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Questions and Answers
What is the general form of a quadratic equation?
What is the sum of the angles in a triangle?
What is the notation for a translation of 3 units right and 2 units up?
What is the sine of an angle in a right triangle?
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What is the formula for the area of a rectangle?
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What is the result of a 90-degree rotation about the origin?
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What is the term for a statement that two expressions are equal?
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What is the ratio of the opposite side to the hypotenuse in a right-angled triangle?
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What is the name of the theorem that relates the lengths of the sides of a right-angled triangle?
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What is the result of a reflection of a shape over a line or axis?
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What is the angle between the line of sight and the horizontal when looking up at an object?
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What is the term for the lowest or highest point of a parabola?
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Study Notes
Algebra
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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 systems of linear equations using substitution and elimination methods
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Functions:
- Domain and range of a function
- Composition of functions
- Inverse functions
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Graphing:
- Graphing linear and quadratic equations
- Identifying key features of graphs (x-intercepts, y-intercepts, vertex, etc.)
Trigonometry
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Angles and Triangles:
- Measuring angles in degrees and radians
- Properties of triangles (Pythagorean identity, sum of angles)
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Trigonometric Functions:
- Sine, cosine, and tangent functions
- Graphs and identities of trigonometric functions
- Solving trigonometric equations
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Applications:
- Modeling periodic phenomena (sound waves, light waves)
- Solving problems involving right triangles
Geometry
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Points, Lines, and Planes:
- Properties of points, lines, and planes (midpoint, distance, slope)
- Relationships between points, lines, and planes (parallel, perpendicular, intersecting)
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Angles and Measurement:
- Measuring angles and arcs in degrees and radians
- Properties of angles (complementary, supplementary, corresponding)
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Shapes and Solids:
- Properties of 2D and 3D shapes (perimeter, area, volume)
- Calculating perimeter, area, and volume of various shapes
Transformations
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Translations:
- Translating shapes and points on a coordinate plane
- Notation and formulas for translations
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Rotations:
- Rotating shapes and points on a coordinate plane
- Notation and formulas for rotations
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Reflections:
- Reflecting shapes and points on a coordinate plane
- Notation and formulas for reflections
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Combinations:
- Combining multiple transformations to create a new transformation
- Notation and formulas for combined transformations
Algebra
- Linear equations are represented as ax + by = c, where a, b, and c are constants.
- Quadratic equations are represented as ax^2 + bx + c = 0, where a, b, and c are constants.
- Systems of linear equations can be solved using substitution and elimination methods.
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 two or more functions.
- Inverse functions are functions that "undo" the operation of another function.
Graphing
- Linear and quadratic equations can be graphed on a coordinate plane.
- Key features of graphs include x-intercepts, y-intercepts, and vertex.
Trigonometry
Angles and Triangles
- Angles can be measured in degrees and radians.
- Properties of triangles include the Pythagorean identity and the sum of angles.
Trigonometric Functions
- Sine, cosine, and tangent functions relate to the ratios of sides in triangles.
- Graphs of trigonometric functions have unique identities and properties.
- Trigonometric equations can be solved using algebraic methods.
Applications
- Trigonometry can be used to model periodic phenomena, such as sound waves and light waves.
- Right triangles can be used to solve problems involving heights and distances.
Geometry
Points, Lines, and Planes
- Points have no size, lines have no thickness, and planes have no depth.
- Properties of points, lines, and planes include midpoint, distance, and slope.
- Relationships between points, lines, and planes include parallel, perpendicular, and intersecting.
Angles and Measurement
- Angles can be measured in degrees and radians.
- Properties of angles include complementary, supplementary, and corresponding.
Shapes and Solids
- Properties of 2D shapes include perimeter and area.
- Properties of 3D shapes include volume in addition to perimeter and area.
- Formulas exist for calculating perimeter, area, and volume of various shapes.
Transformations
Translations
- Translations involve moving shapes and points on a coordinate plane.
- Notation and formulas exist for translations, such as (x, y) → (x + h, y + k).
Rotations
- Rotations involve turning shapes and points on a coordinate plane.
- Notation and formulas exist for rotations, such as (x, y) → (x cos θ - y sin θ, x sin θ + y cos θ).
Reflections
- Reflections involve flipping shapes and points on a coordinate plane.
- Notation and formulas exist for reflections, such as (x, y) → (x, -y).
Combinations
- Multiple transformations can be combined to create a new transformation.
- Notation and formulas exist for combined transformations, such as a translation followed by a rotation.
Algebra
- Variables represent unknown values, while constants are fixed values
- Algebraic expressions combine variables, constants, and mathematical operations
- Equations and inequalities are statements that two expressions are equal or not equal
- Linear equations have the highest power of the variable(s) as 1, e.g. 2x + 3 = 5
- Quadratic equations have the highest power of the variable(s) as 2, e.g. x^2 + 4x + 4 = 0
Trigonometry
- Angles are measures of rotation, measured in degrees or radians
- Trigonometric ratios relate sides of a right-angled triangle, including sine, cosine, and tangent
- Sine, cosine, and tangent formulas:
- sin(A) = opposite side / hypotenuse
- cos(A) = adjacent side / hypotenuse
- tan(A) = opposite side / adjacent side
- Inverse trigonometric functions (arcsin, arccos, arctan) find the angle given the ratio
Geometry
- Points are locations in space, represented by coordinates
- Lines are sets of points extending infinitely in two directions
- Planes are flat surfaces extending infinitely in all directions
- Angles are measures of rotation, measured in degrees or radians
- Triangles are polygons with three sides and three angles, including right-angled triangles
Transformations
- Translation: moving a shape by a certain distance in a certain direction
- Rotation: rotating a shape by a certain angle around a certain point
- Reflection: flipping a shape over a certain line or axis
- Enlargement: scaling a shape up or down by a certain factor
Pythagoras Theorem
- a^2 + b^2 = c^2, where a and b are the lengths of the legs of a right-angled triangle, and c is the length of the hypotenuse
- Used to find the length of the hypotenuse, or the length of one leg given the other leg and the hypotenuse
Angles of Elevation and Depression
- Angle of elevation: the angle between the line of sight and the horizontal, when looking up at an object
- Angle of depression: the angle between the line of sight and the horizontal, when looking down at an object
- Used to find the height of an object, or the distance to an object, given the angle and other measurements
Graphs
- Cartesian plane: a grid of points with x and y axes, used to plot graphs
- Coordinates: points on the graph, represented by an ordered pair (x, y)
- Linear graphs: graphs of linear equations, which are straight lines
- Quadratic graphs: graphs of quadratic equations, which are parabolas
Quadratic Graphs
- Parabolas: U-shaped curves that open up or down
- Vertex: the lowest or highest point of the parabola
- Axis of symmetry: the vertical line that passes through the vertex
- x-intercepts: the points where the graph crosses the x-axis, found by solving the quadratic equation
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Description
Test your understanding of algebra basics, including linear and quadratic equations, functions, and graphing. Covers topics like solving systems of linear equations, composition of functions, and identifying key features of graphs.
