Lone Star Final Exam Review for Algebra 2 DC 2019-2020 PDF

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 Lone Star Final Exam Review for Algebra 2 DC (1st semester)  Topic Number 1: Absolute Value  What is special about Absolute Value? It’s always positive What do you ALWAYS have to do when solving Absolute Value Equations and inequalities? Isolate What are the 3 different notations we discussed in class when writing out solutions to inequalities? Set notation, interval notation, inequality notation What do you ALWAYS have to do when solving Absolute Value Equations? Check What do we call solutions that do not fit with the original equation? Extraneous solutions When you have Absolute Value on both sides, how many equations do you have to solve? Two What scenario HAS TO HAPPEN in order for your answer to an absolute value equation be “No solution”? The absolute answer turns to a negative answer, but absolute values are always positive so if negative, that is false which means its a no solution. Solve the following Absolute Value Inequalities. Graph the solution and write the solution set in inequality notation, interval notation, and set notation (YES DO ALL 3!): 

| 8x − 3| > 21  2. || 12 y − 3|| ≥ 3  3. 2 |− 4x + 3| − 3 > 13 **  4. 4 |5x + 8| + 7 ≤ 23  1.

 Solve the following Absolute Value Equations and CHECK FOR EXTRANEOUS SOLUTIONS! 

|3 + |4 | 2. |3y − 1.

x|| = 5| = 9|

1 4 4 9

 

3. 6 |8 − 3y | − 1 = 35  4. 2 |5x − 2| + 5 = 15 **   Solve the following equations where the Absolute Value is on BOTH SIDES! 

| 2z + 1| = |3z − 5|  2. |3a − 1| = | 2a + 4|  3. |5x+3| = |2x-1|

1.

4. |3x-4| = |2x+3|  Graph the following absolute value functions given the transformations. 

y = |x| − 4  2. y = − |x − 2| + 2  3. y = 12 |2x + 4| − 6  4. y = − 2 |x − 4| + 2 ** 1.

 Topic Number 2: Complex Numbers  What is the general form for Complex Numbers? What is the real part? What is the imaginary part? A+bi bi: imaginary part a: real part What does “i” represent? (i.e. what is the “value” for i?) √-1 and i^2 = -1 What is the exponent trick related to i? (i.e. list out the values for i, i2 , i3 , and i4 ) i=i reaminders: 0=1 i^2=-1 remainders: 1=i i^3=-i remainders: 2=-1 i^4= 1 remainders: 3=-i How do you add complex numbers? Subtract? Multiply? Divide? Add complex numbers: Real with real and imaginary with imaginary Subtract complex numbers: distribute subtract then combine like terms Multiply complex numbers: FOIL OR BOX Dividing complex numbers: multiply by complex conjugate When dividing complex numbers what do you HAVE to multiply by? (i.e. what is the term called?)

Complex conjugate Perform the following Complex Number operations. 

1. 9i9 + 10i 10 + 11i 11 + 12i 12 **  2. i6 + i7 + i8 + i9  3. (3 + 5i) + (4 − 2i) + (− 2 + i)  4. 5. 6. 7. 8.

(5 + 8i) + (12 + 7i) − (3 − 2i) **  (4 − 5i)2 ** (4 + 2i)(1 − 3i)  (2 − 4i)/(1 + 3i)  (4 − 2i)/(7 − 3i) ** 

 Topic Number 3: Focus and Directrix  What are the two different forms for quadratics? Vertex (y = a(x-h)^2+k and standard form y = ax^2+bx+c What is the focus? Where is it located? focus is a point in the parabola What is the directrix? Where is it located? Directrix is perpendicular to the AOS of a parabola and doesn't touch the parabola so its outside of the parabola, and is a line outside the parabola What is the special property of the focus and directrix? Focus has the same distance from vertex to directrix What is the axis of symmetry? A line that divides the parabola in half (X= h) How do you find the vertex? How do you find “a”? How do you find “p”? Vertex: (h,k) A: ¼(p) P: ¼(a) What are the KEY formulas that will help you find focus, directrix, and the axis of symmetry? AOS: vertex form- the x coordinate of the vertex/ standard form- x=( -b/2a ) Focus: (h,k+p) Directrix: y= k-p  1. For the following quadratic, find the vertex, axis of symmetry, p, focus, and directrix. 

