Questions and Answers
What is another name for the rectangular coordinate system?
What is an equation in two variables?
An equation that can be described with two variables, such as y=3x-5, where x is independent and y is dependent.
Use ( ) for intervals when it overlaps with another interval or there are open circles.
True
What is the formula for the area of a trapezium?
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What is the area of a rhombus?
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What is the formula for the area of a circle?
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What is the area of a sector of a circle?
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What is mathematical modeling?
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What characterizes an even function?
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What characterizes an odd function?
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What does the equation f(-x) = -f(x) imply?
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What is reflection about the x-axis?
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What is a piecewise function?
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What defines a step function?
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What is a regression line?
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What is the slope of a vertical line?
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What is the slope of a horizontal line?
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What is the point-slope form of a linear equation?
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What is the slope-intercept form of a linear equation?
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What is the general form of a linear equation?
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What characterizes parallel lines?
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What characterizes perpendicular lines?
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What does the equation f(x) -> -f(x) represent?
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What does the equation f(x) -> f(-x) represent?
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What is a vertical stretch?
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What is a vertical shrink?
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What does a horizontal stretch refer to?
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Study Notes
Coordinate Systems and Functions
- Rectangular coordinate system is also known as the Cartesian coordinate system, used for graphing equations.
- An equation in two variables, such as y=3x-5, consists of an independent variable (x) and a dependent variable (y).
- Use parentheses for intervals when dealing with overlapping intervals or when open circles are present.
Area Formulas
- Area of a trapezium is calculated using the formula A = 1/2(a + b)h, where a and b are the lengths of the parallel sides.
- The area of a rhombus is determined by A = 1/2 x diagonal 1 x diagonal 2.
- The area of a circle is given by the formula A = πr², where r is the radius.
- Area of a sector of a circle can be calculated using A = (n/360)πr², where n is the angle in degrees.
Mathematical Concepts
- Mathematical modeling involves using equations to represent the relationships between different variables.
- An even function has symmetry with respect to the y-axis, characterized by the property f(x) = f(-x).
- An odd function displays symmetry concerning the origin, following the rule f(-x) = -f(x).
Reflections and Transformations
- The transformation f(-x) = -f(x) results in all components of the equation being sign-changed.
- Reflection about the x-axis transforms (x,y) to (-x,-y) by substituting -y into the equation.
Types of Functions
- A piecewise function is defined using two or more expressions corresponding to different intervals, which may be expressed with parentheses.
- A step function is represented graphically as a series of line segments rather than a continuous curve.
Linear Regression and Line Properties
- A regression line represents the line of best fit for a data set.
- The slope of a vertical line is undefined, while a horizontal line has a slope of m = 0.
- The point-slope form of a line is expressed as y - y₁ = m(x - x₁).
- The slope-intercept form is represented by y = mx + b, where m indicates the slope and b denotes the y-intercept.
- The general form of a linear equation is Ax + By + C = 0, useful for finding intercepts.
Relationship between Lines
- Parallel lines share the same slope.
- Perpendicular lines possess slopes that are negative reciprocals of each other.
Function Transformations
- The transformation f(x) -> -f(x) indicates reflection over the x-axis.
- The transformation f(x) -> f(-x) indicates reflection over the y-axis.
- A vertical stretch is achieved by multiplying all y-values of a function by a factor greater than 1.
- A vertical shrink compresses all y-values by a factor greater than 0 but less than 1.
- A horizontal stretch affects the x-values, typically involving multiplication by a factor less than 1.
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