AATH Comprehensive Study Guide by Abdullah Sultan (made for Jaqua)

ADVANCED ALGEBRA AND TRIGONOMETRY HONORS FINAL EXAM STUDY GUIDE

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Goal: Master key concepts, formulas, graphing techniques, and problem-solving strategies for each unit, including systems and inequalities.

General Study Tips:

• Review Notes: Go through your class notes and homework from each unit.

• Practice Problems: Redo homework problems, quizzes, and tests. Find extra practice problems online or in your textbook.

• Understand Concepts: Don't just memorize formulas. Understand why they work and when to use them.

• Use Flashcards: Create flashcards for formulas, definitions, and unit circle values.

• Teach Someone Else: Explaining concepts to a friend is a great way to solidify your understanding.

• Get Enough Sleep: Crucial for retaining information and performing well on test day.

Unit 1: Exponential Functions and Equations

• Definition: Functions where the variable is in the exponent. Standard form: y=abx.

  • a: initial value (y-intercept when x=0)

  • b: base (growth/decay factor)

    • If b>1: Exponential Growth

    • If 0<b<1: Exponential Decay

    • b=1,b>0

• Graphing:

  • Horizontal Asymptote: Usually y=0 for basic y=abx. Shifts with vertical transformations.

  • Domain: All real numbers (−∞,∞).

  • Range: (0,∞) for basic growth/decay, shifts with vertical transformations.

  • Key Point: (0,a) (y-intercept).

  • Transformations: y=a⋅bx−h+k (horizontal shift h, vertical shift k).

• Special Base e: The natural exponential function y=ex. e≈2.718. Used frequently in growth/decay models (A=Pert).

• Solving Exponential Equations:

  • Common Base: If possible, rewrite both sides with the same base, then set the exponents equal.

  • Using Logarithms: If bases can't be made common, isolate the exponential term, then take the logarithm (usually ln or log) of both sides. Use log properties to bring the exponent down.

  • Example: 5x=12⟹log(5x)=log(12)⟹xlog(5)=log(12)⟹x=log(5)log(12)​.

Unit 2: Square and Cube Root Functions and Equations

• Definition: Functions involving ​ or 3​. Standard forms: y=ax−h​+k, y=a3x−h​+k.

• Graphing:

  • Square Root: Starts at a point (h,k) and extends in one direction.

    • Domain: x−h≥0⟹x≥h.

    • Range: Depends on a and k. If a>0, range is y≥k. If a<0, range is y≤k.

  • Cube Root: Starts at (h,k) and extends in both directions (like a sideways 'S').

    • Domain: All real numbers (−∞,∞).

    • Range: All real numbers (−∞,∞).

  • Transformations: (h,k) is the "starting point" for square root and the "center" for cube root graphs. a affects stretch/shrink and reflection.

• Solving Radical Equations:

  • Isolate the radical term on one side.

  • Raise both sides to the power of the index (square for ​, cube for 3​).

  • Solve the resulting equation (often linear or quadratic).

  • CHECK FOR EXTRANEOUS SOLUTIONS! This is crucial for square root equations, as squaring both sides can introduce solutions that don't work in the original equation. Plug your answers back into the original equation.

Unit 3: Quadratic Functions and Equations

• Definition: Functions of degree 2.

  • Standard Form: y=ax2+bx+c.

    • Vertex: x-coordinate is −b/(2a). Substitute back into the equation to find the y-coordinate.

    • c is the y-intercept.

  • Vertex Form: y=a(x−h)2+k.

    • Vertex: (h,k).

• Graphing (Parabolas):

  • Direction of opening: Up if a>0, Down if a<0.

  • Vertex: The minimum or maximum point.

  • Axis of Symmetry: Vertical line x=h (or x=−b/(2a)).

  • Intercepts:

    • y-intercept: Set x=0.

    • x-intercepts (Roots/Zeros): Set y=0 and solve the quadratic equation.

• Solving Quadratic Equations (ax2+bx+c=0):

  • Factoring: If possible, factor the quadratic and set each factor to zero.

  • Completing the Square: Useful for converting to vertex form or solving equations.

  • Quadratic Formula: Always works! x=2a−b±b2−4ac​​.

• The Discriminant (D=b2−4ac): Tells you the nature of the roots without solving.

  • D>0: Two distinct real roots (parabola crosses x-axis twice).

  • D=0: Exactly one real root (a double root; parabola touches x-axis at the vertex).

