Notes on Linear vs Nonlinear Functions and Intercepts

Linear functions: key characteristics

  • In a linear function, no variable is raised to a power other than 1. If a variable has a power other than 1, the graph is not a straight line.
  • General form: y=mx+by = mx + b where:
    • $m$ is the slope (rate of change)
    • $b$ is the y-intercept (the value of $y$ when $x = 0$)
  • Intuition: The graph rises or falls by a constant amount for each unit increase in $x$ (constant rate of change).

Example: Simplifying a linear equation

  • Given: y=6x3−53y = \frac{6x}{3} - \frac{5}{3}
  • Simplify: y=2x−53y = 2x - \frac{5}{3}
  • This is a linear function with slope $m = 2$ and y-intercept $b = -\frac{5}{3}$.
  • Graphing note: This can be graphed as a straight line with slope $2$.
  • Quick interpretation: For each increase of $x$ by $1$, $y$ increases by $2$.

Slope and constant rate of change

  • In a linear function $y = mx + b$, the change in $y$ is proportional to the change in $x$: the slope is constant.
  • Definition: m=ΔyΔxm = \frac{\Delta y}{\Delta x}
  • For a change in $x$ by $\Delta x$, the corresponding change in $y$ is $m\Delta x$.
  • Example: For $y = 2x - \frac{5}{3}$, a change in $x$ by $1$ yields a change in $y$ by $2$.

Intercepts: x-intercept and y-intercept

  • Intercepts are where the graph crosses the axes:
    • Y-intercept: where $x = 0$ → point $(0, b)$
    • X-intercept: where $y = 0$ → solve $0 = mx + b$, giving $x = -\frac{b}{m}$ (provided $m \neq 0$) → point $( -\frac{b}{m}, 0)$
  • In the example $y = 2x - \frac{5}{3}$:
    • Y-intercept: $(0, -\frac{5}{3})$
    • X-intercept: $(\frac{5}{6}, 0)$ since $0 = 2x - \frac{5}{3} \Rightarrow x = \frac{5}{6}$
  • Key property for linear functions:
    • The graph crosses the x-axis exactly once (assuming $m \neq 0$).
    • The graph crosses the y-axis exactly once for any non-vertical line (every $x$ has a single $y$ value, so only one $y$ intercept).
  • Additional nuance (noted in the transcript): a function cannot have more than one $y$-intercept
    • This follows from the vertical line test: a given $x$ value maps to a single $y$ value.

What makes a graph linear vs nonlinear

  • Linear: all variables appear to the first power (no squares, cubes, etc.). The graph is a straight line.
  • Nonlinear: at least one variable has power > 1 or the equation involves products, roots, or other nonlinear operations; the graph bends or curves.
  • The transcript emphasizes: if the graph goes up by the same amount for equal steps in $x$, it is linear; otherwise, it is nonlinear.

Special notes on symmetry and line folding

  • Line symmetry (folding onto itself) concept:
    • If a graph has line symmetry about a vertical line, folding along that vertical line would map halves onto each other.
    • The transcript discusses folding along the vertical axis (the $y$-axis) as a common form of symmetry.
    • It also notes that folding along a vertical line other than the $y$-axis would not, in general, map the graph onto itself.
  • Practical takeaway:
    • For a function’s graph, line symmetry, if present, tends to align with a vertical axis (the $y$-axis) in typical even-function cases; folding along other vertical lines is not generally a symmetry of the graph.

Intercept finding practice and checks

  • How to locate intercepts quickly:
    • Y-intercept: set $x = 0$ → $y = b$ → intercept $(0, b)$
    • X-intercept: set $y = 0$ → solve $0 = mx + b$ → $x = -\frac{b}{m}$ (if $m \neq 0$) → intercept $( -\frac{b}{m}, 0)$
  • Quick summary for the line $y = mx + b$:
    • Y-intercept: $(0, b)$
    • X-intercept: $( -\frac{b}{m}, 0)$
  • Edge cases:
    • Horizontal line $y = b$ with $b \neq 0$ has no x-intercept; y-intercept is $(0, b)$.
    • If the line is $y = 0$ (the x-axis), every point on the x-axis is an intercept (degenerate case).

Practical implications and real-world relevance

  • Modeling data:
    • If the observed change in $y$ is proportional to the change in $x$ (constant rate of change), a linear model is appropriate.
    • If the rate of change varies with $x$, nonlinear models are needed.
  • Intercepts provide context:
    • The y-intercept represents the starting value when $x = 0$.
    • The x-intercept represents where the graph crosses the $x$-axis (where the quantity of interest is zero).
  • Symmetry considerations can aid graph analysis and checks, though many real-world graphs are not perfectly symmetric.

Quick reference formulas

  • General linear form: y=mx+by = mx + b
  • Slope: m=ΔyΔxm = \frac{\Delta y}{\Delta x}
  • X-intercept: x=−bm(m≠0)x = -\frac{b}{m}\quad (m \neq 0)
  • Y-intercept: b (so the point is (0,b))b \text{ (so the point is } (0, b)\text{)}
  • Example values for the example line y=2x−53y = 2x - \frac{5}{3}:
    • Y-intercept: (0,−53)(0, -\frac{5}{3})
    • X-intercept: (56,0)(\frac{5}{6}, 0)