Questions and Answers
What is the Graph of a quadratic function?
What is the Graph of an absolute value function?
What is the graph of a linear function?
What is a linear function?
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What is the Graph of a cube root function?
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What is transformation/translation in graphing?
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What is a parent function?
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What is the equation of a quadratic function?
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What is an absolute value function?
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What is the Graph of a square root function?
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What is domain in the context of functions?
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What is range in the context of functions?
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What is the Graph of a cubic equation?
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What is a function with a shift left of 1?
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What is a function with a shift up of 1?
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What is a function with a shift right of 1?
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What does 'All Real Numbers' represent?
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What is a relation in mathematics?
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What is a function?
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What does 'distribute' refer to in solving equations?
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What is the simplified form of: -x > 2?
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What is an exponential function?
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What is a logarithmic function?
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What relationship do inverses have to exponential and logarithmic functions?
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Study Notes
Graph of Functions
- Quadratic functions have a parabolic shape, opening either upwards or downwards based on the coefficient of the squared term.
- Absolute value functions produce a V-shaped graph, reflecting all negative values upwards to positive.
- Linear functions create straight lines, represented by the equation f(x) = mx + b where m is the slope and b is the y-intercept.
- The cube root function has an S-shaped graph that extends in all four quadrants, allowing for both positive and negative outputs.
Types of Functions
- Linear function is defined as f(x) = x, representing a line with a slope of 1.
- Absolute value function can be expressed as f(x) = | x |, which outputs the distance of x from zero, always yielding non-negative results.
- Quadratic functions can be represented in vertex form as f(x) = a(x-h)² + k, where "a" determines the direction and width of the parabola.
- The square root function graphs as a half-parabola only in the positive x-region.
- The domain of any function specifies the possible x-values, while the range determines the possible y-values.
Transformations and Relations
- Transformations refer to modifications in size, shape, position, or orientation of a graph.
- A parent function is the simplest form of a function in its category, serving as the basis for all transformations.
- A function with a left shift of 1 is represented as f(x + 1); shifting right is f(x - 1); and upward shift is denoted by f(x) + 1.
- The domain for all quadratic functions includes all real numbers, allowing for any x-value input.
- A relation is any set of ordered pairs, while a function is a relation where no x-values repeat.
Additional Concepts
- The distributive property is often the first step in simplifying expressions, such as in 2(x + 3) = 9.
- The solution for the inequality - x > 2 simplifies to x < -2.
- Exponential functions grow rapidly and can be expressed in the form f(x) = a(b^x), where a is a constant and b is the base.
- Logarithmic functions are the inverse of exponential functions and can be represented as y = log_b(x), where b is the base.
- Exponential and logarithmic functions are inversely related, meaning they can undo each other's operations.
Studying That Suits You
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Description
Test your knowledge of various graphs with these flashcards focused on quadratic, absolute value, linear, and cube root functions. Each card presents a definition to deepen your understanding of graph transformations and translations. Perfect for students learning about graphing functions in mathematics.
