Algebra FInal
a²+b²
(a-b)(a+b)=?
(f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fg)(x)=f(x)Xg(x) (f/g)(x)=f(x)/g(x) (F₀G)(x)=f(g(x))
(f+g)(x)=? (f-g)(x)=? (fg)(x)=? (f/g)(x)=? (F₀G)(x)=?
-b/2a
A quadratic function is symmetric about this axis.
Divide the factors of the constant term by the factors of the leading coefficient ------------------------ Use synthetic division to test the possible rational zero's. If one has no remainder, then it is a solution. ------------------------- Use the result of the synthetic division to factor the polynomial and solve. (x-2)(x²+3x+1) Solve x²+3x+1=0
According to the Rational Zero Theorem, how do you find all possible rational zero's of a polynomial function? ----------------- After finding all possible rational zero's, how to do you determine which is an actual zero? ----------------- After using the rational zero theorem to find an actual rational zero (through synthetic division), what do you do with this information? x³+x²-5x-2 where 2 is a rational zero
Origin
An odd function is symmetric with respect to what?
If a>0 then f has a minimum value of f(-b/2a) If a<0 then f has a minimum value of f(-b/2a)
Considering the function f(x)=ax²+bx+c, where does the minimum or maximum value of f occur?
Constant Function: f(x)=c Identity Function: f(x)=x Absolute Value Function: f(x)=|x| Quadratic Function: f(x)=x² Square Root Function: f(x)=√x Cubic Function: f(x)=x³ Cube Root Function: f(x)=³√x
Constant Function Identity Function Absolute Function Quadratic Function Square Root Function Cubic Function Cube Root Function
where k is the constant of variation Direct Variation: y=kx Inverse Variation: y=k/x Combined Variation: y=(kn)/p Joint Variation: y=kxz
Direct Variation Inverse variation Combined Variation Joint Variation
Rate X Time = Distance
Equation to a uniform motion problem
f(-x)=f(x)
Even function
y-axis
Even functions are symmetric with respect to which axis.
1. Count the number of sign changes in the equation f(x). The positive real zero's are even possibilities of that number. Ex. 4 changes in sign means there are either 4,2, or 0 positive real zeros. 2. Count the number of sign changes in f(-x). Again, the positive real zeros are even possibilities of that number.
Explain Descarte's Rule of Signs
X-Intercept Solve f(x)=0 (2^x)-3=0 ln2^x=ln3 Take natural log of both sides Xln2=Ln3 Bring X to the front X= Ln2/Ln3 Horizontal Asymptote = the starting point of the vertical shift. Therefore HA: y=-3
Find the x interecept and horizontal asymptote of f(x)=(2^x)-3
Vertex: Use the formula x=-b/2a Therefore x=1 To find the y value, use the result of the vertex formula to find f(resulting value) f(1)=-4 Therefore the vertex is (1,-4) Axis of Symmetry: The axis is the x value of the vertex x=1 X-intercepts: Set the equation to 0 x²-2x-3=0 Use either the quadratic formula or factor y- intercept: Solve f(0)
For the following quadratic equation: x²-2x-3 Find the vertex Axis of Symmetry X and Y intercepts
Vertical Asymptote: set denominator =0 x²-25=0 x=-5 x=5 VA: x=-5 x=5 Horizontal Asymptote: Follow the rules of a horizontal asymptote of the exponents of the leading coefficients. The exponent of the numerator is lower than the exponent of the denominator. therefore HA: y=0 X-intercept: Set the numerator =0 x-2=0 X-intercept: (2,0) Y-intercept: Evaluate f(0) -2/-25 Y-intercept: (0,2/25)
For the following rational function: f(x)= (x-2)/(x²-25) Find Horizontal and vertical asymptotes and graph the x and y intercepts
If positive, the graph rises to the right and the left. If negative, the graph falls to the left and the right.
For the graph of an even polynomial function, where the leading coefficient is positive, what is the graph's shape? What about a negative leading coefficient of an even function?
If positive, the graph falls to the left and rises to the right. If negative, the graph rises to the left and falls to the right.
