Questions and Answers
What are the possible number of solutions to a system of linear equations?
What are the various forms of the equation for a line?
point-slope form, slope-intercept form, and standard form.
What happens to the inequality sign when multiplying or dividing by a negative value?
What features are important to remember for inequalities and their graphs?
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List the critical points/features that should be found before graphing a function.
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How do you find vertical and horizontal asymptotes?
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What is the Law of Sines?
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What are the properties of exponents?
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List the major trigonometric functions.
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Study Notes
Number of Solutions to Linear Equations
- Possible outcomes include no solution, one solution, or infinite solutions.
- Graphically, each solution represents the intersection points of lines.
Forms of Equation for a Line
- Various forms include slope-intercept, point-slope, and standard form.
- Parallel lines have equal slopes; intersecting/perpendicular lines have slopes that are negative reciprocals.
Key Features of Inequalities and Graphs
- Multiplying/dividing by a negative flips the inequality sign.
- Dotted lines represent strict inequalities (below/above), while solid lines indicate inclusive inequalities (includes the line).
- For multiple inequalities, the solutions satisfy all conditions simultaneously.
Factoring Quadratic and Cubic Functions
- Common methods for quadratic functions include factoring, completing the square, and quadratic formula.
- Cubic functions can often be factored by grouping or synthetic division.
Critical Points and Features for Graphing
- Y-intercept: Found by setting x to 0.
- X-intercept(s): Found by setting the function to 0.
- Domain: The input values where the function is defined; undefined for division by 0 or imaginary results.
- Range: The resulting output values of the function.
- Consider end behavior and vertical asymptotes when analyzing the function's graph.
Relation of Pythagorean Theorem to Other Concepts
- The Pythagorean theorem (a² + b² = c²) links to the distance formula and is foundational in deriving trigonometric identities.
General and Standard Form Equations
- Line: y = mx + b
- Absolute value: y = |x|
- Quadratic: y = ax² + bx + c
- Cubic: y = ax³ + bx² + cx + d
- Exponential: y = ab^x
- Logarithmic: y = log_b(x)
- Rational: y = p(x)/q(x)
- Radical: y = √x
- Sinusoidal: y = A sin(B(x - C)) + D
- Circle: (x - h)² + (y - k)² = r²
Transformations and Properties
- Dilation: Changing size proportionally.
- Rotation: Pivoting around a fixed central point.
- Translation: Shifting a figure without altering its orientation.
- Reflection: A mirrored counterpart across a line.
- Similarity: Identical shapes with proportional dimensions.
- Congruency: Identical shapes and sizes.
Perimeter and Area Formulas
- Rectangle: Perimeter = 2(l + w), Area = l × w
- Trapezoid: Perimeter = sum of all sides, Area = ½(b₁ + b₂)h
- Circle: Perimeter (Circumference) = 2πr, Area = πr²
- Cylinder: Surface Area = 2πr² + 2πrh, Volume = πr²h
- Sphere: Surface Area = 4πr², Volume = (4/3)πr³
Asymptotes and Undefined Portions
- Vertical asymptotes arise from equation settings that involve division by zero.
- Horizontal asymptotes relate to the degrees of polynomials in the numerator and denominator.
- Undefined portions also occur in radical functions with negative values leading to imaginary results.
Operations on Functions
- Addition, subtraction, multiplication, and division of functions can create new functions.
- Composition of functions is another fundamental operation.
General Forms of Function Types
- Linear: y = mx + b
- Absolute Value: y = |x - h| + k
- Quadratic: y = ax² + bx + c
- Cubic: y = ax³ + bx² + cx + d
- Exponential: y = ab^x
- Logarithmic: y = log_b(x)
- Sinusoidal: y = A sin(B(x - C)) + D
Properties of Exponents and Logarithms
- Exponent rules include multiplication (add exponents), division (subtract exponents), and raising to a power (multiply exponents).
- Logarithmic properties encompass the product, quotient, and power rules, linking them to their corresponding exponential forms.
Law of Sines and Law of Cosines
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) for any triangle.
- Law of Cosines: c² = a² + b² - 2ab cos(C) relates side lengths to angles in a triangle.
Major Trigonometric Functions
- Functions include sine, cosine, and tangent, with respective opposite, adjacent, and hypotenuse side lengths related to angles in right triangles.
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Test your knowledge with these flashcards on advanced algebra and functions concepts. Explore the number of solutions to linear equations, different forms of line equations, and the properties of inequalities. Perfect for students looking to enhance their understanding of algebraic principles.
