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Questions and Answers
What is the value of x in the equation 2x + 5 = 11?
What is the value of x in the equation 2x + 5 = 11?
What is the composition of functions f(x) = 2x and g(x) = x^2?
What is the composition of functions f(x) = 2x and g(x) = x^2?
What is the equation of a line that passes through the points (2,3) and (4,5)?
What is the equation of a line that passes through the points (2,3) and (4,5)?
What is the value of sin(2x) in terms of sin(x) and cos(x)?
What is the value of sin(2x) in terms of sin(x) and cos(x)?
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What is the derivative of the function f(x) = 3x^2 + 2x - 5?
What is the derivative of the function f(x) = 3x^2 + 2x - 5?
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What is the value of the limit as x approaches 2 of the function f(x) = (x^2 - 4) / (x - 2)?
What is the value of the limit as x approaches 2 of the function f(x) = (x^2 - 4) / (x - 2)?
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What is the equation of a circle with center (3, -2) and radius 4?
What is the equation of a circle with center (3, -2) and radius 4?
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What is the graph of the function f(x) = |x|?
What is the graph of the function f(x) = |x|?
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What is the value of cos(2x) in terms of cos(x)?
What is the value of cos(2x) in terms of cos(x)?
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Study Notes
Algebra
-
Equations and Inequalities
- Linear equations: ax + by = c, where a, b, and c are constants
- Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
- Solving linear and quadratic equations using various methods (e.g. substitution, elimination, quadratic formula)
- Graphing linear and quadratic equations on the coordinate plane
-
Functions
- Domain and range of a function
- Composition of functions
- Inverse functions
- Graphing functions, including identifying maxima and minima
Geometry
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Points, Lines, and Planes
- Midpoint and distance formulas
- Slope of a line
- Equation of a line (point-slope and slope-intercept forms)
- Parallel and perpendicular lines
-
Triangles and Quadrilaterals
- Properties of congruent and similar triangles
- Types of triangles (e.g. isosceles, right, equilateral)
- Properties of quadrilaterals (e.g. rectangles, squares, trapezoids)
Trigonometry
-
Angles and Triangles
- Degree and radian measures of angles
- Trigonometric ratios (sine, cosine, tangent)
- Solving right triangles using trigonometric ratios
- Graphing trigonometric functions
-
Identities and Formulas
- Pythagorean identity
- Sum and difference formulas
- Double and half-angle formulas
Calculus (Introduction)
-
Limits and Continuity
- Basic limit properties and rules
- Continuity of functions
- Intermediate Value Theorem
-
Derivatives
- Basic derivative rules (e.g. power rule, product rule)
- Geometric interpretation of derivatives (e.g. tangent lines, slopes)
Algebra
-
Equations and Inequalities
- Linear equations have the form ax + by = c, where a, b, and c are constants
- Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants
- Methods for solving linear and quadratic equations include substitution, elimination, and the quadratic formula
- Graphing linear and quadratic equations involves plotting points on the coordinate plane
-
Functions
- The domain of a function is the set of input values, while the range is the set of output values
- Composition of functions involves combining functions by applying one function to the output of another
- Inverse functions "undo" each other, and can be used to solve equations
- Graphing functions helps identify maxima and minima
Geometry
-
Points, Lines, and Planes
- The midpoint formula calculates the midpoint of a line segment: M = ((x1+x2)/2, (y1+y2)/2)
- The distance formula calculates the distance between two points: √((x2-x1)^2 + (y2-y1)^2)
- The slope of a line is a measure of its steepness, calculated as m = (y2-y1)/(x2-x1)
- The equation of a line can be written in point-slope or slope-intercept form
- Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals
-
Triangles and Quadrilaterals
- Congruent triangles have equal corresponding sides and angles, while similar triangles have proportional corresponding sides
- There are different types of triangles, including isosceles, right, and equilateral
- Quadrilaterals can be classified as rectangles, squares, trapezoids, and more, each with unique properties
Trigonometry
-
Angles and Triangles
- Angles can be measured in degrees or radians
- Trigonometric ratios include sine, cosine, and tangent, which are used to solve right triangles
- Graphing trigonometric functions involves plotting periodic curves
-
Identities and Formulas
- The Pythagorean identity states sin^2(θ) + cos^2(θ) = 1
- The sum and difference formulas are used to find trigonometric ratios for sums and differences of angles
- The double and half-angle formulas are used to find trigonometric ratios for double and half angles
Calculus (Introduction)
-
Limits and Continuity
- The basic limit properties include the sum and product rules, and the squeeze theorem
- Continuity of functions is important for many calculus concepts
- The Intermediate Value Theorem states that a continuous function on a closed interval takes on every value between its maximum and minimum
-
Derivatives
- The power rule is a basic derivative rule, stating that if f(x) = x^n, then f'(x) = nx^(n-1)
- The product rule states that if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x)
- Geometrically, derivatives represent the slope of a tangent line to a curve
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Description
Test your understanding of algebraic equations, inequalities, and functions, including linear and quadratic equations, graphing, and function composition.
