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Questions and Answers
A polynomial function can be represented in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
A polynomial function can be represented in the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
True (A)
The equation for a quadratic function is given by $y = mx + b$.
The equation for a quadratic function is given by $y = mx + b$.
False (B)
In trigonometry, the cosine function is defined as $\cos(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.
In trigonometry, the cosine function is defined as $\cos(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.
False (B)
The Pythagorean Theorem states that $a^2 + b^2 = c^2$ for any triangle.
The Pythagorean Theorem states that $a^2 + b^2 = c^2$ for any triangle.
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In a geometric sequence, each term is obtained by adding a fixed number to the previous term.
In a geometric sequence, each term is obtained by adding a fixed number to the previous term.
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The product rule for logarithms is expressed as $\log_b(MN) = \log_b(M) + \log_b(N)$.
The product rule for logarithms is expressed as $\log_b(MN) = \log_b(M) + \log_b(N)$.
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The distance formula is defined as $d = \sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}$.
The distance formula is defined as $d = \sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}$.
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The mean, median, and mode are all measures of central tendency.
The mean, median, and mode are all measures of central tendency.
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Study Notes
Math 11 Key Concepts and Topics
Functions
- Definition: A relation where each input has a single output.
- Types:
- Linear Functions: Graph is a straight line, defined by the equation (y = mx + b).
- Quadratic Functions: Graph is a parabola, defined by (y = ax^2 + bx + c).
- Polynomial Functions: Functions of the form (f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0).
- Exponential Functions: Form (y = a \cdot b^x), where (b) is a positive constant.
- Logarithmic Functions: Inverse of exponential functions, defined by (y = \log_b(x)).
Trigonometry
- Basic Ratios: Sine, Cosine, Tangent.
- Sine: ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
- Cosine: ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
- Tangent: ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )
- Unit Circle: Points correspond to angles and ratios.
Algebra
- Factoring: Breaking down expressions into simpler factors, e.g., (x^2 - 9 = (x - 3)(x + 3)).
- Solving Equations:
- Linear Equations: Use addition, subtraction, multiplication, and division to isolate variable.
- Quadratic Equations: Use factoring, completing the square, or the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
Geometry
- Basic Shapes: Properties of triangles, quadrilaterals, circles.
- Theorems:
- Pythagorean Theorem: (a^2 + b^2 = c^2) in right triangles.
- Area and perimeter formulas for various shapes.
Probability and Statistics
- Basic Definitions:
- Probability: Measure of likelihood of an event occurring.
- Mean, Median, Mode: Measures of central tendency.
- Standard Deviation: Measure of the amount of variation in a set of values.
Sequences and Series
- Arithmetic Sequences: Series where the difference between consecutive terms is constant.
- Geometric Sequences: Series where each term after the first is found by multiplying the previous term by a fixed, non-zero number.
Logarithms
- Properties:
- Product Rule: ( \log_b(MN) = \log_b(M) + \log_b(N) )
- Quotient Rule: ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) )
- Power Rule: ( \log_b(M^p) = p \cdot \log_b(M) )
Important Formulas
- Distance Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
- Midpoint Formula: (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
Graphing
- Understanding transformations: shifts, reflections, stretches.
- Identifying intercepts and discontinuities in functions.
Exam Preparation
- Practice problem-solving: Focus on a variety of problems from each topic.
- Review past tests and quizzes for common themes.
- Utilize study groups for collaborative learning.
Functions
- Definition: A function is a relationship between inputs and outputs where each input has only one output.
- Types:
- Linear Functions: Their graph is always a straight line. The equation for a linear function is (y = mx + b), where m is the slope and b is the y-intercept.
- Quadratic Functions: Their graph is a parabola. The equation for a quadratic function is (y = ax^2 + bx + c).
- Polynomial Functions: These are functions of the form (f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0). They can have various shapes, depending on the number of terms and the exponents.
- Exponential Functions: Their equation is (y = a \cdot b^x) where 'b' is a positive constant. These functions can either grow or decay rapidly.
- Logarithmic Functions: They are the inverse of exponential functions. The equation is (y = \log_b(x)).
Trigonometry
- Basic Ratios: Sine, Cosine, and Tangent are fundamental trigonometric ratios that relate the sides of a right triangle to the angles.
- Sine: ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} ).
- Cosine: ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
- Tangent: ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )
- Unit Circle: The unit circle is a circle with a radius of 1, used to represent the values of the trigonometric ratios for any angle.
Algebra
- Factoring: It is the process of breaking down an expression into simpler expressions (factors). For example, (x^2 - 9 = (x - 3)(x + 3)).
- Solving Equations:
- Linear Equations: Use addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.
- Quadratic Equations: There are three main methods: factoring (if possible), completing the square, and using the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
Geometry
- Basic Shapes: Understand the properties of triangles, quadrilaterals, and circles (e.g., area, perimeter, types of triangles).
- Theorems:
- Pythagorean Theorem: This theorem applies to right triangles: (a^2 + b^2 = c^2) where 'c' is the hypotenuse.
Probability and Statistics
- Basic Definitions:
- Probability: The likelihood of an event occurring, expressed as a number between 0 and 1.
- Mean: The average of a set of values.
- Median: The middle value in a sorted set of data.
- Mode: The value that appears most frequently in a set of data.
- Standard Deviation: This measures the spread or variation of data around the mean.
Sequences and Series
- Arithmetic Sequences: Each term is found by adding a constant value (the common difference) to the previous term.
- Geometric Sequences: Each term is found by multiplying the previous term by a constant value (the common ratio).
Logarithms
- Properties:
- Product Rule: ( \log_b(MN) = \log_b(M) + \log_b(N) )
- Quotient Rule: ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) )
- Power Rule: ( \log_b(M^p) = p \cdot \log_b(M) )
Important Formulas
- Distance Formula: This formula helps calculate the distance between two points on a coordinate plane: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
- Midpoint Formula: This formula finds the midpoint of a line segment: (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
Graphing
- Transformations: Be familiar with transformations that affect a graph, such as shifting, reflecting, and stretching.
- Intercepts and Discontinuities: Understand how to identify intercepts (where the graph crosses the axes) and discontinuities (gaps or breaks in the graph).
Exam Preparation
- Practice: Practice solving a variety of problems from each topic.
- Review: Review past tests and quizzes to identify areas for improvement.
- Study Groups: Utilize study groups for collaborative learning and problem-solving.
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