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1.4.3: Graphs of Exponential Functions
https://math.libretexts.org/Workbench/Book-_Precalculus_I_for_Highline_College_w/Rational_Inequalities_and_Equations_of_Circles/1.04%3A_Exponential_and_Logarithmic_Functions/1.4.03%3A_Graphs_of_Exponential_Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an eq...As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.
4.2: Graphs of Exponential Functions
https://math.libretexts.org/Courses/Hartnell_College/MATH_25%3A_PreCalculus_(Abramson_OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.02%3A_Graphs_of_Exponential_Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an eq...As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.
4.2: Graphs of Exponential Functions
https://math.libretexts.org/Courses/Truckee_Meadows_Community_College/TMCC%3A_Precalculus_I_and_II/Under_Construction_test2_04%3A_Exponential_and_Logarithmic_Functions/Under_Construction_test2_04%3A_Exponential_and_Logarithmic_Functions_4.2%3A_Graphs_of_Exponential_Functions
For example, if we begin by graphing the parent function $f(x)=2^x$ ,we can then graph the stretch, using $a=3$ ,to get $g(x)=3{(2)}^x$ as shown on the left in Figure $\PageIndex{8}$ , and the co...For example, if we begin by graphing the parent function $f(x)=2^x$ ,we can then graph the stretch, using $a=3$ ,to get $g(x)=3{(2)}^x$ as shown on the left in Figure $\PageIndex{8}$ , and the compression, using $a=\dfrac{1}{3}$ ,to get $h(x)=\dfrac{1}{3}{(2)}^x$ as shown on the right in Figure $\PageIndex{8}$ .
4.3: Graphs of Exponential Functions
https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/04%3A_Exponential_and_Logarithmic_Functions/4.03%3A_Graphs_of_Exponential_Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an eq...As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.
4.2: Graphs of Exponential Functions
https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.02%3A_Graphs_of_Exponential_Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an eq...As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.
5.2: Transformation of Graphs
https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Tradler_and_Carley)/05%3A_Basic_Functions_and_Transformations/5.02%3A_Transformation_of_Graphs
Similarly, to get from the graph of $y=\sqrt{x}$ to the graph of $y=\sqrt{x+1}$ , we shift the graph to the left, and then for $y=2\cdot \sqrt{x+1}$ , we need to stretch the graph by a factor \(2\...Similarly, to get from the graph of $y=\sqrt{x}$ to the graph of $y=\sqrt{x+1}$ , we shift the graph to the left, and then for $y=2\cdot \sqrt{x+1}$ , we need to stretch the graph by a factor $2$ away from the $x$ -axis. (Alternatively, we could first stretch the the graph away from the $x$ -axis, then shift the graph by $1$ to the left.)
5.4E: Exercises for Section 5.4
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.04%3A_Special_Linear_Transformations_in_Two_and_Three_Dimensions/5.4E%3A_Exercises_for_Section_5.4
This page features exercises on deriving transformation matrices for linear operations in $\mathbb{R}^2$ and $\mathbb{R}^3$ , focusing on rotations, reflections, projections, shears, and dilations....This page features exercises on deriving transformation matrices for linear operations in $\mathbb{R}^2$ and $\mathbb{R}^3$ , focusing on rotations, reflections, projections, shears, and dilations. It includes specific angles for rotations and combinations of transformations, with answers demonstrating the relationships between trigonometric functions and geometric effects.
4.2: Graphs of Exponential Functions
https://math.libretexts.org/Courses/Highline_College/Math_141%3A_Precalculus_I_(old_edition)/04%3A_Exponential_and_Logarithmic_Functions/4.02%3A_Graphs_of_Exponential_Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an eq...As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.
4.4: Transformations
https://math.libretexts.org/Courses/Mount_Royal_University/Mathematical_Reasoning/4%3A_Basic_Concepts_of_Euclidean_Geometry/4.4%3A_Transformations
A translation is a transformation that moves a figure (without altering dimensions) to a new position.
7.1: Introducing Rational Functions
https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/07%3A_Rational_Functions/7.01%3A_Introducing_Rational_Functions
In this section, our study will lead us to the rational functions. Note the root word “ratio” in the term “rational.” Does it remind you of the word “fraction”? It should, as rational functions are fu...In this section, our study will lead us to the rational functions. Note the root word “ratio” in the term “rational.” Does it remind you of the word “fraction”? It should, as rational functions are functions in a very specific fractional form.
11.1: Sets of Real Numbers and the Cartesian Coordinate Plane
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/11%3A_Appendix_-_Prerequisite_Function_Material/11.01%3A_Sets_of_Real_Numbers_and_the_Cartesian_Coordinate_Plane
This section introduces the sets of real numbers, their properties, and the Cartesian coordinate plane. It covers the classification of numbers (natural, whole, integers, rational, and irrational) and...This section introduces the sets of real numbers, their properties, and the Cartesian coordinate plane. It covers the classification of numbers (natural, whole, integers, rational, and irrational) and explains how to plot points and graph equations on the Cartesian plane. The section also discusses intervals and the distance formula, providing foundational knowledge for working with functions and graphs in Algebra.

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