Linear functions

Linear functions represent relationships with a constant rate of change, creating straight lines when graphed. They can be modeled as $f(x) = mx + b$ or $y = mx + b$, where $f(x)$ represents the output value for any input $x$.

Function form: $f(x) = mx + b$

The term $f(x)$ replaces $y$ to emphasize that the output value depends on the input $x$.

• Evaluating $f(a)$ means substituting $x = a$ into the function rule to find the corresponding output value.
  
• The notation $f(x) = k$ means finding the input value $x$ that yields an output of $k$.
  
• On a graph, $(x, f(x))$ represents an ordered pair $(x, y)$ on the line.
  
• In modeling, $f(x)$ represents the total amount or final state after $x$ units of change.

Transformations of Linear Functions

Transformations alter the position, steepness, or orientation of a base linear function, such as $f(x) = 3x + 7$.

• Vertical Stretch/Compression: $g(x) = a \cdot f(x)$ multiplies all output values by $a$.

  Example: Stretch $f(x) = 3x + 7$ vertically by a factor of $2$:

  $$g(x) = 2 \cdot f(x) = 2(3x + 7) = 6x + 14$$

• Reflection Across the $x$-Axis: $g(x) = -f(x)$ flips the entire graph vertically.

  Example: Reflect $f(x) = 3x + 7$ across the $x$-axis:

  $$g(x) = -f(x) = -(3x + 7) = -3x - 7$$

• Horizontal Shift: $g(x) = f(x - h)$ shifts the graph right by $h$ units ($h > 0$) or left ($h < 0$).

  Example: Shift $f(x) = 3x + 7$ right by $4$ units:

  $$g(x) = f(x - 4) = 3(x - 4) + 7 = 3x - 12 + 7 = 3x - 5$$

• Vertical Shift: $g(x) = f(x) + k$ shifts the graph up by $k$ units ($k > 0$) or down ($k < 0$).

  Example: Shift $f(x) = 3x + 7$ down by $5$ units:

  $$g(x) = f(x) - 5 = (3x + 7) - 5 = 3x + 2$$

Comparing Linear Functions

Linear functions can be compared across different representations, such as equations, tables, graphs, or verbal descriptions.

• Comparing Rates of Change: Determine which function grows or decays faster by comparing the absolute values of their slopes ($\vert{}m\vert{}$).

• Comparing Initial Values: Identify which function starts higher or lower by comparing their $y$-intercepts ($b = f(0)$).

• Finding Intersections: Setting $f(x) = g(x)$ finds the input value $x$ where both functions produce identical outputs.

1. Evaluate $f(x)$ carefully when nested or given algebraic inputs.

   Example: If $f(x) = 3x - 4$, what is $f(2a + 1)$? Substitute $(2a + 1)$ for $x$: $f(2a + 1) = 3(2a + 1) - 4 = 6a + 3 - 4 = 6a - 1$.

2. Identify vertical shifts directly by comparing $y$-intercepts.

   Example: The graph of $g(x)$ is formed by shifting $f(x) = 2x + 5$ down by $3$ units. What is $g(x)$? Subtract $3$ from the constant term: $g(x) = f(x) - 3 = (2x + 5) - 3 = 2x + 2$.

3. Distinguish between input shifts $f(x - h)$ and output shifts $f(x) + k$. 

   Example: If $f(x) = 4x$, what equation represents $g(x) = f(x - 2)$? Replace $x$ with $(x - 2)$: $g(x) = 4(x - 2) = 4x - 8$.

4. Compare slopes from tables by finding $\frac{\Delta f(x)}{\Delta x}$. 

   Example: Function $A$ has $f(x) = 5x + 1$. Function $B$ is given by points $(0, 2)$ and $(4, 18)$. Which has a greater rate of change? Slope of $B$: $m = \frac{18 - 2}{4 - 0} = \frac{16}{4} = 4$. Comparison: Function $A$ has a greater rate of change ($5 > 4$).

5. Find the break-even or equivalence point by setting $f(x) = g(x)$. 

   Example: Plan $A$ costs $f(x) = 20x + 50$ and Plan $B$ costs $g(x) = 30x + 10$. At how many hours $x$ do both plans cost the same? Set equal: $20x + 50 = 30x + 10 \implies 10x = 40 \implies x = 4\text{ hours}$.

6.  Interpret domain and range restrictions in real-world contexts. 

   Example: A car's gas tank function is $g(m) = 15 - 0.05m$, where $m$ is miles driven. What is the practical domain? Tank empty condition: $15 - 0.05m = 0 \implies 0.05m = 15 \implies m = 300$. Practical domain: $0 \le m \le 300\text{ miles}$.