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- 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_FunctionsAs 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.
- 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_FunctionsAs 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.
- 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_FunctionsFor 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 \(\PageIndex{8}\).
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_375%3A_Pre-Calculus/13%3A_Exponential_and_Logarithmic_Functions/13.02%3A_Graphs_of_Exponential_FunctionsThis section explores the graphs of exponential functions, detailing key features such as domain, range, asymptotes, and intercepts. It explains how transformations like shifts, reflections, and stret...This section explores the graphs of exponential functions, detailing key features such as domain, range, asymptotes, and intercepts. It explains how transformations like shifts, reflections, and stretches affect the shape of the graph. The section includes examples demonstrating how to graph exponential functions and analyze their behavior.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/04%3A_Exponential_and_Logarithmic_Functions/4.03%3A_Graphs_of_Exponential_FunctionsAs 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.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.02%3A_Graphs_of_Exponential_FunctionsAs 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.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Tradler_and_Carley)/05%3A_Basic_Functions_and_Transformations/5.02%3A_Transformation_of_GraphsSimilarly, 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.)
- 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_5.4This 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.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/11%3A_Exponential_and_Logarithmic_Functions/11.02%3A_Graphs_of_Exponential_FunctionsThis section explores the graphs of exponential functions, detailing key features such as domain, range, asymptotes, and intercepts. It explains how transformations like shifts, reflections, and stret...This section explores the graphs of exponential functions, detailing key features such as domain, range, asymptotes, and intercepts. It explains how transformations like shifts, reflections, and stretches affect the shape of the graph. The section includes examples demonstrating how to graph exponential functions and analyze their behavior.
- 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_FunctionsAs 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.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_2e_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.03%3A_Graphs_of_Exponential_FunctionsAs 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.