y = 2(x − 6)2 − 4 ** 

 Vertex: ______________**  Axis of Symmetry: ____________**  P: _______________**  Focus:__________________**  Directrix:___________________**      2. Given the following characteristics, find the equation of the parabola.  Focus: (0, -2) Vertex: (0, -6)  3. Vertex is located at the origin, Focus: (0, -1/32)  4. Vertex is located at the origin, Directrix: y = ¼  Topic Number 4: 3x3 Systems of Equations  What are the two methods that I went over as to how to solve 3x3 Systems of Equations? Traditional method/ Gaussian method (matrix) Solve the following systems using the methods asked in the previous question. Be able to use BOTH METHODS!  1. -2x+3y+4z = 1 3x-4y+z = -7 x+2y+3z = 5  2. x+3y+4z = 14 2x-3y+2z = 10 3x-y+z = 9  3.

x+y+z = 15** x-2y-2z= -15

2x+2y-z = 15  4.

x+y+z = 6 -3x+2y+3z = 10 -2x-4y+3z = -1

 Topic Number 5: Section 3.2 (Polynomial Functions)  1. Know the definition of a polynomial function. Function of degree n 2. Know the difference between what is a polynomial versus what is not. 3. What is a degree? What other terms can we use to define degree? Degree is the exponent or power 4. The values of x in the domain of a polynomial function f for which f(x)=0 are called the z  eros of the function. 5. What are the maximum number of turning points of the graph of f(x) = 2x4 − 4x 3 + 6x 2 − 5x + 10?  n-1= 3 turning points 6. If the graph of a polynomial function has 2 turning points, what is the minimum degree of the function? The degree is 3 7. If c is a real zero of a polynomial function and the multiplicity is 5, does the graph of the function cross or touch the x-axis at (c,0)?** cross 8. If c is a real zero of a polynomial function and the multiplicity is 8, does the graph of the function cross or touch the x-axis at (c,0)?** touch 9. Suppose that f is a polynomial function and that a < b. If f(a) and f(b) have opposite signs, then f has at least one zero on the interval [a,b]. What theorem is this? Intermediate value theorem 10. What is the leading term of a polynomial? Coefficient 11. What are the two ways we use to label end behavior? (i.e. what is the notation and how do we write it out?) Leading term test, notation for infinite behavior 12. How do you find zeros and multiplicities? -the values of x in the domain of f / f(0) or degree factors 13. List the first 5 steps to how you SKETCH a polynomial function. Use the leading term test to determine End behavior Determine y intercept by evaluating f(0)

Determine real zeros and multiplicities Plot x & y intercepts and note end behavior. Plot in the order that they appear from left to right using the rules - Odd multiplicities - is cross point - Even multiplicities - is touch point 14. Determine the END BEHAVIOR of the graph of the function h(x) = − 2(x + 4)(3x − 1) 3(x + 5)  15. Determine the END BEHAVIOR of the graph of the function k(x) = − 5x 4(2 − x) 3(2x + 5)  16. Find the zeros and state the multiplicities for the function p(x) = 4x 4 − 65x 2 + 16  17. Find the zeros and state the multiplicities for the function s(x) = 9x5 + 18x4 − 4x3 − 8x 2  18. Determine whether the Intermediate Value Theorem guarantees that the function has a zero on the given interval. d(x) = 9x3 − 18x2 − 100x + 200 for the interval [1,2].  19. Determine whether the Intermediate Value Theorem guarantees that the function has a zero on the given interval. b(x) = 2x3 − 13x2 + 18x + 5 for the interval [1,2]. 20. Sketch the following functions: a. k(x) = 2x2 + 8x − 24  b. t(x) = x4 − 10x2 + 9   Topic Number 6: Section 3.3 (Division of Polynomials/Factor and Remainder Theorems)  1. Given the division algorithm, identify the polynomials representing the dividend, divisor, quotient, and remainder. (i.e. f(x) = d(x) * q (x) + r(x) ) 

2. 3. 4.

