2.4.5 Recognizing, interpret graphs, exponential functions

• What is the function f(x) = 2^x?

  $f(x) =$$2^x$
• What is the formula for the exponential function given in the image?

  y = a·b^x
• What is the y-intercept of an exponential function in the form $y =$$ka^{x}$?


  (0, k)
• What are the constraints on the constants a and b in the exponential function formula?

  a ≠ 0, b > 0, and b ≠ 1
• What is the equation for growth shown in the image?

  $y=$$2^x$
• What are the key features of the exponential function?

  • Domain: All real numbers
  • Range: (0, ∞)
  • y-intercept: (0, 1)
  • x-intercept: None
  • Asymptote: y = 0 (x-axis)
  • End Behavior: As x → -∞, y → 0

  As x → ∞, y → ∞
• What are the three different colored lines in the graph?

  • Blue line
  • Pink line
  • Yellow line
• How can the concepts of growth and decay be applied in real-world scenarios?

  • Growth can model population growth, compound interest, technology adoption
  • Decay can model radioactive decay, depreciation of assets, signal attenuation
• How do you evaluate the function f(x) = 2^x at different values of x?

  • When x = -2, f(x) = 2^(-2) = 1/4 = 0.25
  • When x = -1, f(x) = 2^(-1) = 1/2 = 0.5
  • When x = 0, f(x) = 2^0 = 1
  • When x = 1, f(x) = 2^1 = 2
  • When x = 2, f(x) = 2^2 = 4
  • When x = 3, f(x) = 2^3 = 8
• What is the domain of a basic exponential function graph?

  All real numbers
• What is the function g(x) = 5(1/2)^x?

  $g(x) =$$5(1/2)^x$
• What is an example of an exponential decay function?

  $y =$$(\frac{1}{2})^{x}$
• What is an exponential function?

  Variable in the exponent
• What are the key features of the exponential function shown in the image?

  • Asymptote at y = 0
  • Passes through the point (0, 1)
  • Increasing function
  • Concave up
• What is the equation for decay shown in the image?

  $y=$$\frac{1}{2^x}$
• What is the base in the exponential function \( y = 2^x \)?


  2
• What is the difference between growth and decay as shown in the image?

  • Growth shows an exponential increase over time, with the function $y=$$2^x$
  • Decay shows an exponential decrease over time, with the function $y=$$\frac{1}{2^x}$
• How can the exponential function be used to model real-world phenomena?

  • Population growth
  • Radioactive decay
  • Compound interest
  • Spread of infectious diseases
• What is an example of an exponential growth function?

  $y =$$2^{x}$
• How do the three colored lines differ in their exponential growth rates?

  • The blue line has the slowest exponential growth rate
  • The pink line has a faster exponential growth rate than the blue line
  • The yellow line has the fastest exponential growth rate of the three
• What is the title of the graph shown in the image?

  Horse Two: Exponential Model
• What is the x-axis label of the graph?

  Time (seconds)
• What is the y-intercept if $k =$$1$ in the function $y =$$ka^{x}$?


  1
• How do you evaluate the function g(x) = 5(1/2)^x at different values of x?

  • When x = -2, g(x) = 5(1/2)^(-2) = 5(4) = 20
  • When x = -1, g(x) = 5(1/2)^(-1) = 5(2) = 10
• Match the base size with its growth characteristics:

  Small (e.g., 2) ↔️ Gradual growth

  Medium (e.g., 3) ↔️ Moderate growth

  Large (e.g., 10) ↔️ Rapid growth
• What type of model is being used to represent the data in the graph?

  Exponential model
• What is the equation of the exponential function shown in the image?

  $y =$$2^x$
• What are the key features of the exponential model shown in the graph?

  • Starts off slowly, then increases rapidly over time
  • Approaches an asymptotic limit as time increases
  • Represents an accelerating growth pattern
• How does the exponential model differ from a linear model in representing the horse's motion?

  • Exponential model shows accelerating growth over time
  • Linear model would show constant velocity/rate of change
  • Exponential better captures the dynamic nature of the horse's motion
• How can you calculate the distance traveled by the horse at any given time using the exponential model?

  • The exponential model is given by the equation:

  $d =$$1.57 \cdot e^{t/4.57}$

  • Where:
  • d is the distance traveled (in inches)
  • t is the time (in seconds)
  • Plug in the time value to calculate the corresponding distance
• Why does the graph show an exponential shape?

  The graph represents a process that grows or decays at a rate proportional to its current value
• Match the growth rate with the corresponding base:

  50% growth rate ↔️ 1.50

  75% growth rate ↔️ 1.75

  25% growth rate ↔️ 1.25

  100% growth rate ↔️ 2.00
• What is the shape of the graph shown in the image?

  Exponential
• What does the base of an exponential function directly affect?

  Steepness and growth rate
• Match the component of the exponential function $y =$$ka^{x}$ with its description:


  y-intercept ↔️ (0, k)

  k ↔️ Initial value

  a ↔️ Base of the exponent

  x ↔️ Exponent
• What does the green arrow at the bottom of the graph represent?

  The initial starting point or value of the exponential process
• If the horse travels for 10 seconds, what is the distance it has traveled according to the exponential model?

  $21.7 \text{ inches}$
• The graph of $y =$$2^{x}$ crosses the x-axis as x becomes very negative


  False
• What restriction applies to the base \( a \) in an exponential function?

  Must be positive and not equal to 1
• What do exponential models represent in real-world contexts?

  Increasing growth rates