Overview
This lecture covers major function types—parabola, circle, ellipse, hyperbola, absolute value, square root, reciprocal, and greatest integer—by analyzing their standard equations, describing vertical/horizontal shifts, and outlining graphing techniques and common features.
The Absolute Value Function
- The function f(x) = |x| forms a V-shaped graph symmetric about the y-axis.
- The domain is all real numbers; the range is y ≥ 0.
- A graph of f(x) = |x - h| + k shifts horizontally by h and vertically by k.
- Horizontal shifts occur inside the function, vertical shifts occur outside.
The Square Root Function
- The function f(x) = √x has domain x ≥ 0 and range y ≥ 0.
- The graph starts at (0,0) and curves upward/right.
- Shifting follows the same pattern: f(x) = √(x - h) + k moves right by h, up by k.
The Reciprocal Function
- The function f(x) = 1/x has domain x ≠0 and range y ≠0.
- The graph consists of two branches in opposite quadrants, approaching both axes but never touching (asymptotes).
- Shifts: f(x) = 1/(x - h) + k shifts right by h and up by k.
The Greatest Integer Function (Step Function)
- The function f(x) = ⎣x⎦ (or [x], or int(x)) outputs the largest integer ≤ x.
- The graph is a series of horizontal steps (step function).
- Domain is all real numbers; range is all integers.
- f(x) = ⎣x - h⎦ + k shifts horizontally by h, vertically by k.
Parabola, Circle, Ellipse, and Hyperbola
- A parabola: f(x) = x², U-shaped, vertex at (0,0), shifts and stretches as f(x) = a(x - h)² + k.
- A circle: (x - h)² + (y - k)² = r²; center (h,k), radius r.
- An ellipse: (x - h)²/a² + (y - k)²/b² = 1; stretches more in the direction of the larger denominator.
- A hyperbola: (x - h)²/a² – (y - k)²/b² = 1 (horizontal) or vice versa (vertical), with asymptotes guiding the branches.
Shift Rules Summary
- Inside the function (e.g., x - h): horizontal shift right by h (if h > 0), left by |h| (if h < 0).
- Outside the function (e.g., +k): vertical shift up by k (if k > 0), down by |k| (if k < 0).
Key Terms & Definitions
- Domain — Set of allowable input (x) values for a function.
- Range — Set of output (y) values a function can take.
- Vertex — The highest/lowest point of a parabola or V-shape.
- Asymptote — A line the graph approaches but never touches.
- Step Function — Function with piecewise constant values, graph is a series of steps.
Action Items / Next Steps
- Practice sketching graphs of all function types using shifts.
- Understand which changes to an equation cause vertical or horizontal shifts.
- For each function, identify domain, range, intercepts, and symmetry.