5. 6.

f(x)= dividend d(x)= divisor q(x)=quotient r(x)=remainder Know how to use the division algorithm to check that you did the division correctly. The remainder theorem indicates that if a polynomial f(x) is divided by x - c, then the remainder is f(c). What goes in the blank? Given a polynomial f(x), the factor theorem indicates that if f(c) = 0, then x - c is a f actor of f(x). Furthermore, if x - c is a factor of f(x), then f(c) =. What goes in the blanks? If 4 is a zero of a polynomial, then (x-4) is a factor, TRUE OR FALSE? T  rue If 6 is a zero of a polynomial, then (x+6) is a factor, TRUE OR FALSE? False

7. Know how to build a polynomial given the zeros of a function. 8. What are the two different methods we use to divide polynomials? Synthetic divison, and long division 9. When can you use the Synthetic Division? (i.e. What are the conditions that HAVE to be met?) When its divided by x-c 10. Use Long Division to divide then identify the dividend, divisor, quotient, and remainder and finally check the result with the Division Algorithm. (6x 2 + 9x + 5) ÷ (2x − 5)  11. Use Long Division to divide (x 5 − 2x 4 + x3 − 8x + 18) ÷ (x 2 − 3)  12. Use Long Division to divide (2x 3 + x2 + 1) ÷ (3x + 1)  13. Use Synthetic Division to divide the polynomials (x 4 − 81) ÷ (x + 3)  14. Use Synthetic Division to divide the polynomials (2x 5 + 13x 4 − 3x 3 − 58x 2 − 20x + 24) ÷ (x − 2)  15. Use Remainder Theorem to determine if the given number c is a zero of the polynomial p(x) = 2x 3 + 3x 2 − 22x − 33 f or c = − 2  16. Use Remainder Theorem to determine if the given number c is a zero of the polynomial f(x) = x 4 + 3x 3 − 7x 2 + 13x − 10 f or c = 2  17. Use Factor Theorem to determine if the given binomial is a factor of f(x). f(x) = x 4 + 11x 3 + 41x 2 + 61x + 30; binomials are x + 5 and x − 2  18. Use Factor Theorem to determine if the given binomial is a factor of f(x). f (x) = x3 + 64; binomials are x − 4 and x + 4  19. Factor and Solve f (x) = 3x3 + 16x2 − 5x − 50, given − 2 is a zero.  20. Factor and Solve f(x) = 4x3 − 20x 2 + 33x − 18, given 2 is a zero.  21. Write a polynomial f(x) that meets the given conditions: Degree 3 polynomial with zeros 1, -6, and 3. 22. Write a polynomial f(x) that meets the given conditions: Degree 5 polynomial with zeros 2, 5/2 (each with multiplicity 1), and 0 (with multiplicity 3).  Topic Number 7: Section 3.4 (Zeros of a Polynomial Function)  1. Know the Rational Zero Theorem, Fundamental Theorem of Algebra, Linear Factorization Theorem, and the Conjugate Zeros Theorem. (This means you MUST KNOW THE DEFINITION/THEOREM FORWARDS AND BACKWARDS!!) - Rational Zero Theorem- If f(x) has integer coefficients and “an” doesn’t equal 0,a dn if p/q (written in lowest terms) is a rational zero of f, then: p is a factor of the constant term a0, q is a factor of the leading coefficient an. - Fundamental Theorem of Algebra- If f(x) is a polynomial of degree n greater than or equal to 1 with complex coefficients, then f(x) has at least one complex zero

-

2. 3. 4. 5.

Linear factorization Theorem- If f(x) where n greater or equal to 1 and “an” not equal to 1, then f(x)=an(X-C1)(X-C2)...(X-Cn) where C1, C2, … Cn are complex numbers - Conjugate Zeros Theorem- If f(x) is a polynomial with real coefficients and if a+bi (doesn’t equal 0) is a zero of f(x), then its conjugate a-bi is also a zero of f(x) List the possible rational zeros for the following polynomial: f (x) = x5 − 2x3 + 7x 2 + 4  List the possible rational zeros for the following polynomial: f (x) = 2x3 − 5x2 + 12  Find ALL the zeros for the following polynomial: f (x) = 5x3 − x2 − 35x + 7  Find ALL the zeros for the following polynomial: f (x) = x 3 − 7x 2 + 6x + 20 

6. Find all the zeros, write out the factors using the Linear Factorization Theorem, and solve the equation f (x) = 0 . f (x) = x 4 − 4x 3 + 22x 2 + 28x − 203; 2 − 5i is a zero  7. Find all the zeros, write out the factors using the Linear Factorization Theorem, and solve the equation f (x) = 0 . f (x) = 3x3 − 28x2 + 83x − 68; 4 + i is a zero  8. Write a polynomial f(x) that satisfies the given conditions: Degree 3

