Glossary
Practice Quizzes

6.2 Graphs of Exponential Functions

3 min read•june 24, 2024

Exponential functions are powerful tools for modeling growth and decay. They have a unique shape, with a , , and distinct . Understanding these key features helps us analyze real-world scenarios like population growth or radioactive decay.

Transformations of exponential functions allow us to shift, , or reflect their graphs. By adjusting parameters, we can model various situations, from delayed growth to accelerated decay. Comparing different exponential functions helps us understand their relative behaviors and applications.

Graphing Exponential Functions

Key features of exponential graphs

Top images from around the web for Key features of exponential graphs

  • Basic Algebra/Exponential Functions/Graphs of Exponential Functions - Wikibooks, open books for ... View original

    Is this image relevant?

  • Graphs of Exponential Functions | Algebra and Trigonometry View original

    Is this image relevant?

  • Basic Algebra/Exponential Functions/Graphs of Exponential Functions - Wikibooks, open books for ... View original

    Is this image relevant?

1 of 3

Top images from around the web for Key features of exponential graphs

  • Basic Algebra/Exponential Functions/Graphs of Exponential Functions - Wikibooks, open books for ... View original

    Is this image relevant?

  • Graphs of Exponential Functions | Algebra and Trigonometry View original

    Is this image relevant?

  • Basic Algebra/Exponential Functions/Graphs of Exponential Functions - Wikibooks, open books for ... View original

    Is this image relevant?

1 of 3
  • of exponential functions: f(x)=abxf(x) = ab^x
    • aa represents or and across y-axis
      • ∣a∣>1|a| > 1 vertically stretches the graph
      • 0<∣a∣<10 < |a| < 1 vertically compresses the graph
      • a<0a < 0 reflects the graph across x-axis
    • bb represents the and determines growth or decay
      • b>1b > 1 indicates (doubles every day)
      • 0<b<10 < b < 1 indicates ( of radioactive elements)
  • y-intercept: point where the graph crosses y-axis found by evaluating f(0)f(0)
  • : line the graph approaches as xx approaches positive or negative infinity
    • functions have horizontal asymptote y=0y = 0 as x→−∞x \to -\infty
    • Exponential decay functions have horizontal asymptote y=0y = 0 as x→∞x \to \infty
  • End behavior: trend of the graph as xx approaches positive or negative infinity
    • Exponential growth functions have f(x)→∞f(x) \to \infty as x→∞x \to \infty and f(x)→0f(x) \to 0 as x→−∞x \to -\infty (population growth)
    • Exponential decay functions have f(x)→0f(x) \to 0 as x→∞x \to \infty and f(x)→∞f(x) \to \infty as x→−∞x \to -\infty (medication concentration in body)

Transformations of exponential functions

  • : moves graph up or down by adding or subtracting constant kk to function
    • f(x)=abx+kf(x) = ab^x + k shifts graph vertically by kk units
      • k>0k > 0 shifts graph up (increased initial investment)
      • k<0k < 0 shifts graph down (lower starting population)
  • : moves graph left or right by adding or subtracting constant hh to input variable xx
    • f(x)=abx−hf(x) = ab^{x-h} shifts graph horizontally by hh units
      • h>0h > 0 shifts graph right (delayed start of exponential growth)
      • h<0h < 0 shifts graph left (earlier onset of exponential decay)
  • or compression: multiplies output values by constant aa affecting steepness of graph
    • ∣a∣>1|a| > 1 stretches graph vertically (faster growth rate)
    • 0<∣a∣<10 < |a| < 1 compresses graph vertically (slower decay rate)
  • : flips graph across x-axis or y-axis
    • Reflection across x-axis occurs when a<0a < 0 in general form
    • Reflection across y-axis occurs when input variable xx is negated f(x)=ab−xf(x) = ab^{-x}

Analyzing and Comparing Exponential Functions

Comparison of exponential graphs

  • Changes in aa:
    • Increasing ∣a∣|a| makes graph steeper while decreasing ∣a∣|a| makes it less steep (comparing growth rates)
    • Changing sign of aa reflects graph across x-axis (growth vs decay)
  • Changes in bb:
    • Increasing bb (b>1b > 1) makes graph grow faster while decreasing bb (0<b<10 < b < 1) makes graph decay slower (base represents growth/)
  • Comparing y-intercepts:
    • y-intercept affected by value of aa and any kk (different initial values)
  • Comparing asymptotes:
    • Exponential functions with same base bb share same horizontal asymptote y=0y = 0 (common long-term behavior)
  • Comparing end behavior:
    • Functions with same base bb exhibit similar end behavior depending on b>1b > 1 (growth) or 0<b<10 < b < 1 (decay)

Properties of Exponential Functions

  • : All real numbers, as exponential functions are defined for every input value
  • : All positive real numbers for exponential functions with positive base (excluding zero)
  • : Exponential functions are smooth and unbroken, with no gaps or jumps in their graphs
  • : An equation where the variable appears in the exponent (e.g., 2x=82^x = 8)
  • : A special case where the base is the mathematical constant e (approximately 2.71828)

Key Terms to Review (38)