  • D<0: Two complex conjugate roots (parabola does not cross x-axis).

Unit 4: Logarithmic Functions and Equations

• Definition: The inverse of an exponential function.

  • Logarithmic form: logb​x=y

  • Exponential form: by=x

  • Base b, argument x.

• Common Logarithm: logx=log10​x.

• Natural Logarithm: lnx=loge​x.

• Graphing:

  • Vertical Asymptote: Usually x=0 for basic y=logb​x. Shifts with horizontal transformations.

  • Domain: (0,∞) for basic logb​x. Argument must be positive.

  • Range: All real numbers (−∞,∞).

  • Key Point: (1,0) (x-intercept).

  • Transformations: y=alogb​(x−h)+k.

• Properties of Logarithms: Essential for simplifying and solving.

  • logb​(MN)=logb​M+logb​N (Product Rule)

  • logb​(NM​)=logb​M−logb​N (Quotient Rule)

  • logb​(Mp)=plogb​M (Power Rule)

  • logb​b=1

  • logb​1=0

  • logb​(bx)=x

  • blogb​x=x (if x>0)

  • Change of Base: logb​M=loga​bloga​M​ (commonly logblogM​ or lnblnM​).

• Solving Logarithmic Equations:

  • Condense logarithms on each side using properties if necessary.

  • If logb​M=logb​N, then M=N.

  • If you have a single log term, convert to exponential form.

  • CHECK FOR EXTRANEOUS SOLUTIONS! The argument of a logarithm must be positive.

Unit 5: Absolute Value Functions and Equations

• Definition: ∣x∣=x if x≥0, and ∣x∣=−x if x<0. Represents distance from zero.

• Graphing:

  • Basic shape is a "V".

  • Vertex: The point where the graph changes direction. For y=a∣x−h∣+k, the vertex is (h,k).

  • Transformations: a affects the width and direction the V opens (up if a>0, down if a<0). h and k are horizontal and vertical shifts.

• Solving Absolute Value Equations: ∣ax+b∣=c

  • If c<0, there is no solution. Absolute value cannot be negative.

  • If c≥0, set up two equations:

    1. ax+b=c

    2. ax+b=−c

  • Solve both linear equations.

• Solving Absolute Value Inequalities:

  • ∣ax+b∣<c: −c<ax+b<c (Solve as a compound inequality). "Less ThAND"

  • ∣ax+b∣>c: ax+b>c OR ax+b<−c (Solve as two separate inequalities). "GreatOR Than"

Unit 6: Finding the Roots of a Function / Odd or Even Functions

• Roots, Zeros, x-intercepts: These all refer to the values of x where f(x)=0 (where the graph crosses the x-axis).

• Finding Roots from a Graph: Look at the graph and visually identify the x-coordinates where the graph intersects the x-axis.

• Finding Roots using Synthetic Division:

  • Used to test potential rational roots of a polynomial function.

  • Rational Root Theorem: (Might be covered, good to know) If a polynomial has integer coefficients, any rational root p/q must have p as a factor of the constant term and q as a factor of the leading coefficient.

  • Process: To test if c is a root, perform synthetic division with c. If the remainder is 0, then c is a root, and (x−c) is a factor. The result of the synthetic division is a polynomial of one degree lower, which you can then try to factor or use the quadratic formula on to find the remaining roots.

• Odd and Even Functions: Determine symmetry.

  • Even Function: f(−x)=f(x) for all x in the domain. Symmetric about the y-axis. (Examples: y=x2,y=cosx,y=∣x∣)

  • Odd Function: f(−x)=−f(x) for all x in the domain. Symmetric about the origin. (Examples: y=x3,y=sinx,y=1/x)

  • Neither: If neither of the above conditions holds.

  • How to Test: Substitute −x into the function and simplify. Compare f(−x) to the original f(x) and to −f(x).

Unit 7: Basic Trigonometry and The Unit Circle

• Right Triangle Trigonometry (SOH CAH TOA):

  • sinθ=HypotenuseOpposite​

  • cosθ=HypotenuseAdjacent​

  • tanθ=AdjacentOpposite​

  • Reciprocal Identities: cscθ=sinθ1​, secθ=cosθ1​, cotθ=tanθ1​.

• Radians and Degrees:

  • π radians = 180∘.

  • Degrees to Radians: Multiply by 180∘π​.

  • Radians to Degrees: Multiply by π180∘​.

