2.4 ext Linear vs. Exponential Functions

Introduction to Functions

• The session covered the introduction of both linear and exponential functions, including comparisons and differentiation between the two types.

Bell Work Question

• Key Question: How can you tell if a function is linear?

• Answer Explanation:

  • A linear function has a constant rate of change.

  • This means the function increases at a constant unchanging rate.

Linear Functions

• General Form:

  • The equation of a linear function can be expressed as: $f(x) = mx + b$

  • Where:

    • $m$ = slope (the rate of change)

    • $b$ = y-intercept (the starting point)

• Similarities with Previous Learning:

  • The structure $mx + b$ has been covered earlier in the course as the equation for a line.

• Determinants:

  • The notation $f(x)$ is similar to $y$, indicating that the function takes some input and produces some output.

Finding Outputs in Linear Functions

• Example given:

  • For an equation $f(x) = 2x + 3$, if we let $x = 4$:

  • Calculation:

    • $f(4) = 2(4) + 3$ results in $f(4) = 8 + 3 = 11$.

Components of the Linear Function

• Starting Point (b):

  • Also known as the y-intercept.

• Rate of Change (m):

  • Defined as the constant rate of change.

  • Positive $m$ indicates growth; negative $m$ indicates decay.

Exponential Functions

• General Form:

  • The equation of an exponential function can be expressed as: $f(x)=b\cdot m^{x}$

  • Where:

    • $b$ = starting point (initial value)

    • $m$ = growth factor (not constant rate of change)

Key Differences from Linear Functions

• In exponential functions, while $b$ is still the starting point, $m$ is referred to as the growth factor rather than a rate of change.

• The growth factor $m$ indicates how much the function grows for each unit increase in $x$.

Choosing Between Linear and Exponential Functions

• Linear function model:

  • $y = mx + b$ (where $m$ = rate and $b$ = starting amount)

• Exponential function model:

  • $f(x)=b\cdot m^{x}$ (where $b$ = starting amount and $m$ = growth factor)

Analysis Techniques for Differentiation

• For Exponential Functions:

  • The growth factor $m$ can be determined by dividing the second output by the first output.

Example of Finding Growth Factor: Exponential

• Given data points:

  • (2, 6) and (3, 12)

  • Division:

  • $\frac{12}{6}=2$ thus growth factor $m = 2$.

Example of Finding Rate of Change: Linear

• For a linear function showing constant increase of 1, the method for finding the slope is Rise over Run.

• Formula:

  • Linear slope =(y2 - y1)/(x2 - x1)

Practical Examples

• Library Example: A library has 8,000 books, adding 500 books each year.

  • Function Type: Linear.

  • Formula: $y = 500x + 8000$

• Bank Account Example: Starting with $10, tripling each month.

  • Function Type: Exponential.

  • Formula: $f(x)=10\cdot3^{x}$ .

• Owls Example: 20,000 owls halving every decade.

  • Function Type: Exponential.

  • Formula: $f(x)=20000\cdot\frac12^{x}$ .

Combining Both Functions in Evaluations

• Important Recognition:

  • Recognizing keywords indicating growth or decay (e.g., ‘double,’ ‘triple,’ or ‘half’) can help identify whether a function is linear or exponential.

• Using Tables for Determining Function Type:

  • Linear Example: A table showing constant changes between outputs would signify a linear function.

  • Exponential Example: A table showing outputs increasing by a constant factor (e.g., dividing outputs would yield the same result) signifies an exponential function.