⅘

polynomial with integer coefficients with zeros 6i and  . 9. Write a polynomial f(x) that satisfies the given conditions: Polynomial of lowest degree with zeros of -4 (multiplicity 1), 2 (multiplicity 3) and with f(0) = 160. 10. Write a polynomial f(x) that satisfies the given conditions: Polynomial of lowest degree with zeros of 7-4i and 0 (multiplicity 4)  Topic Number 8: Section 3.5 (Rational Functions)  1. Know the definition of a rational function. Let p(x) and q(x) be polynomials where q(x) doesn’t equal 0. A function f defined by f (x) = p(x)/q(x) is called rational function 2. How do you find the Vertical Asymptotes? Horizontal Asymptotes? Slant Asymptotes? (What is the strategy/technique for finding EACH one?) Vertical asymptotes: set bottom to zero (x = 0) Horizontal asymptotes: compare top and bottoms highest degree *if degree is the same the look at the constant *If bottom bigger than top y = 0 *If top bigger no horizontal asymptotes Slant Asymptotes: use long division (divide the numerator of the function by the denominator 3. What are the steps to sketching/graphing a rational function? 1. Determine y-intercept

2. Determine x-intercept 3. Identify the vertical asymptotes 4. Identify horizontal asymptotes 5. If not horizontal and slant asymptotes 6. Determine behavior on each interval 4. How do you find the domain of a rational function? The range? The domain of a rational function is all real numbers excluding the real zeros of q(x). The range 5. What are the 4 notations you need to know? What is the strategy/technique you need to use to find the result for f(x)? (i.e. As x ---> ∞ or x ----> -∞, what does f(x) approach? As x approaches the vertical asymptote from the left or from the right, what does f(x) approach?)  6. Given the following graph, answer the following questions:

 a. b. c. d.

As x ----> -∞, f(x) ----> _____________ As x ----> ∞, f(x) ----> _____________ − As x ----> − 2 , f(x) ----> _____________ + As x ----> − 2 , f(x) ----> _____________ − e. As x ----> 2 , f(x) ----> _____________ + f. As x ----> 2 , f(x) ----> _____________  

7. Given the following graph, answer the following questions:

 a. As x ----> -∞, f(x) ----> _____________ b. As x ----> ∞, f(x) ----> _____________ − c. As x ----> − 3 , f(x) ----> _____________ + d. As x ----> − 3 , f(x) ----> _____________   8. Determine the Vertical Asymptotes from the Rational Functions given: a. 8/(x − 4)  b. (x − 3)/(2x 2 − 9x − 5)  c. x/(x2 + 5)  9. Determine the Horizontal Asymptotes from the Rational Functions given: a. 5/(x2 + 2x + 1)  b. 3x 2 + 8x − 5/(x 2 + 3)  c. x4 + 2x + 1/(5x + 2)  10. Determine the Slant Asymptotes from the Rational Functions given: a. (2x 2 + 3)/(x)  b. (− 3x 2 + 4x − 5)/(x + 6)  c. (x 3 + 5x 2 − 4x + 1)/(x 2 − 5)  11. Graph the Rational Functions by using the Transformations of the graphs of y = 1/x and y = 1/ x2  a. 1/(x − 3)  b. 1/x2 + 2  c. 1/(x + 4) 2 − 3  d. − 1/x 

12. For the following f(x) find the x-intercepts, vertical asymptotes, horizontal or slant asymptotes, and y-intercepts: a. f(x) = (x + 3)(2x − 7)/(x + 2)(4x + 1)  b. f (x) = (4x − 9)/(x2 − 9)  13. Sketch a Rational Function subject to the given conditions - Horizontal asymptote: y = 2; Vertical Asymptote: x = 3; y-intercept: (0, 8/3); x-intercept: (4,0) 14. Sketch a Rational Function subject to the given conditions - Horizontal asymptote: y = 0; Vertical Asymptote: x = -2 and x = 2; y-intercept: (0, 1); no

-⅘)

x-intercepts; symmetric to the y-axis; passes through the point (3,  15. Sketch the following rational functions: a. − 3/(2x + 7)  b. (x − 4)/(x − 2)  c. (2x − 4)/(x + 3)  d. 6/(x2 − 9)  e. 5x/(x2 − x − 6)  f. (3x 2 − 5x − 2)/(x 2 + 1)  g. (x 2 + 7x + 10)/(x + 3) 

