Graphing Exponential Functions With e, Transformations, Domain and Range, Asymptotes, Precalculus

The Organic Chemistry Tutor・2 minutes read

Plot exponential functions by creating a table with key points and understanding shifts in the graph. Domains and ranges vary depending on the function's equation and shifts made.

Insights

  • The domain of exponential functions extends from negative infinity to infinity, while the range starts at 0 and goes up to infinity, excluding 0, showcasing the broad scope of possible values for x and y.
  • Shifting exponential functions by adding or subtracting values alters their graphs horizontally or vertically, impacting the location of asymptotes and the range of the function, exemplifying how simple adjustments can lead to significant changes in the visualization of the function.

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Recent questions

  • How do you graph an exponential function?

    To graph an exponential function like y = 2^x, create a table by plugging in values like 0 and 1. The horizontal asymptote (ASM) is the x-axis, unless shifted by a number like +3, which moves the ASM to y = 3. Plotting the points (0, 1) and (1, 2) creates the graph.

  • What is the domain and range of exponential functions?

    The domain of the function, representing possible x values, is from negative infinity to infinity. The range, representing y values, starts at 0 and goes up to infinity, excluding 0.

  • How do you shift an exponential function graph?

    For a function like y = 3^(x+1) - 2, shift the graph down by 2 units and left by 1 unit. Choose x values like -1 and 0, plot the points, and connect them to graph the function. The domain remains from negative infinity to infinity, while the range is from -2 to infinity.

  • What is the base value of 'e' in exponential functions?

    When dealing with e as the base in a function like y = e^(x+1), e is approximately 2.7. Plug in x values of 0 and 1 to find corresponding y values. The domain is still from negative infinity to infinity, and the range is from 1 to infinity, excluding 1.

  • How do you shift the graph of a function with a negative exponential term?

    In a function like y = 3 - e^(x-2), shift the graph two units to the right. The horizontal ASM is at y = 3, and the graph reflects below it due to the negative sign. Plugging in x values of 2 and 3, the range goes from negative infinity to 3, excluding 3, while the domain remains from negative infinity to infinity.

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Summary

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Graphing Exponential Functions: Shifts and Ranges

  • To graph an exponential function like y = 2^x, create a table by plugging in values like 0 and 1. The horizontal asymptote (ASM) is the x-axis, unless shifted by a number like +3, which moves the ASM to y = 3. Plotting the points (0, 1) and (1, 2) creates the graph.
  • The domain of the function, representing possible x values, is from negative infinity to infinity. The range, representing y values, starts at 0 and goes up to infinity, excluding 0.
  • For a function like y = 3^(x+1) - 2, shift the graph down by 2 units and left by 1 unit. Choose x values like -1 and 0, plot the points, and connect them to graph the function. The domain remains from negative infinity to infinity, while the range is from -2 to infinity.
  • When dealing with e as the base in a function like y = e^(x+1), e is approximately 2.7. Plug in x values of 0 and 1 to find corresponding y values. The domain is still from negative infinity to infinity, and the range is from 1 to infinity, excluding 1.
  • In a function like y = 3 - e^(x-2), shift the graph two units to the right. The horizontal ASM is at y = 3, and the graph reflects below it due to the negative sign. Plugging in x values of 2 and 3, the range goes from negative infinity to 3, excluding 3, while the domain remains from negative infinity to infinity.
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