Questions and Answers
What is the Midpoint Formula?
m = ((x1+x2)/2,(y1+y2)/2)
What is the Distance Formula?
d = √((x1-x2)^2 + (y1-y2)^2)
What is the equation for vertical lines?
x = k
What is the equation for horizontal lines?
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What is the x-intercept?
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What is the y-intercept?
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What is the Standard Form of a line?
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What is the slope of a line?
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What is the Slope-Intercept Form of a line?
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What is the Point-Slope Form of a line?
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What do parallel lines have in common?
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What do perpendicular lines do?
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Define 'Relation' in mathematics.
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What is a 'Function'?
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What is 'Domain' in a function?
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What is 'Range' in a function?
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What is the Difference Quotient?
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What is the equation of a Quadratic Function?
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What is the equation of a Square Root Function?
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What is the equation of a Cubic Function?
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What is a Nameless Function?
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What is the equation of an Absolute Value Function?
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What is a Vertical Translation?
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What is a Horizontal Translation?
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How do you flip a graph over the x-axis?
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How do you flip a graph over the y-axis?
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What happens to a graph when it widens?
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What happens to a graph when it narrows?
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What is K - constant variation?
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What is Direct Variation?
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What is Inverse Variation?
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What is Joint Variation?
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What is the Standard Form of the equation of a circle?
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Study Notes
Midpoint and Distance Formulas
- Midpoint Formula: m = ((x1 + x2) / 2, (y1 + y2) / 2) calculates the midpoint of two points.
- Distance Formula: d = √((x1 - x2)² + (y1 - y2)²) determines the distance between two points in the Cartesian plane.
Line Characteristics
- Vertical Line Equation: x = k is the equation representing vertical lines.
- Horizontal Line Equation: y = k represents horizontal lines.
Intercepts
- X-Intercept: Point where the graph crosses the x-axis, represented as (x, 0). Found by setting y = 0.
- Y-Intercept: Point where the graph crosses the y-axis, represented as (0, y). Found by setting x = 0.
Standard Form and Slope Information
- Standard Form of a Line: Ax + By = C. To graph, find intercepts by setting variables to zero.
- Slope of a Line: Calculated as (rise/run) or (y1 - y2)/(x1 - x2). Vertical lines have an undefined slope, while horizontal lines have a slope of zero.
Line Forms
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
- Point-Slope Form: y - y1 = m(x - x1), using a specific point (x1, y1) on the line.
Line Relationships
- Parallel Lines: Have identical slopes and never intersect, denoted as ||.
- Perpendicular Lines: Have slopes that are opposite reciprocals and meet at a right angle, denoted as |.
Relations and Functions
- Relation: A set of ordered pairs.
- Function: A relation where each first component (x) is unique.
- Domain: The set of all possible first components (x-values) in a function.
- Range: The set of all possible second components (y-values) in a function.
Special Function Types
- Quadratic Function: Defined as y = x².
- Square Root Function: Denoted as y = √x.
- Cubic Function: Represented by y = x³.
- Nameless Function: Defined as y = (1/x).
- Absolute Value Function: Denoted by y = |x|.
Transformations of Graphs
-
Vertical Translation:
- y = f(x) + c shifts the graph up by c units.
- y = f(x) - c shifts the graph down by c units.
-
Horizontal Translation:
- y = f(x + c) shifts left by c units.
- y = f(x - c) shifts right by c units.
- Flip Over X-axis: y = -f(x) reflects the graph across the x-axis.
- Flip Over Y-axis: y = f(-x) reflects the graph across the y-axis.
Graph Shape Adjustments
- Graph Widens: y = (fraction < 1) f(x) increases the width of the graph.
- Graph Narrows: y = (whole number) f(x) decreases the width of the graph.
Variations
- K-constant Variation: K ≠ 0 indicates a non-zero constant.
- Direct Variation: y = Kx shows a direct proportion between y and x.
- Inverse Variation: y = (K/x) shows inverse proportionality.
- Joint Variation: y = Kx^n z^n incorporates multiple variables raised to powers.
Circle Equation
- Standard Form of a Circle: (x - h)² + (y - k)² = r², where (h, k) denotes the center and r is the radius, r > 0.
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Prepare for your Algebra 2 Unit 3 test with these comprehensive flashcards. Each card features essential formulas and concepts, including the Midpoint and Distance formulas, vertical and horizontal lines, and x and y intercepts. Master these topics to excel in your upcoming exam!