For the graph of an odd polynomial function, where the leading coefficient is positive, what is the graph's shape? what about a negative leading coefficient of an odd function?
m=(y₂-y₁)/(x₂-x₁)
Formula for Slope
x²+y²+Dx+Ey+F=0
General Form of the Equation of a Circle
Ax+By=C
General form of a line
a. Matrix form (Note that there should be only one set of parenthesis per set (vertically) and one equal sign) (3 -7)(x)=(10) (11 8)(y)=(31) b. Either use the formula or elementary row of operations (3 -7)⁻¹=1/24-(-77) X (8 7) (11 8) (-11 3) Answer (8/101 7/101) (-11/101 3/101)
Given the following system {3x-7y=10 {11x-8y=31 a. Write the system in matrix form b. Find the inverse of the coefficient matrix
Add (b/2)² to both sides.
How do you complete the square?
If a>0, it opens upward If a<0, it opens downward
How do you determine if a parabola opens upward or downward using the equation f(x)=a(x-h)²+k or f(x)=ax²+bx+c?
(h,k)
How do you determine the vertex of a quadratic equation using the equation f(x)=a(x-h)²+k?
fing g(x) and then find f(g).
How do you find the composite function (F₀G)(x)?
Step 1: Replace f(x) with y: y=2x+7 Step 2: Interchange x and y x=27+7 Step 3: Solve for y y=(x-7)/2 Step 4: Replace y with f⁻¹(x)
How do you find the inverse of a function? Use f(x)=2x+7 as an example.
(-b/2a, f(-b/2a))
How do you find the vertex of a parabola whose equation is f(x)=ax²+bx+c?
Solve f(x)=0 to find the function's real 0's --------------- solve f(0)
How do you find the x-intercepts of the equation f(x)=a(x-h)²+k or f(x)=ax²+bx+c? --------------- How do you find the y-intercepts of f(x)=a(x-h)²+k or f(x)=ax²+bx+c?
Divide factors of the constant by factors of the leading coefficient. 1, -1, 2, -2
How do you find the zeros of a polynomial function using the rational zero theorem? x³-5x²+7x-2
Use synthetic division until one results with a quotient of 0. Set the result to the opposite sign. (x-2) 2/ 1 -5 7 -2 1 -3 1 0 Use the result numbers to create a new second zero. (x-2)(x²-3x+1) Then solve the resulting equation either by factoring or using the quadratic formula.
How do you find the zeros, real and complex, of a polynomial function? x³-5x²+7x-2
Eliminate the same variable from two equations to form a new system of equations a. x-y-z=3 + c. x+3y+x=-1 ---------------- d. 2x+2y=2 b. 3x-2y-2z=5 + c(2). 2x+6y+2z ______________________ e. 5x+4y=3 Use d and e to eliminate one more variable One the variable is eliminated, solve for the remaining variable. Plug that variable back into d or e to find the other variable. Augmented Matrix: (Only one set of parenthesis) (1 -1 -1 | 3) (3 -2 -2 | -5) (1 3 1 | 1)
How do you solve a system of equations with three variables? a. x-y-z=3 b. 3x-2y-2z=5 c. x+3y+z=-1 When complete, write the augmented matrix
f(x)=3^x-1
How would you write the exponential function, base 3, phase shifted one unit right?
n-1
If f is a polynomial function of degree n, then the graph of f has at most, this many turning points.
There are no points on the real number line where the graph changes sign and thus the inequality will be satisfied by either all real numbers or no real numbers. Solve by testing for any value of x. If x>0 then all real numbers satisfy. If x<0 then no real numbers satisfy.
If there are no real solutions to a polynomial inequality, then you must conclude what and solve how?
Conjugate pairs a+bi a-bi
Imaginary roots, if they exist, occur in what? a+bi
Reverse the signs
In order to find (h,k) in the standard form of a circle (x-h)²+(y-k)²=r², what must you do to h and k?
Complete the square by adding (b/2)² to both sides.
In order to find the center and radius of a circle that is not in standard form, such as x²+bx, you must do what?
Take the square root
In order to find the r of a circle, when the equation is in standard form (x-h)²+(y-k)²=r², what must you do to r?
Even Multiplicity: touch the x-axis at r but not cross it. Odd Multiplicity: go across the x-axis at r
In regards to polynomial functions if r is a zero of even multiplicity what will it do? What about odd multiplicity?