Base: The base is a fundamental component in various mathematical concepts, serving as a reference point or starting value. It is a crucial element in understanding exponents, exponential functions, logarithmic functions, and geometric sequences, among other topics.
Change-of-base formula: The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.
Compression: Compression refers to a transformation that reduces the distance between points in a graph. It often results in the graph appearing 'squeezed' either horizontally or vertically.
Continuous function: A continuous function is a function where small changes in the input result in small changes in the output. Mathematically, a function $f(x)$ is continuous at a point $x = c$ if $\lim_{{x \to c}} f(x) = f(c)$.
Continuous Function: A continuous function is a mathematical function that has no abrupt changes or jumps in its graph. It is a function where small changes in the input result in small changes in the output, with no sudden or drastic changes. Continuity is an important property that allows for the smooth and predictable behavior of functions, which is essential in various mathematical and scientific applications.
Decay Factor: The decay factor, also known as the damping factor, is a crucial parameter that describes the rate of exponential decay in various mathematical and scientific contexts. It represents the degree to which a quantity diminishes over time or with successive iterations, and it plays a vital role in understanding the behavior of exponential functions and fitting exponential models to data.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Doubling time: Doubling time is the period it takes for a quantity to double in size or value at a constant growth rate. It is commonly used in exponential growth models.
Doubling Time: Doubling time is the amount of time it takes for a quantity to double in value. It is a crucial concept in the study of exponential growth and decay, and is closely tied to the understanding of exponential functions, their graphs, logarithmic functions, and their applications in various models.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction or diminishment of a quantity over time. It is characterized by an initial value that decreases by a constant proportion during each successive time interval, resulting in an exponential decrease. This concept is fundamental to understanding various phenomena in fields such as physics, chemistry, biology, and finance.
Exponential equation: An exponential equation is an equation in which the variables appear as exponents. These equations often take the form $a^{x} = b$ where $a$ and $b$ are constants.
Exponential Equation: An exponential equation is a mathematical equation in which the unknown variable appears as the exponent of another quantity. These equations model situations where a quantity grows or decays at a constant rate over time, and are closely related to the behavior of exponential functions.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
General Form: The general form of an equation is a standardized way of expressing the equation that reveals its underlying structure and characteristics. This term is particularly relevant in the context of various mathematical functions and conic sections, as it allows for a concise and informative representation of these entities.
Growth Factor: A growth factor is a naturally occurring substance capable of stimulating cellular growth, proliferation, healing, and cellular differentiation. Growth factors are important for regulating a variety of cellular processes, including cell growth, cell differentiation, cell migration, and cell survival, which are crucial for the development and maintenance of tissues and organs.
Half-life: Half-life is the time it takes for a radioactive or other substance to decay to half of its initial value. This concept is central to understanding exponential functions, their graphs, logarithmic functions, and how these models are applied to real-world situations involving growth and decay.
Horizontal asymptote: A horizontal asymptote is a horizontal line that a graph approaches as the input values go to positive or negative infinity. It indicates the end behavior of a function's output.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function's graph approaches as the input value (x) approaches positive or negative infinity. It represents the limit of the function as it gets closer and closer to this line, without ever touching it.
Horizontal reflection: A horizontal reflection is a transformation that flips a function's graph over the y-axis. It changes the sign of the x-coordinates of all points on the graph.
Horizontal Shift: A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This concept is applicable to various types of functions, including transformations of functions, absolute value functions, exponential functions, trigonometric functions, and the parabola.
Natural Exponential Function: The natural exponential function, denoted as $e^x$, is a fundamental function in mathematics that describes continuous exponential growth or decay. It is the base-$e$ exponential function, where $e$ is the mathematical constant approximately equal to 2.718. The natural exponential function is a crucial concept in the study of exponential functions and their graphs.
One-to-one: A one-to-one function is a function in which each element of the range is paired with exactly one element of the domain. This implies that no two different inputs produce the same output, ensuring the function passes the horizontal line test.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Reflection: Reflection is a transformation that flips a graph over a specified axis, creating a mirror image. In algebra, this often involves reflecting exponential and logarithmic functions over the x-axis or y-axis.
Reflection: Reflection is a transformation of a function that creates a mirror image of the original function across a specified axis. This concept is fundamental in understanding the behavior and properties of various mathematical functions.
Stretch: A stretch is a transformation that scales a graph horizontally or vertically. It changes the shape of the graph without altering its basic structure.
Vertical compression: A vertical compression is a transformation that scales a function's graph towards the x-axis. This is achieved by multiplying the function by a constant factor between 0 and 1.
Vertical Compression: Vertical compression is a transformation of a function that scales the function vertically, effectively shrinking or stretching the function along the y-axis. This transformation can impact the amplitude, range, and behavior of the function.
Vertical shift: A vertical shift is a transformation that moves a graph up or down in the coordinate plane by adding or subtracting a constant to the function's output. It does not affect the shape of the graph, only its position.
Vertical Shift: Vertical shift refers to the movement of a graph or function up or down the y-axis, without affecting the shape or orientation of the graph. This transformation changes the y-intercept of the function, but leaves the x-intercepts and the overall shape unchanged.
Vertical stretch: A vertical stretch is a transformation that scales a function's graph away from the x-axis by multiplying all y-values by a factor greater than 1. It does not affect the x-values of the function.
Vertical Stretch: Vertical stretch is a transformation of a function that involves scaling the function vertically, either by expanding or compressing the function along the y-axis. This transformation affects the amplitude or the range of the function, without changing its basic shape or period.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Glossary
Glossary
6.3 Logarithmic Functions