Key Concepts in Functions: Intercepts, Domain/Range, Monotonicity, Transformations, and Average Rate of Change

Average Rate of Change

• Definition over interval [a, b]: the slope of the secant line
• Formula: $\text{ARC} = \frac{f(b) - f(a)}{b - a}$
• Interpretation: total change in output divided by interval length; sign indicates increasing/decreasing trend over [a, b]; magnitude indicates average rate of change per unit input

Increasing / Decreasing Intervals

• Increasing: f(x) increases as x increases; graph goes up from left to right
• Decreasing: f(x) decreases as x increases; graph goes down from left to right
• How to determine:
  • If differentiable, check sign of derivative: if f'(x) > 0 then increasing; if f'(x) < 0 then decreasing
  • Sign-chart or vertex analysis for common shapes
• Common examples:
  • Linear: increasing everywhere if slope > 0
  • Quadratic: increasing on one side of the vertex and decreasing on the other
  • Absolute value: decreasing on ((-\infty, 0]), increasing on ([0, \infty))

Domain and Range

• Domain: set of all x-values for which f is defined
• Range: set of all possible y-values produced by f
• For continuous functions (e.g., quadratic):
  • If a > 0 (parabola opens up): Domain $(-\infty, \infty)$; Range $[k, \infty)$ where k is the y-coordinate of the vertex
  • If a < 0 (parabola opens down): Domain $(-\infty, \infty)$; Range $(-\infty, k]$
• Example:
  • Quadratic $y = x^2$: Domain $(-\infty, \infty)$, Range $[0, \infty)$
• Discontinuous or restricted functions may have domains like $(-3, 5]$ etc.; ranges adjust accordingly

Positive / Negative Intervals

• Positive Interval: where f(x) > 0 (graph above x-axis)
• Negative Interval: where f(x) < 0 (graph below x-axis)
• Determine using sign analysis or zeros of f
• Example: for $f(x) = x^2 - 9$
  • X-intercepts at $x = -3, 3$
  • Positive on $(-\infty, -3) \cup (3, \infty)$
  • Negative on $(-3, 3)$

Intercepts (X and Y)

• X-intercepts: points where $f(x) = 0$ (graph crosses or touches x-axis)\
• Y-intercept: point where x = 0, i.e., $y = f(0)$; coordinate is ((0, f(0)))
• Algebraic identification:
  • Example: for $f(x) = 2x - 2$
  • X-intercept: set $0 = 2x - 2\Rightarrow x = 1$ → $(1,0)$
  • Y-intercept: $f(0) = -2$ → $(0,-2)$
• Graphical identification aligns with algebraic computation

Set Builder Notation vs Interval Notation

• Set Builder Notation: describes the set of elements with a property, e.g., {x \mid x > 10}
• Interval Notation: describes a continuous range of real numbers, e.g., $(10, \infty)$ or $(-\infty, 5]$
• Use:
  • Domain and Range of continuous functions are often given in interval notation
  • Set-builder is useful for discrete sets or conditions

Transformation Rules for Functions

• Vertical translation up by k units: $f(x) \rightarrow f(x) + k$
• Vertical translation down by k units: $f(x) \rightarrow f(x) - k$
• Horizontal translation left by h units: $f(x) \rightarrow f(x + h)$
• Horizontal translation right by h units: $f(x) \rightarrow f(x - h)$
• Reflection over x-axis: $f(x) \rightarrow -f(x)$
• Reflection over y-axis: $f(x) \rightarrow f(-x)$
• Vertical stretch for |a| > 1: $f(x) \rightarrow a\,f(x)$
• Vertical compression for 0 < |a| < 1: $f(x) \rightarrow a\,f(x)$
• Horizontal compression for |b| > 1: $f(bx)$ with |b| > 1
• Horizontal stretch for 0 < |b| < 1: $f(bx)$ with 0 < |b| < 1

Key Features of Functions

• Intercepts:
  • X-intercepts: points where the graph crosses the x-axis; solve $f(x) = 0$
  • Y-intercept: point where the graph crosses the y-axis; evaluate $f(0)$
• Intercept identification:
  • Graphical: read coordinates where graph intersects axes
  • Algebraic: set y = 0 for x-intercepts; set x = 0 for y-intercepts
• Intercepts examples (types):
  • X-intercepts: e.g., $(-3,0), (5,0)$
  • Y-intercept: e.g., $(0, 4)$

Practice concepts (brief recap)

• Domain: x-values for which function is defined
• Range: y-values produced by the function
• Monotonicity: intervals where function is increasing or decreasing
• Intercepts: locations where graph crosses axes
• Transformations: shifting, reflecting, and scaling of graphs
• Notation: Set Builder vs Interval Notation usage depending on context