If p and q have no common factors, and the degree of p is one greater than the degree of q. The slant asymptote can be found by dividing q(x) into p(x).
In what scenario is there a slant asymptote, and how is it located?
i⁰=0, i¹=i, i²=-1, i³-i, i⁴=1
List the rules for exponents of imaginary numbers up to i⁴
1. If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f. 2. If n=m, the line y=a/b is the horizontal asymptote of the graph of f. 3. If n>m, the graph of f has no horizontal asymptote.
List the rules for locating horizontal asymptote's where n is the degree of the numerator and m is the degree of the denominator.
1. Determine symmetry 2. Find the y-intercept by solving f(0) 3. Find the x-intercepts by solve the equation p(x)=0 4. Find any vertical asymptote's by solving the equation q(x)=0 5. Find the horizontal asymptote using the rule for determining the horizontal asymptote of a rational function. 6. Pl0t at least one point and beyond each x-intercept and the vertical asymptote. DON'T FORGET THAT THE FINAL GRAPH WILL INCLUDE THREE PIECES; THE CENTER, AND ONE CURVE AT EACH END
List the steps in graphing p(x)/q(x)
Point Slope Form: y-y₁=m(x-x₁) Slope Intercept Form: y=mx+b General Form: Ax+By=C
List the three equations of a line.
x The natural log of the variable e, is equal to the exponent of e
Ln(e^x)=?
(x₁+x₂/2 , y₁+y₂/2)
Midpoint Formula; This finds the exact midpoint between two points.
f(-x)=-f(x)
Odd function
Step 1: Find f(g(x)) 4((x+7)/4)-7=x Step 2: Find g(f(x)) (4x-7+7)/4=x Step 3: Do they both equal the same? If so, then they are inverse
Prove that f(x)=4x+7 is an inverse function of g(x)=(x+7)/4
x= (-b±√(b²-4ac))/2a
Quadratic formula
y=mx+b
Slope Intercept Form
Divide left side by the coefficient 2=e^.1468t take natural log of both sides Ln2=lne^1468t (remember that the Ln of e is equal to the exponent) Ln2=.01468t t=Ln2/.1468
Solve 18000=9000e^.1468t
(3²)^3x-2=3⁻³ 3²^(3x-2)=3⁻³ 2(3x-2)=-3 6x-4=-3 6x=1 x=1/6
Solve 9^(3x-2)=1/27
f(4x²+7) 5(4x²+7)-3 Etc...
Solve f(g(x)) where f(x)=5x-3 and g(x)=4x²+7
Step 1: Grouping (x³+x²)+(5x+5)=0 Step 2: Factor x²(x+1)+5(x+1)=0 Step 3: Regroup by factors (x²+5)(x+1)=0 Step 4: Solve using the Zero Product Property x²+5=0 and x+1=0 x²=-5 and x=-1 x=±√-5 x=±i√5
Solve the cubic equation: x³+x²+5x+5=0
Step 1: Factor out a variable to get to a sum of cubes (a³+b³) 2x(x³+8x)=0 Step 2: Put into the Sum of Cubes equation: 2x(x+2)(x²-2x+2²)=0 Step 3: Use the zero product property: 2x=0 and x+2=0 and x²-2x+4=o x=0 and x=-2 and use not solved Step 4: Solve the remaining equation using the quadratic formula and simplify. x=(2±√-12)/2 Step 5: Solve x=1±i√3
Solve the following equation using the sum of cubes equation: 2x⁴+16x=0
Step 1: Change to a positive exponent by taking the inverse 1/(x³/²)=1/27 Step 2: Cross multiply x³/²=27 Step 3: Solve by setting both sides equal to the index of the radical and then simplify x³=27² x³=729 Step 4:Take the cube root of both sides (remember that a ± is not needed as the index is even.) x=³√729 x=9
Solve x-³/²=1/27
First graph the equation and find the x-intercepts. Do this by setting the equation=0 and solving, either by factoring or using the quadratic formula. Check where the equation meets the requirement (here it is >0) by testing each of the three sections. Solution set is those numbers where the inequality is satisfied
Solve x²+2x-15>0
Step 1: Set both sides equal to the index of the radical x²=4³ x²=64 Step 2: Use the square root property to solve x=±√64 Step 3: Simplify if needed x=±8
Solve x²/³=4
(x-h)²+(y-k)²=r²
Standard form of a Circle
(x-3)²+(y+4)²=17 Step 1: Factor x²-6x+9+y²+8y+16=17 Step 2: Rearrange into general form x²+y²-6x+8y+8=0
Steps required to convert from Standard Form to General Form of a circle
x²+y²+4x-4y-1=0 Step 1: Regroup (x²+4x) +(y²-4y) = 1 Step 2: Complete the square (x²+4x+4)+(y²-4y+4)=1+4+4 Step 3: Factor (x+2)²+(y-2)²=9 --> (x+2)²+(y-2)²=3² Step 4: Find Center and R Center=(2,-2) and r=3
Steps to convert General form to Standard form of a circle.
