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Questions and Answers
What are the domain and range of the relation defined by the set {(5,-2), (6,-5), (7,-2), (8,-1)} and is it a function?
What are the domain and range of the relation defined by the set {(5,-2), (6,-5), (7,-2), (8,-1)} and is it a function?
Domain: {5, 6, 7, 8}; Range: {-5, -2, -1}; Function? Yes.
Calculate the value of the function 𝑓(𝑥) = 2𝑥² - 3 at 𝑓(−4).
Calculate the value of the function 𝑓(𝑥) = 2𝑥² - 3 at 𝑓(−4).
𝑓(−4) = 2(−4)² - 3 = 32 - 3 = 29.
Determine the degree of the polynomial function 𝑓(𝑥) = 2𝑥(𝑥 + 4).
Determine the degree of the polynomial function 𝑓(𝑥) = 2𝑥(𝑥 + 4).
Degree: 2.
Based on the equation 𝑦 = −2(𝑥 + 3)² + 4, what are its vertex coordinates?
Based on the equation 𝑦 = −2(𝑥 + 3)² + 4, what are its vertex coordinates?
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What is the minimum value of the function defined by 𝑓(𝑥) = 3(𝑥 + 1)² - 8?
What is the minimum value of the function defined by 𝑓(𝑥) = 3(𝑥 + 1)² - 8?
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If a parabola has an axis of symmetry at 𝑥 = −4 and is congruent in shape to 𝑦 = 5𝑥², write its equation.
If a parabola has an axis of symmetry at 𝑥 = −4 and is congruent in shape to 𝑦 = 5𝑥², write its equation.
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What is the range of the football height function ℎ(𝑡) = −5(𝑡 − 2)² + 21?
What is the range of the football height function ℎ(𝑡) = −5(𝑡 − 2)² + 21?
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Define the transformation from 𝑦 = 𝑥² to obtain the function 𝑓(𝑥) = 3(𝑥 + 1)² - 8.
Define the transformation from 𝑦 = 𝑥² to obtain the function 𝑓(𝑥) = 3(𝑥 + 1)² - 8.
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Study Notes
Relations and Functions
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Determining if a relation is a function: A relation is a function if each input (x-value) corresponds to exactly one output (y-value). A vertical line drawn through the graph crosses it only once at any given x-coordinate for a function.
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Example a (graph): The input 1 maps to the output 2 and 3. In turn, the input 2 maps to outputs 1 and 3. This is NOT a function as one input maps to more than one output.
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Example b (set of ordered pairs): {(5,-2), (6,-5), (7,-2), (8,-1)} This is function since each x-value maps to a unique y-value.
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Domain: The set of all possible input values (x-values) in a relation or function.
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Range: The set of all possible output values (y-values) in a relation or function.
Quadratic Functions and Transformations
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Function notation: f(x) = 2x² - 3. This represents the output value (y) of the function based on input (x).
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Evaluating a Function: To find f(-4), substitute -4 for x: f(-4) = 2*(-4)² - 3 = 28-3 = 25
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Function Rules and Values: Given f(x) = 2x² - 3, to find f(3m), replace x with 3m: f(3m)= 2(3m)² – 3 = 18 m²–3
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Multiple of a Function: To find 2f(-1), first calculate f(-1) and then double the result. f(-1) = 2(-1)² – 3 = -1 then 2f(-1) = 2(-1)= -2
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Transformations of y = x²: Transformations affect the shape, position, and orientation of the graph of a function. Example transformation would be translation, stretching, or reflecting.
Quadratic Function Characteristics
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Degree: The highest power of the variable in a polynomial function.
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Example 3 a (f(x) = 2x(x + 4)): degree is 2.
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Example 3 b (f(x) = (x − 3)(x + 2) − x): degree is 2
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Vertex: The turning point of a parabola. The coordinates are (h, k)
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Axis of symmetry: A vertical line that passes through the vertex of the parabola. Equation is x = h
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x-intercepts: The points where the graph of the function crosses the x-axis. These are solutions to f(x) = 0.
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Parabola Transformations: Understanding how different parameters (like 2 or 3 in a given example) influence the shape and position of the parabola is key.
Applications of Quadratic Functions
- Modelling a physical situation: For example, a projectile's height can be modelled by a quadratic function. (Example in the question)
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Description
Explore the concepts of relations and functions, including how to determine if a relation is a function. Delve into quadratic functions and transformations, learning about function notation and evaluation. This quiz will solidify your understanding of domain and range in mathematical functions.
