Study sheet: Graphing Exponential Functions

📋 Course Outline

1. Exponential Function Foundations
2. Growth and Decay Models
3. Basic Exponential Graph Features
4. Transformations of Exponential Graphs
5. Piecewise Exponential Comparisons
6. Applications and Average Change
7. Analyzing Exponential Statements

📖 1. Exponential Function Foundations

🔑 Key Concepts & Definitions

• Exponential function : a function with the form $f(x)=b^x$, where the base b is a constant and the independent variable x is the exponent
• Asymptote : a line that the graph of a function approaches

📝 Essential Points

📌 An exponential growth function has base b>1, whereas an exponential decay function has base 0<b<1.

💡 Memory Hook

Growth uses b > 1, whereas decay uses 0 < b < 1.

📖 2. Growth and Decay Models

★ Must-know

📐 Formula — Exponential growth can be modeled by $A(t)=a(1+r)^t$, where A(t) is the amount after t time periods, a is the initial amount, and r is the percent increase per period.

📐 Formula — Exponential decay with an initial amount a and a decay rate r can be modeled by $A(t)=a(1-r)^t$.

Further detail

• The growth factor for an exponential growth model is 1+r.

📐 Formula — A $50 million investment earning 5% annual interest is modeled, in millions of dollars, by $A(t)=50(1.05)^t$.

💡 Memory Hook

A constant percent change each period causes multiplication by a constant factor.

📖 3. Basic Exponential Graph Features

📝 Essential Points

• For $f(x)=2^x$, the domain is all real numbers, the range is all positive real numbers, the y-intercept is (0,1), the asymptote is y=0, and as x approaches negative infinity f(x) approaches 0 while as x approaches positive infinity f(x) approaches infinity.

• The graph of a basic exponential function has a horizontal asymptote at y=0 and never reaches zero.

📌 For a positive base, exponential growth increases as x increases, whereas exponential decay decreases as x increases.

💡 Memory Hook

A curve approaches a horizontal floor or ceiling without touching it.

📖 4. Transformations of Exponential Graphs

★ Must-know

📐 Formula — A transformed exponential function can be written as $g(x)=ab^{x-h}+k$, where a controls vertical reflection and stretch or compression, h controls horizontal translation, and k controls vertical translation.

Further detail

• To transform the graph of $f(x)=ab^x$ into $g(x)=ab^{x-h}+k$, apply the vertical effect from a, translate horizontally by h units, and translate vertically by k units.

• The function $g(x)=-2(1/4)^{x-4}+3$ is obtained from $f(x)=(1/4)^x$ by reflecting across the x-axis, stretching vertically by a factor of 2, translating 4 units right, and translating 3 units up.

💡 Memory Hook

Scale or reflect, shift horizontally, then shift vertically.

📖 5. Piecewise Exponential Comparisons

★ Must-know

• 🔄 Graphing a piecewise function requires these steps:
  1. Graph each formula on its specified interval
  2. Compare the domains and ranges
  3. Compare intercepts, monotonicity, extrema, symmetry, sign, and end behavior

Further detail

• For the piecewise function with exponential piece $(1/2)^x$ for x<-1 and linear piece $2x+4$ for x≥-1, the exponential piece has y-intercept 4, while the linear piece has y-intercept 3 in the comparison shown.

💡 Memory Hook

One piece controls one interval, while another piece controls the remaining interval.

📖 6. Applications and Average Change

★ Must-know

📐 Formula — A substance that starts at 27.3 grams and decreases by 10% each year is modeled by $A(t)=27.3(0.9)^t$.

📐 Formula — A bacteria population that starts at 20,000 and reaches 30,000 after one day is modeled by $P(t)=20000(1.5)^t$.

Further detail

📌 For a real-world exponential model, the appropriate domain is restricted to the time values that make sense in the situation, such as t≥0 for elapsed time.

📖 7. Analyzing Exponential Statements

📝 Essential Points

📌 The statement that an exponential function of the form $y=ab^x-h+k$ has a y-intercept is always true when the function is defined at x=0.

📌 The statement that every exponential function of the form $y=ab^x-h+k$ has an x-intercept is not always true because an exponential graph may approach its asymptote without crossing the x-axis.

• If $f(x)=(8/b)^x$ represents exponential decay, then its base must satisfy $0<8/b<1$.