Graphing Exponential Functions - Varsity Tutors

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Graphing Exponential Functions

Study Guide

Key Definition

An exponential function is defined as $y = a^x$, where $a$ is a constant, a > 0 and a ≠ 1.

Important Notes

  • The graph of $y = 2^x$ has the x-axis as an asymptote.
  • Changing the base $a$ changes the shape of the graph.
  • Replacing $x$ with $-x$ reflects the graph across the y-axis.
  • Replacing $y$ with $-y$ reflects it across the x-axis.
  • Translating $x$ inside the exponent by $+h$ ($y = a^{x + h}$) moves the graph $h$ units to the left; translating the output ($y = a^x + h$) shifts it up/down by $h$ units.

Mathematical Notation

$y = a^x$ defines an exponential function$+$ represents addition$-$ represents subtraction$\times$ or $*$ represents multiplication$\div$ or $/$ represents divisionRemember to use proper notation when solving problems

Why It Works

Exponential functions model growth and decay processes, with key properties shown in their graphs.

Remember

The x-axis acts as an asymptote for exponential graphs like $y = 2^x$.

Quick Reference

Reflection across y-axis:$y = a^{-x}$Translation up:$y = a^x + k$

Understanding Graphing Exponential Functions

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

To graph y = 2^x, start by making a table of values: for x = -2, -1, 0, 1, 2, compute y = 1/4, 1/2, 1, 2, 4. Plot these points on the coordinate plane. Observe that as x → -∞, y approaches 0 (the horizontal asymptote), and as x → ∞, y grows rapidly.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Which of the following formulas represents exponential growth with base 2?

A$y = 2^x$B$y = 3^x$C$y = -2^x$D$y = 2^{-x}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine you deposit $100 in a savings account where the money doubles each year following $y = 100 \times 2^x$. How much money will you have after 5 years?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Consider how the graph of $y = 2^{x+3}$ differs from $y = 2^x$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Which transformation results in $y = 2^{-x}$?

AReflection across the y-axisBReflection across the x-axisCTranslation 3 units upDTranslation 3 units downCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

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