symmetric over the y axis (points are equidistant)
Tip for finding the domain:
find the domain before you simplify
Principal square root of a negative number
for any positive real number b, the principal square root of the negative number -b is defined by √-b = i√b
Horizontal Line Test
functions that do not pass the Horizontal Line Tests means that the function does not have an inverse
Inverse of f(x) is written as
f⁻¹(x)
(g o f)(x)=
g(f(x))
Vertical Stretching or Shrinking
given f(x) is a function cf(x) is a vertical stretch when c>1 cf(x) is a vertical shrink when 0<c<1 (where c is a proper fraction) -In a vertical stretching or shrinking, multiply the y values of the previous step by c
Vertical Shift
given f(x) is a function f(x)+c is a vertical shift c units up f(x) -c is a vertical shift c units down Add c to the y values of the previous step to go up or subtract c from the y values of the previous step to go down
Horizontal shift
given f(x) is a function f(x-c) shifts f(x) c units right if c>0 f(x+c) shifts f(x) c units left if c<0 In a horizontal shift the x values of the control shift right or left
The imaginary unit i is defined as...
i=√-1 where i²=-1
How can you tell if an equation is a function
if the exponent on the y is odd
Two functions are inverses...
if they both equal x when you plug them into each other (find the compositions of them) Like this: f(f⁻¹(x)) and f⁻¹(f(x))
Evaluating a function
in f(x)=x-3 find f(-2) you plug -2 in for x, when x is negative two y is...
The axis of symmetry...
is always an equation in the form x=
What does the graph of a quadratic function look like?
it is always a parabola
In your steps...
make sure to refer to what graph you are transforming
When writing out the steps of the transformation...
make sure to write the transformations you do not perform as well
How to go from vertex form to standard form
multiply out vertex form
Composition of is expressed by a
o
The product rule for radicals...
only applies to real numbers
When finding the intervals for increasing, decreasing, and constants of a function...
only look at the x-values, not the y-values
Slope in standard form (ax+by=c)
opposite of A over B -A/B
When determining if a function is even or odd you must plug in numbers that are...
opposites of one another, like 2 and -2
Intercepts are listed as
ordered pairs
Extraneous solution (journal)
solutions that you obtain when solving the ration equation but do not result in a true statement when plugged back in to the original equation
If b=0 (in a+bi) then... (in complex number)
the complex number is a real number
If b≠0 (in a+bi) then... (for complex numbers)
the complex number is called and imaginary number
The Remainder Theorum
the remainder in synthetic division by "c" is the value of the function at the number "c". f(c).
Slope of a line definition
the steepness of a line
If A is negative...
the vertex is a maximum and the parabola opens downward
If A is negative
the vertex is a maximum and the parabola opens downwards
If A is positive
the vertex is a minimum and the parabola opens upward
If A is positive...
the vertex is a minimum and the parabola opens upward
The relative max or min is **at**
the x value
domain
the x's, set of all first components of the ordered pairs
The relative max or min **is**
the y value
range
the y's, set of all second components of the ordered pairs
If a function does not contain a radical or have a denominator...
then x can be anything (-∞,∞)
If a functions is neither even nor odd...
there is no symmetry
Relative minimum
turning point, where the function changes from decrease to increase
Relative maximum
turning point, where the function changes from increase to decrease
How to find the vertex from vertex form
vertex is (h,k) *Remember signs, h is already being subtracted*
Domain of a function definition
what x can equal
Equation of a vertical line
x=c c can be any number c=x intercept of line slope= undefined
Point slope form
y-y₁= m(x-x₁)
Equation of a horizontal line
y=b (or y=c) c can be any number slope= 0
Slope-intercept form
y=mx+b m=slope b= y intercept f(x)=mx+b <- linear function
If the discriminant is a negative number...
you get 2 imaginary solutions
If the discriminant is zero...
you get one real rational solution
If the discriminant is a positive perfect square...
you get two real rational solutions
If the discriminant is a positive non-perfect square...
you will get two real irrational solutions
Formula for finding slope of a line
y₂-y₁ −−−− = change in y over change in x x₂-x₁
vertex form of quadratic functions
ƒ(x)= a(x-h)²+k
general/standard form of quadratic functions
ƒ(x)=ax²+bx+c ax² is the quadratic term bx is the linear term c is the constant term
Slope as Average rate of change equation
∆Y/∆X f(x₂) -f(x₁) --------- x₂-x₁