 Topic Number 9: Section 3.6 (Polynomial and Rational Inequalities)  1. What is the definition of a Polynomial Inequality? Let f(x) be a polynomial. Then an inequality of the form f(x) < 0. f(x) > 0, f(x) ≤ 0, or f(x) ≥ 0. 2. What is the Procedure to Solve a Nonlinear Inequality? (i.e. what are the steps?) 1. Express the inequality as f(x) < 0, f(x) > 0, or f(x) ≥ 0. That is, rearrange the terms of the inequality so that one side is set to zero 2. Find the real solutions of the related equation f(x) = 0 and any values of x that make f(x) undefined. These are the “boundary” points for the solution set to the inequality. 3. Determine the sign of f(x) on the intervals defined by the boundary points a. If f(x) is positive, then the values of x the interval are solutions to f(x) > 0. b. If f(x) is negative, then the value of x interval are solutions to f(x) < 0. 4. Determine whether the boundery points are included in the solution set 5. Write the solution set in interval notation and set- builder notation 3. Solve the following inequalities: a. 4w2 − 9 ≥ 0 

b. 3w2 + w < 2(w + 2)  c. a2 ≥ 3a  d. (x + 4)(x − 1)(x − 3) ≥ 0  4.

e. − 5c(c + 2) 2(4 − c) > 0  Solve the following Rational Inequalities: a. (x + 2)/(x − 3) ≤ 0  b. 5/(2t − 7) > 1  c. 2x/(x − 2) ≤ 2  d. 4/(y + 3) > − 2/y 

 Topic Number 10: Section 3.1 (Quadratic Functions)  1. What is the definition of a Quadratic Function? A function defined by f(x) = ax^2+ bx+c is (a≠0) 2. What is Vertex Form? Standard Form? Vertex form: f(x)= a(x-h)^2+k Standard form: f(x) = ax^2+ bx+c 3. What is the Focus? Directrix? Focus: always in parabola Directrix: outside of the parabola ● Focus has the same distance from vertex to directrix 4. Can you find the following key attributes of a Quadratic Function: 1.) vertex: (h,k) 2) x-intercept(s): Set problem equal to zero and find the x 3) y-intercept: plug in x’s as zeros 4) axis of symmetry: vertex form- the x coordinate of the vertex/ standard form- x=-b2a 5) the max or min value: standard form: -b/2a vertex form: look at k if negative min if positive its max/ max= mountain min= valley 6) domain and range in set and interval notation: 5. Can you determine if the parabola opens upward or downward? If a is positive, the parabola opens up. If a is negative, the parabola opens down 6. What is the Vertex Formula to Find the Vertex of a Parabola? f(x)=a(x-h)^2+k 7. Can you use the Discriminant to determine the # of x-intercepts? Yes 8. What is the Discriminant? 1. If b^2 -4ac= 0, the graph of y = f(x) has one x-intercept 2. If b^2 ﹥4ac= 0, the graph of y = f(x) has two x intercepts 3. If b^2﹤4ac= 0, the graph of y = f(x) has NO REAL x-intercepts  9. Graph the following Quadratic Functions: a. f (x) = − (x − 4)2 + 1 

b. h(x) = 2(x + 1)2 − 8  10. Write the function in Vertex Form and determine the vertex, x-intercept(s), y-intercept, sketch the function, determine the axis of symmetry, minimum or maximum?, write the domain and range in set and interval notation: a. x2 + 6x + 5  b. x2 + 9x + 17  11. Find the vertex of the parabola by applying the vertex formula. a. 3x2 − 42x − 91  b. − 1/3a2 + 6a + 1  12. Find the focus and directrix for the following quadratic functions and also put the quadratic function in Standard Form (refer to Topic Number 3): a. f (x) = − (x − 4)2 + 1  b. h(x) = 2(x + 1)2 − 8  

 Lone Star Final Exam Review for Algebra 2 DC (2nd Semester)  Topics that We Have Covered: 1. Inverse Functions 2. Exponential Functions 3. Logarithmic Functions 4. Properties of Logarithms  An In-Depth Look at Topic #1 (Inverse Functions)  1. Go back to the Guiding Questions in D2L (Google Classroom) and see if you have a better understanding of this section. The next set of questions will check said understanding.   2. What does it mean if a function is one-to-one in YOUR OWN words, not the definition?   3. How do you determine if a function is one-to-one? ...