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
- 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