(a³+b³)=(a+b)(a²-ab+b²)
Sum of cubes
√(x₂-x₁)²+(y₂-y₁)²
The Distance Formula; The distance between two points (x,y)
x²=k is equal to x=±√k
The Square Root Property
Amount X Concentration = Amount of Substance
The equation to a mixture question.
0
The factor theorem states that if f(c)=this then x-c is a factor of f(x).
f(c)
The remainder theorem states that if f(x) is divided by x-c, then the remainder is this.
ax²+bx+c=0
The standard form of a quadratic equation.
Step 1: Leading coefficient of 1 Step 2: Isolate x variables x²-4x=-1/3 Step 3: Complete the square x²-4x+4=-1/3 + 4 ---> (x-2)²=-1/3+4 Step 4: Use the square root property x-2=±√-11/3 ---> x=2±i√11/3) Step 5: Rationalize if needed x=2±i√(33)/3
What are the steps in solving a quadratic by completing the square? Use the equation of 3x²-12x+1=0 as reference
Vertical asymptote's are zero's of the denominator of a rational function.
What are the vertical asymptote's of a rational function?
1. Find what makes the denominator 0 2. Find what causes a square root of a negative number
What do you exclude from the domain of functions?
Exponential Function F(x)=(1^x)+0
What does this graph represent?
Constant Function
What function does this represent?
Cubic Function
What function does this represent?
Quadratic Function
What function does this represent?
Square Root Function
What function does this represent?
Square Root Function
What function is this?
(f(x+h)-f(x))/h where h≠0
What is the equation for a difference quotient?
A=P(1+r/n)^nt where A is balance of P principal after t years at r rate
What is the formula for compounding interest in t years?
A=Pe^rt Where A is the amount after interest, P is the principal, e is constant, r is rate of interest, and t is time in years.
What is the formula for continually compounding interest and what does each variable mean?
A=Pe^rt
What is the formula for continuous compounding interest?
e
What is the natural base of an exponential function?
a+bi
What is the standard form of a complex number.
f(x)=a(x-h)²+k, a≠0
What is the standard form of a quadratic function?
Function f has an inverse if there is no horizontal line that intersects the graph of the function f at more than one point.
What test can be used to graphically prove that f has an inverse or not?
Use the factor theorem to state that since f(-1) is a zero then you divide the equation by x+1.
When told that -1 is a zero of a polynomial function, how can you solve the function?
x=0
Where is the horizontal asymptote of an exponential function?
Absolute Value Function
Which function is this?
Cube Root Function
Which function is this?
exponential function
Which function is this?>
(a+bi)(a-bi)=a²+b²
a²+b²=?
y₁-mx₁
b in Slope Intercept form y=mx+b
y=logb(x)
b^y=x is equivalent to what?
y=3x+7 Let y = f(x) x=3y+7 Switch y and x (x-7)/3=y Solve for y therefore f⁻¹(x)=(x-7)/3
f(x)=3x+7 find f⁻¹(x)
2 because log means 10 to what power equals 100.
log base10(100)=?
0
logb(1)=? logbase10(1)
1
logb(b)=? logbase10(10)
5^-3
logbase5(1/125)=?
b^y=x
y=logb(x) is equivalent to what?