• The Unit Circle: A circle with radius 1 centered at the origin.

  • Points on the circle are (x,y)=(cosθ,sinθ), where θ is the angle from the positive x-axis.

  • Allows you to find exact values of trigonometric functions for key angles in all four quadrants.

  • Know the values for angles like 0,π/6,π/4,π/3,π/2,π,3π/2,2π (and their degree equivalents) and their reflections.

  • ASTC Rule (All Students Take Calculus): Helps remember which trig functions are positive in each quadrant (Quadrant I: All, II: Sine, III: Tangent, IV: Cosine).

  • Reference Angles: The acute angle formed by the terminal side of an angle and the x-axis. Used to find trig values in different quadrants.

Unit 8: Trigonometric Identities and Equations

• Trigonometric Identities: Equations that are true for all values of the variable for which both sides are defined. Used to simplify expressions and solve equations.

  • Reciprocal Identities: cscθ=1/sinθ, etc.

  • Quotient Identities: tanθ=sinθ/cosθ, cotθ=cosθ/sinθ.

  • Pythagorean Identities:

    • sin2θ+cos2θ=1

    • tan2θ+1=sec2θ

    • 1+cot2θ=csc2θ

  • (Other identities like Double Angle, Half Angle, Sum/Difference might be covered - check your notes!)

• Verifying Identities: Start with one side (usually the more complicated one) and use algebraic manipulation and identities to transform it into the other side. Do NOT treat it like an equation where you do the same thing to both sides.

• Solving Trigonometric Equations: Find the angle(s) that satisfy the equation.

  • Use identities to simplify or get the equation in terms of a single trig function.

  • Use algebraic techniques (factoring, quadratic formula, isolating the trig function).

  • Find the angles in the specified interval (usually [0,2π) or [0,360∘]) or find the general solution (add 2πk or 360∘k for sin/cos, πk or 180∘k for tan/cot, where k is an integer).

  • Be aware of multiple angles (e.g., sin(2θ)=1/2). You'll need to solve for 2θ first, then divide by 2, making sure to find all solutions in the interval.

Unit 9: Sin/Cos Wave Functions (Graphing)

• Standard Form: y=Asin(B(x−C))+D and y=Acos(B(x−C))+D.

• Key Parameters:

  • ∣A∣: Amplitude. Vertical distance from the midline to the max/min.

  • B: Affects the period.

  • Period: The horizontal length of one complete cycle. Period =∣B∣2π​ (for radians) or ∣B∣360∘​ (for degrees).

  • C: Phase Shift. Horizontal shift. Shift is C units to the right if C>0, left if C<0. (Be careful if B is factored out vs. not!)

  • D: Vertical Shift (Midline). The horizontal line y=D around which the wave oscillates.

• Graphing:

  • Identify the midline y=D.

  • Identify the amplitude ∣A∣. Max value is D+∣A∣, min value is D−∣A∣.

  • Identify the period.

  • Identify the phase shift C. This is the starting point of a standard cycle.

  • Divide the period into four equal intervals. These marks correspond to the key points (midline, max, min).

  • Plot the key points starting from the shifted starting point C.

  • Draw the smooth curve.

• Finding the Equation from a Graph: Determine the midline, amplitude, period, and phase shift from the graph, then substitute the values into the standard form. Decide if it's a sine or cosine function based on the starting point relative to the phase shift.

Unit 10: Combinations and Permutations

• Factorials: n!=n×(n−1)×⋯×2×1. 0!=1.

• Permutations (nPr): The number of ways to arrange r items from a set of n distinct items where order matters.

  • Formula: nPr=(n−r)!n!​

  • Think "arrangement," "order," "position."

• Combinations (nCr): The number of ways to choose r items from a set of n distinct items where order does not matter.

  • Formula: nCr=r!(n−r)!n!​

  • Think "group," "selection," "subset."

• Key Distinction: Read problems carefully to determine if the order of selection is important.

• Finding the nth Term (of sequences): This usually refers to Arithmetic or Geometric sequences.

  • Arithmetic Sequence: Each term is found by adding a constant difference (d).

    • Formula for the nth term: an​=a1​+(n−1)d

  • Geometric Sequence: Each term is found by multiplying by a constant ratio (r).

    • Formula for the nth term: an​=a1​⋅rn−1

  • To find the nth term, you need the first term (a1​) and the common difference (d) or ratio (r).

