Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 22: Volume As a Function of . . .

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Closed Captioned Video: Geometry Applications: Area and Volume, 1	Closed Captioned Video: Geometry Applications: Area and Volume, Segment 1: Volume and Density. The sinking of the Titanic provides an opportunity to explore volume, density, and buoyancy. Students construct a mathematical model of the Titanic to determine why it sank and what could have been done to prevent it from sinking.	Applications of Surface Area and Volume, Surface Area and Volume
Closed Captioned Video: Geometry Applications: Area and Volume, 1	Closed Captioned Video: Geometry Applications: Area and Volume, Segment 1: Volume and Density. The sinking of the Titanic provides an opportunity to explore volume, density, and buoyancy. Students construct a mathematical model of the Titanic to determine why it sank and what could have been done to prevent it from sinking.	Applications of Surface Area and Volume, Surface Area and Volume
Closed Captioned Video: Geometry Applications: Area and Volume, 2	Closed Captioned Video: Geometry Applications: Area and Volume, Segment 2: Surface Area. The glass-paneled pyramid at the Louvre Museum in Paris is a tessellation of rhombus-shaped glass panels. Students create a model of the pyramid to calculate the number of panels used to cover the surface area of the pyramid.	Applications of Surface Area and Volume, Surface Area and Volume
Closed Captioned Video: Geometry Applications: Area and Volume, 3	Closed Captioned Video: Geometry Applications: Area and Volume, Segment 3: Ratio of Surface Area to Volume. The Citibank Tower in New York City presents some unique design challenges. In addition it has to cope with a problem that all tall structure have to deal with: heat loss. By managing the ratio of surface area to volume, a skyscraper can effective manage heat loss.	Applications of Surface Area and Volume, Surface Area and Volume
Definition--Functions and Relations Concepts--Absolute Value Function	Absolute Value Function Topic Functions and Relations Definition An absolute value function is a function that contains an algebraic expression within absolute value symbols. It is defined as $$ f(x) = \|x\| $$ where \|x\| denotes the absolute value ofx. Description The absolute value function is significant in mathematics because it measures the distance of a number from zero on the number line, regardless of direction. This function is linear in nature, as it can be broken into two linear pieces: one for $$ x \geq 0 $$ and one for $$ x < 0 $$	Special Functions
Definition--Functions and Relations Concepts--Composite Function	Composite Function Topic Functions and Relations Definition A composite function is a function that is formed when one function is applied to the result of another function. Description Composite functions are significant in mathematics because they allow the combination of two functions to form a new function. This is denoted as (f∘g)(x) = f(g(x)). Composite functions are widely used in various fields, including computer science for function composition in programming and in calculus for chain rule applications. For example, if f(x) = 2x and g(x) = x + 3, then the composite function (f∘g)(x) = f(g(x)) = 2(x + 3) = 2x+6	Composite Functions
Definition--Functions and Relations Concepts--Conic Sections	Conic Sections Topic Functions and Relations Definition Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Description Conic sections include circles, ellipses, parabolas, and hyperbolas, which are fundamental in the study of geometry and algebra. These shapes are described by quadratic equations and have numerous applications in physics, engineering, and astronomy. For example, the orbits of planets are ellipses, and parabolic mirrors are used in telescopes and satellite dishes. The general quadratic equation for conic sections is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0	Conic Sections
Definition--Functions and Relations Concepts--Constant Function	Constant Function Topic Functions and Relations Definition A constant function is a function that always returns the same value, no matter the input. Description The constant function is one of the simplest types of functions in mathematics, expressed as f(x) = c	Relations and Functions
Definition--Functions and Relations Concepts--Continuous Function	Continuous Function Topic Functions and Relations Definition A continuous function is a function that does not have any breaks, holes, or gaps in its domain. Description Continuous functions are fundamental in calculus and mathematical analysis because they allow for the application of limits, derivatives, and integrals. A function f(x) is continuous if, for every point 𝑐 c in its domain, lim x→c f(x) = f(c)	Relations and Functions
Definition--Functions and Relations Concepts--Decreasing Function	Decreasing Function Topic Functions and Relations Definition A decreasing function is a function where the value of the function decreases as the input increases. Description Decreasing functions are important in mathematics because they describe scenarios where an increase in one variable leads to a decrease in another. This is mathematically represented as f(x1) > f(x2) for any 𝑥1 < 𝑥2	Relations and Functions
Definition--Functions and Relations Concepts--Dependent Variable	Dependent Variable Topic Functions and Relations Definition A dependent variable is a variable whose value depends on one or more other variables.	Relations and Functions
Definition--Functions and Relations Concepts--Discontinuous Function	Discontinuous Function Topic Functions and Relations Definition A discontinuous function is a function that has one or more points where it is not continuous.	Relations and Functions
Definition--Functions and Relations Concepts--Discrete Functions	Discrete Functions Topic Functions and Relations Definition Discrete functions are functions that are defined only for a set of discrete points.	Special Functions
Definition--Functions and Relations Concepts--Distorting a Function Horizontally	Distorting a Function Horizontally Topic Functions and Relations Definition Distorting a function horizontally involves stretching or compressing the graph of the function along the x-axis. Description Horizontal distortions of functions are significant because they alter the input values while maintaining the overall shape of the graph. This is mathematically represented as f(kx) where k is a constant. If k > 1, the function compresses horizontally, and if 0 < 𝑘 < 1, it stretches.	Relations and Functions
