| Term | Definition / Description |
|---|---|
| Intercepts | Points where graph crosses axes. • x-intercepts: where y=0. • y-intercept: where x=0. |
| Vertex | Turning point of a curve (e.g., maximum or minimum point of a parabola). Found using symmetry or calculus. |
| Symmetry | Graph is symmetric about y-axis if f(−x) = f(x) (even function). Symmetric about origin if f(−x) = −f(x) (odd function). |
| Asymptotes | Lines the graph approaches but never touches:• Vertical: denominator = 0 (rational functions).
• Horizontal: determined by end behaviour (limits as x → ±∞). • Oblique: occurs if degree of numerator > degree of denominator by 1. |
| Zeros / Roots | x-values for which f(x)=0. Found algebraically or using GDC “intersect” or “zero” functions. |
| Intersection points | Where two graphs meet: coordinates satisfy both equations simultaneously. Found by solving f(x) = g(x) or using graphing technology. |
📌 Key features of graphs — systematic approach
- Intercepts: Substitute x=0 (for y-intercept) and y=0 (for x-intercepts).
- Symmetry: Test if f(−x) = f(x) or f(−x) = −f(x).
- Vertex: For y = ax2 + bx + c, vertex = (−b ÷ 2a, f(−b ÷ 2a)).
- Extrema: Local maximum/minimum identified via vertex (quadratic) or using graphing technology (for non-quadratic).
- Asymptotes: Check denominator = 0 (vertical), and limits as x → ±∞ (horizontal/oblique).
- End behaviour: Examine leading term of f(x) for general trend as x → ±∞.
🧮 GDC Tips
- Use Calc → zero to find intercepts where f(x)=0.
- Use Calc → minimum / maximum to locate extrema quickly.
- Use Calc → intersect to find intersection points of f and g.
- Set proper window (xmin/xmax, ymin/ymax) to ensure all key features visible.
- Use sliders to explore how changing parameters (e.g., a in y = ax2) affects shape and position
Example 1 — Quadratic function:
f(x) = x2 − 4x + 3 → Vertex: (2, −1); Axis of symmetry x = 2; x-intercepts: (1,0), (3,0); y-intercept: (0,3).
Sketch shows parabola opening upwards with minimum point at (2, −1).
Example 2 — Rational function:
f(x) = (2x + 1)/(x − 1) → Vertical asymptote at x = 1; Horizontal asymptote y = 2 (degrees equal → divide leading coefficients).
x-intercept: (−½, 0), y-intercept: (0, −1). Graph approaches y = 2 as x → ±∞.
Example 3 — Symmetry & extrema:
f(x) = cos x is an even function (symmetric about y-axis).
Maxima: (0,1), minima: (π, −1), periodic pattern.
Useful in modelling waves, sound, and seasonal data.
📌 Finding intersection points
- Analytically: Solve f(x) = g(x) → gives x-coordinate(s) of intersection; substitute back to find y.
- Using technology: Use “intersect” or “solve” functions on GDC to find intersection coordinates quickly.
- Interpreting intersections:
- In economics: market equilibrium → demand = supply.
- In physics: time when two objects are at same position.
- In geography: intersection of contour curves represents equal elevation points.
Example 4 — Intersection:
f(x) = 2x + 1, g(x) = x2 + 1.
Set equal: 2x + 1 = x2 + 1 ⇒ x2 − 2x = 0 ⇒ x(x − 2)=0 ⇒ x=0 or 2.
Intersection points: (0,1) and (2,5). Confirm visually with GDC “intersect” tool..
🧠 Examiner Tip
Label all key features clearly: intercepts, extrema, asymptotes, and intersections. When using GDC, copy approximate coordinates to 3 s.f. and indicate that they were obtained using technology. Avoid vague sketches—clarity and precision are key.
🔍 TOK Perspective
The Bourbaki group promoted abstraction and symbolic precision, while Mandelbrot emphasized visual patterns like fractals. How does visualization complement analytical methods? Does “seeing” patterns count as mathematical knowledge, or only algebraic proof?
📝 Paper tips
- For Paper 1: show analytical steps when finding intercepts, symmetry, or vertex.
- For Paper 2: use GDC effectively but annotate output with feature labels and reasoning.