Unit 11: Systems of Nonlinear Equations

• Definition: A set of two or more equations where at least one equation is not linear (e.g., quadratic, circle, exponential, absolute value, etc.).

• Solutions: The points (x,y) where the graphs of the equations intersect. There can be 0, 1, 2, or more solutions depending on the types of equations involved.

• Solving Methods:

  • Substitution: This is the most common method.

    1. Solve one equation for one variable (e.g., solve for y in terms of x).

    2. Substitute that expression into the other equation(s).

    3. Solve the resulting equation (which may be quadratic, etc.).

    4. Substitute the values back into one of the original equations to find the corresponding values of the other variable.

    5. Write solutions as ordered pairs (x,y).

  • Elimination: Sometimes possible if equations are structured similarly (e.g., two circle equations, two quadratic equations in vertex form). Align terms and add/subtract equations to eliminate a variable.

  • Graphing: Sketch the graphs of all equations on the same coordinate plane. The intersection points are the solutions. This is good for visualizing the number of solutions or estimating solutions, but often doesn't give exact answers unless the points are easy to read.

• Checking Solutions: Always substitute your potential solution(s) back into all original equations to verify they work.

Unit 12: Circle Equations

• Definition: The set of all points equidistant from a central point.

• Standard Form: (x−h)2+(y−k)2=r2

  • Center: (h,k)

  • Radius: r

• General Form: Ax2+Ay2+Dx+Ey+F=0

  • Key: The coefficients of x2 and y2 must be the same (A).

• Converting from General to Standard Form: Use the process of completing the square for both the x terms and the y terms.

  1. Group x terms and y terms. Move the constant term to the other side.

  2. If A=1, factor out A from the x terms and the y terms.

  3. Complete the square for x: Take half of the coefficient of x, square it, and add it to both sides (inside the parenthesis if factored out A, remember to multiply by A before adding to the other side).

  4. Complete the square for y: Take half of the coefficient of y, square it, and add it to both sides (inside the parenthesis if factored out A, remember to multiply by A before adding to the other side).

  5. Rewrite the perfect square trinomials as squares: (x−h)2 and (y−k)2. Combine constants on the other side to get r2.

• Graphing: Plot the center (h,k) and use the radius r to find points up, down, left, and right from the center. Draw the circle.

• Circles as Functions: A full circle is NOT a function (fails the vertical line test). Semi-circles (top or bottom half y=k±r2−(x−h)2​ or left/right half x=h±r2−(y−k)2​) can be functions of x or y.

Unit 13: Linear and Nonlinear Inequalities

• Concept: Representing regions on a coordinate plane where the inequality holds true.

• Steps for Graphing a Single Inequality:

  1. Graph the Boundary: Treat the inequality as an equation and graph the resulting line or curve.

     • Use a solid line/curve if the inequality includes "equal to" (≤ or ≥). Points on the boundary are solutions.

     • Use a dashed line/curve if the inequality does NOT include "equal to" (< or >). Points on the boundary are NOT solutions.

  2. Choose a Test Point: Pick a point not on the boundary line/curve (the origin (0,0) is often easiest if it's not on the boundary).

  3. Substitute and Test: Plug the coordinates of the test point into the original inequality.

  4. Shade the Region:

     • If the test point makes the inequality true, shade the region containing the test point.

     • If the test point makes the inequality false, shade the region that does not contain the test point.

• Types of Boundaries:

  • Linear: Straight lines (y=mx+b, x=c, y=c).

  • Nonlinear: Parabolas (y=x2), absolute value V's (y=∣x∣), exponentials (y=ex), logarithms (y=logx), circles ((x−h)2+(y−k)2=r2), etc.

• Systems of Inequalities:

  • Graph each inequality on the same coordinate plane, shading the solution region for each inequality.

  • The solution to the system is the region where the shaded areas overlap.

  • If there are three or more inequalities, the solution is the region where all shaded areas overlap.

  • This region might be bounded or unbounded.

Final Thoughts:

This is a comprehensive list! Take your time working through each section. Practice graphing, solving equations, and applying the formulas. Pay special attention to the conditions for extraneous solutions in radical and log equations, and the domain/range restrictions for those functions. For trig, make sure you know your unit circle values and identities cold. And for systems/inequalities, focus on the graphical interpretation and the different solution methods.

Deep breaths, break it down into smaller chunks, and tackle it one unit at a time. You've got the tools now. Go ace that final! Good luck, bro!