Definition--Functions and Relations Concepts--Distorting a Function Vertically	Distorting a Function Vertically Topic Functions and Relations Definition Distorting a function vertically involves stretching or compressing the graph of the function along the y-axis. Description Vertical distortions of functions are significant because they alter the output values while maintaining the overall shape of the graph. This is mathematically represented as kf(x)	Relations and Functions
Definition--Functions and Relations Concepts--Domain of a Function	Domain of a Function Topic Functions and Relations Definition The domain of a function is the set of all possible input values (x-values) for which the function is defined.	Relations and Functions
Definition--Functions and Relations Concepts--Even Function	Even Function Topic Functions and Relations Definition An even function is a function that satisfies the condition f(x) = f(−x) for all x in its domain. Description Even functions are important in mathematics because they exhibit symmetry about the y-axis. This property is useful in various fields, including physics and engineering, where symmetry simplifies analysis and problem-solving. For example, the function f(x)=x2	Relations and Functions
Definition--Functions and Relations Concepts--Function	Function Topic Functions and Relations Definition A function is a relation that uniquely associates each element of a set with exactly one element of another set.	Relations and Functions
Definition--Functions and Relations Concepts--Function Machine	Function Machine Topic Functions and Relations Definition A function machine is a conceptual tool used to understand how functions work by visualizing inputs and outputs. Description The function machine is a useful educational tool that helps students grasp the concept of functions by visualizing the process of converting inputs into outputs. It emphasizes the idea that a function takes an input, processes it according to a specific rule, and produces an output. For example, if the function is f(x) = x + 2	Relations and Functions
Definition--Functions and Relations Concepts--Function Notation	Function Notation Topic Functions and Relations Definition Function notation is a way to represent functions in the form f(x), where f denotes the function and x denotes the input variable. Description Function notation is a standardized way to write functions, making it easier to understand and communicate mathematical relationships. It is widely used in algebra, calculus, and other branches of mathematics. For example, the notation f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Function Table	Function Table Topic Functions and Relations Definition A function table is a table that lists input values and their corresponding output values for a given function. Description Function tables are useful tools in mathematics for organizing and analyzing the relationship between inputs and outputs of a function. They help in visualizing how a function behaves and in identifying patterns. For example, a function table for f(x) = 2x + 1	Relations and Functions
Definition--Functions and Relations Concepts--Graphs of Functions	Graphs of Functions Topic Functions and Relations Definition Graphs of functions are visual representations of the relationship between input values and output values of a function. Description Graphs of functions are essential tools in mathematics for visualizing how a function behaves. They provide a clear picture of the relationship between the input and output values, making it easier to analyze and interpret the function. For example, the graph of f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Graphs of Relations	Graphs of Relations Topic Functions and Relations Definition Graphs of relations are visual representations of the relationship between two sets of values, not necessarily functions.	Relations and Functions
Definition--Functions and Relations Concepts--Increasing Function	Increasing Function Topic Functions and Relations Definition An increasing function is a function where the value of the function increases as the input increases.	Relations and Functions
Definition--Functions and Relations Concepts--Independent Variable	Independent Variable Topic Functions and Relations Definition An independent variable is a variable that represents the input or cause and is manipulated to observe its effect on the dependent variable.	Relations and Functions
Definition--Functions and Relations Concepts--Inverse Function	Inverse Function Topic Functions and Relations Definition An inverse function is a function that reverses the effect of the original function, such that f(f−1(x)) = x.	Functions and Their Inverses
Definition--Functions and Relations Concepts--Odd Function	Odd Function Topic Functions and Relations Definition An odd function is a function that satisfies the condition f(−x) = −f(x) for all x in its domain.	Relations and Functions
Definition--Functions and Relations Concepts--Parametric Equations	Parametric Equations Topic Functions and Relations Definition Parametric equations are a set of equations that express the coordinates of the points of a curve as functions of a variable called a parameter.	Relations and Functions
Definition--Functions and Relations Concepts--Piecewise Functions	Piecewise Functions Topic Functions and Relations Definition Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the domain.	Special Functions
Definition--Functions and Relations Concepts--Range of a Function	Range of a Function Topic Functions and Relations Definition The range of a function is the set of all possible output values (y-values) that the function can produce. Description Understanding the range of a function is crucial in mathematics because it defines the scope of possible outputs. The range is determined by the function's rule and the domain. For example, the range of the function f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Recursive Function	Recursive Function Topic Functions and Relations Definition A recursive function is a function that calls itself in its definition. Description Recursive functions are important in mathematics and computer science because they provide a way to solve problems by breaking them down into simpler sub-problems. They are defined by a base case and a recursive case. For example, the factorial function f(n) = n⋅f(n−1) with f(0)=1	Relations and Functions
