Study sheet: Graphing Exponential Functions

📋 Course Outline

1. Exponential Function Foundations
2. Graphs of Growth and Decay
3. Transformations of Exponential Graphs
4. Exponential Growth Models
5. Exponential Decay Models
6. Applications of Exponential Functions
7. Analyzing Function Properties
8. Piecewise and Comparative Graphs

📖 1. Exponential Function Foundations

🔑 Key Concepts & Definitions

• Exponential function : has the form $f(x)=b^x$, where b is a positive constant and x is the exponent

★ Must-know

📌 For $y=ab^x$ with a>0, the function represents exponential growth when b>1 and exponential decay when 0<b<1.

Further detail

📌 When b=1, $y=ab^x$ is constant rather than exponential growth or decay, and a negative base does not produce a real-valued function for every real x.

💡 Memory Hook

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

📖 2. Graphs of Growth and Decay

📝 Essential Points

• The graph of $f(x)=2^x$ has domain all real numbers, range all positive real numbers, y-intercept (0,1), and horizontal asymptote y=0.

• For $f(x)=2^x$, as x approaches negative infinity, f(x) approaches 0, and as x approaches positive infinity, f(x) approaches positive infinity.

📌 A function with a base greater than 1 is increasing, while a function with a base between 0 and 1 is decreasing.

💡 Memory Hook

A curve approaches the horizontal asymptote without touching it.

📖 3. Transformations of Exponential Graphs

★ Must-know

📐 Formula — The transformed exponential function has the form $f(x)=ab^{x-h}+k$, where a controls reflection and vertical scaling, h controls horizontal translation, and k controls vertical translation.

• To transform $g(x)=b^x$ into $f(x)=ab^{x-h}+k$, first apply the vertical scale or reflection given by a, then translate h units horizontally, and finally translate k units vertically.

Further detail

• For $g(x)=-\frac12 3^{x+4}+1$, the graph of $3^x$ is reflected across the x-axis, vertically compressed by a factor of one-half, translated 4 units left, and translated 1 unit up.

💡 Memory Hook

Scale or reflect, shift horizontally, then shift vertically.

📖 4. Exponential Growth Models

★ Must-know

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

• 🔄 The modeling procedure is:
  1. Define the initial amount and time variable.
  2. Convert the percentage increase to a decimal.
  3. Add the decimal rate to 1.
  4. Use the resulting factor as the exponential base.

Further detail

• An investment of 50 million dollars earning 5% annually is modeled by $A(t)=50(1.05)^t$ when amounts are measured in millions of dollars.

💡 Memory Hook

A repeated percentage increase produces multiplication by the growth factor 1 + r.

📖 5. Exponential Decay Models

★ Must-know

📐 Formula — A substance that decreases by r percent per time period is modeled by $A(t)=a(1-r)^t$, where a is the initial amount and r is written as a decimal.

Further detail

• A substance with initial amount 27.3 grams that decreases by 10% each year is modeled by $A(t)=27.3(0.90)^t$.

• If a substance decays by 35% each day and 8 milligrams remain after 8 days, its amount satisfies $8=a(0.65)^8$, where a is the initial amount.

💡 Memory Hook

A repeated percentage decrease produces multiplication by the decay factor 1 − r.

📖 6. Applications of Exponential Functions

★ Must-know

• A bacteria population that increases from 20,000 to 30,000 in one day is modeled by $P(t)=20{,}000(1.5)^t$.

Further detail

• An investment of $22,000 earning 5% annually from age 28 to age 65 is modeled by $A(t)=22{,}000(1.05)^t$ over 37 years.

• A car purchased for $28,000 that depreciates by 15% annually is modeled by $V(t)=28{,}000(0.85)^t$.

📖 7. Analyzing Function Properties

★ Must-know

• For a basic exponential function $b^x$ with b>0 and b≠1, the domain is all real numbers and the range is all positive real numbers.

📌 An exponential function can have a y-intercept because x=0 is in its domain, but it may or may not have an x-intercept depending on its transformations.

Further detail

📌 A positive coefficient preserves the vertical orientation of an exponential graph, whereas a negative coefficient reflects the graph across the x-axis.

💡 Memory Hook

Domain is unrestricted for basic exponentials, whereas application domains are restricted by time or context.

📖 8. Piecewise and Comparative Graphs

★ Must-know

• 🔄 The graphing procedure is:
  1. Create a table for each formula.
  2. Graph each formula on its specified interval.
  3. Use an open endpoint for an excluded boundary.
  4. Use a closed endpoint for an included boundary.

Further detail

• For the piecewise function $f(x)=\begin{cases}(3/4)^x&x<0\\3x+1&x\ge 0\end{cases}$, the exponential rule applies for x<0 and the linear rule applies for x≥0.

📊 Synthesis Tables

Growth and Decay Models

Model	Base	Behavior
Growth	1+r	Increases by a fixed percentage
Decay	1-r	Decreases by a fixed percentage