9 Questions
What is the value of x in the equation 2x + 5 = 11?
3
What is the composition of functions f(x) = 2x and g(x) = x^2?
f(g(x)) = 4x^2
What is the equation of a line that passes through the points (2,3) and (4,5)?
y = 2x + 1
What is the value of sin(2x) in terms of sin(x) and cos(x)?
2sin(x)cos(x)
What is the derivative of the function f(x) = 3x^2 + 2x - 5?
f'(x) = 6x + 2
What is the value of the limit as x approaches 2 of the function f(x) = (x^2 - 4) / (x - 2)?
4
What is the equation of a circle with center (3, -2) and radius 4?
(x - 3)^2 + (y + 2)^2 = 16
What is the graph of the function f(x) = |x|?
A V-shaped graph with a minimum at x = 0
What is the value of cos(2x) in terms of cos(x)?
2cos^2(x) - 1
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
-
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
Test your understanding of algebraic equations, inequalities, and functions, including linear and quadratic equations, graphing, and function composition.
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