Definition--Functions and Relations Concepts--Reflecting a Function	Reflecting a Function Topic Functions and Relations Definition Reflecting a function involves flipping the graph of the function over a specified axis. Description Reflecting functions is significant in mathematics because it helps in understanding the symmetry and transformations of functions. A function can be reflected over the x-axis or y-axis, changing its orientation. For example, reflecting the function f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Relation	Relation Topic Functions and Relations Definition A relation is a set of ordered pairs, where each element from one set is paired with an element from another set.	Relations and Functions
Definition--Functions and Relations Concepts--Step Function	Step Function Topic Functions and Relations Definition A step function is a function that increases or decreases abruptly at certain points, creating a series of steps.	Special Functions
Definition--Functions and Relations Concepts--Translating a Function	Translating a Function Topic Functions and Relations Definition Translating a function involves shifting the graph of the function horizontally, vertically, or both, without changing its shape. Description Translating functions is significant in mathematics because it helps in understanding how functions behave under shifts. A function can be translated horizontally by adding or subtracting a constant to the input, and vertically by adding or subtracting a constant to the output. For example, translating the function f(x) = x2 horizontally by 2 units results in	Relations and Functions
Definition--Functions and Relations Concepts--Vertical Line Test	Vertical Line Test Topic Functions and Relations Definition The vertical line test is a method used to determine if a graph represents a function by checking if any vertical line intersects the graph more than once. Description The vertical line test is important in mathematics because it helps in identifying whether a given graph represents a function. If a vertical line intersects the graph at more than one point, then the graph does not represent a function. This test is used in various fields, including computer science for validating functions in programming and in mathematics for analyzing graphs. For example, the graph of	Relations and Functions
Definition--Functions and Relations Concepts--Visual Models of Functions	Visual Models of Functions Topic Functions and Relations Definition Visual models of functions are graphical representations that illustrate the relationship between input and output values of functions. Description Visual models of functions are important in mathematics because they provide a clear and intuitive way to understand how functions behave. These models include graphs, tables, and diagrams that show the relationship between input and output values. For example, the graph of f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Visual Models of Relations	Visual Models of Relations Topic Functions and Relations Definition Visual models of relations are graphical representations that illustrate the relationship between two sets of values, not necessarily functions.	Relations and Functions
Definition--Functions and Relations Concepts--Visualization of the Dependent Variable	Visualization of the Dependent Variable Topic Functions and Relations Definition Visualization of the dependent variable involves graphically representing the outcomes or responses that depend on the independent variable.	Relations and Functions
Definition--Functions and Relations Concepts--Visualization of the Independent Variable	Visualization of the Independent Variable Topic Functions and Relations Definition Visualization of the independent variable involves graphically representing the variable that is manipulated to observe its effect on the dependent variable.	Relations and Functions
Formulas--Volume of a Cube	Formulas--Volume of a Cube The formula for the Volume of a Cube. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Volume
Formulas--Volume of a Cylinder	Formulas--Volume of a Cylinder The formula for the Volume of a Cylinder. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Volume and Cylinders
Formulas--Volume of a Rectangular Prism	Formulas--Volume of a Rectangular Prism The formula for the Volume of a Recantular Prism. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Volume
Formulas--Volume of a Sphere	Formulas--Volume of a Sphere The formula for the Volume of a Sphere. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Volume
Formulas--Volume of a Square Pyramid	Formulas--Volume of a Square Pyramid The formula for the Volume of a Square Pyramid. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Volume
Formulas--Volume of a Triangular Prism	Note: The download is a JPG file. Related Resources To see the complete collection of Formulas, click on this link: https://media4math.com/math-formulas	Volume
Formulas--Volume of a Triangular Pyramid	Volume of a Triangular Pyramid Watch this video to learn about the volume of a pyramid. (The video transcript is also included.) Your browser does not support the video tag. Video Transcript Pyramids are majestic and a bit mysterious. Arising from deserts and jungles, they are grand in size and sleek in appearance. Mathematically, pyramids also have an air of mystery. Let’s take a closer look. Here is a cube of side s. Its volume couldn’t be more straightforward: s • s •s, or s-cubed. Its name, its shape, its volume are straightforward.	Volume
Geometry Applications Teachers Guide: 3D Geometry	Geometry Applications Teachers Guide: 3D Geometry This is the Teacher's Guide that accompanies Geometry Applications: 3D Geometry. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of 3D Geometry
Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 1	Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 1 Topic Geometric Models	Applications of Surface Area and Volume and Rational Functions and Equations
Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2	Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2 Topic Geometric Models	Surface Area, Volume and Rational Functions and Equations