Questions and Answers
What is the distance formula?
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
What is the midpoint formula?
(x₁+x₂)/2, (y₁+y₂)/2
What is the equation of a circle?
(x-h)²+(y-k)²=r²
How do you identify the center of the circle from the equation (x-3)²+(y+8)²=9?
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How do you identify the radius of the circle from the equation (x-3)²+(y+8)²=9?
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If two pairs have the same X but different Y, then it is a function.
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What is the vertical line test?
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How do you find the domain and range of a graph?
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What is the equation for a linear function?
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What is the slope equation?
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What is the point-slope form?
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Vertical lines are parallel.
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Horizontal lines and vertical lines are opposite.
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What is the perpendicular line formula?
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If the slopes are not the same or -1, then the lines are either parallel or perpendicular.
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What formula do you use to find if a line is parallel or perpendicular through the point (x,y)?
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What is a coordinate system?
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Study Notes
Distance and Midpoint Formulas
- Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] calculates the distance between two points in a coordinate plane.
- Midpoint Formula: Find the midpoint between two points using (x₁ + x₂)/2, (y₁ + y₂)/2.
Circle Equations
- Equation of a Circle: The general form is (x-h)² + (y-k)² = r², where (h, k) is the center and r is the radius.
- Center Identification: For example, in (x-3)² + (y+8)² = 9, the center is at (3, -8).
- Radius Identification: From the equation (x-3)² + (y+8)² = 9, the radius r is derived from r² = 9, resulting in r = 3.
Functions and Relations
- A function requires each domain member to have exactly one corresponding range member. Two pairs, like (9,5) and (9,-5), share the same x-value but differ in y-values, indicating it's not a function.
- Vertical Line Test: A function's graph passes this test if any vertical line crosses it at only one point.
Domain and Range
- Domain: Refers to the set of possible x-values (horizontal).
- Range: Refers to the set of possible y-values (vertical).
- Use parentheses for open intervals, brackets for closed intervals, and infinity symbols when relevant.
Linear Functions
- Linear Function Equation: y = mx + b, where m is the slope and b is the y-intercept.
- Slope Equation: Calculated as (y₂ - y₁) / (x₂ - x₁) to determine the steepness of the line.
- Point-Slope Form: Expressed as y - y₁ = m(x - x₁), useful for writing equations from a point and slope.
Line Properties
- Vertical Lines: Have the same slope and are parallel.
- Horizontal Lines: Perpendicular to vertical lines, with slopes of -1.
- Perpendicular Line Formula: For two lines, m₁ * m₂ = -1 indicates that they are perpendicular.
Determining Relationships Between Lines
- Lines that are not parallel or perpendicular do not share the same slope or a negative reciprocal relationship.
- Use y - y₁ = m(x - x₁) to find if a line through a given point is parallel or perpendicular to another line.
Coordinate System
- A grid featuring intersecting lines that help locate points and features in a two-dimensional space.
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