H2 Maths Notes (JC 1-2): 1.2) Graphs and Transformations

Before you revise\ Keep a doodle pad next to your GC. Sketch every curve by hand even if technology can plot it—the mark scheme rewards annotations and reasoning, not just the final picture.

Core Sketching Strategy

1. Intercepts: Solve \( f(x) = 0 \) for x-intercepts; evaluate \( f(0) \) for the y-intercept.
2. Asymptotes: Inspect denominators for vertical asymptotes and examine end behaviour (long division) for horizontal or oblique asymptotes.
3. Stationary Points: Solve \( f'(x) = 0 \), determine nature via \( f''(x) \) or sign chart.
4. Symmetry: Test \( f(-x) = f(x) \) (even) or \( f(-x) = -f(x) \) (odd). For rational graphs, look for rotational symmetry.
5. Sign Diagram: Tabulate sign changes across critical points to understand where the graph lies above or below the x-axis.

Common Function Families

• Polynomials: End behaviour determined by degree and leading coefficient; turning points up to \( n - 1 \) for degree \( n \).
• Rational functions: Simplify to remove removable discontinuities; locate vertical, horizontal, and oblique asymptotes.
• Exponentials and logarithms: \( y = a^x \) always positive; \( y = \ln x \) defined for \( x > 0 \).
• Trigonometric graphs: Document period, amplitude, phase shift, and vertical displacement before sketching.

Example -- Rational sketch

Sketch \( y = \frac{2x - 3}{x + 1} \).

1. Vertical asymptote at \( x = -1 \); horizontal asymptote from division: \( y = 2 - \frac{5}{x + 1} \) so \( y = 2 \).
2. Intercepts: x-intercept at \( x = \tfrac{3}{2} \), y-intercept \( y = -3 \).
3. Sketch the hyperbola considering symmetry about the intersection of asymptotes.

Transformation Toolkit

Apply transformations in reverse order of operations \( inside-out for \times changes, outside-in for y changes \).

Example -- Transformation chain

Given \( y = f(x) \), sketch \( y = -3 f\bigl(2(x - 1)\bigr) + 2 \).

1. Shift right by 1 (replace \( x \) with \( x - 1 \)).
2. Compress horizontally by factor 2 (replace \( x \) with \( 2x \)).
3. Stretch vertically by 3 and reflect in x-axis (multiply by \( -3 \)).
4. Shift up by 2.

Document each step and track how intercepts/asymptotes move.

Parametric and Polar Sketches

• When \( x = g(t) \) and \( y = h(t) \), produce a parameter table and arrow the direction of increasing \( t \).
• To convert to Cartesian form, eliminate \( t \) algebraically or use trigonometric identities.
• For polar coordinates \( (r, \theta) \), remember \( x = r \cos\theta \), \( y = r \sin\theta \). Sketch over one full period.

Example -- Parametric curve

For \( x = 2 \cos t \), \( y = 3 \sin t \), eliminating \( t \) gives \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), an ellipse centred at the origin.

Graphs of Inverses and Moduli

• Plot the line \( y = x \) to reflect the original graph when sketching \( y = f^{-1}(x) \).
• For \( y = \lvert f(x) \rvert \), copy the portions above the axis and reflect negative sections.
• For \( y = f(\lvert x \rvert) \), ignore the left-half of the original and mirror the right-half.

Example -- Modulus manipulation

Sketch \( y = \lvert 2x - 5 \rvert - 3 \).

1. Sketch \( y = 2x - 5 \) first.
2. Reflect the negative portion above the axis to get \( y = \lvert 2x - 5 \rvert \).
3. Translate down by 3.
4. Locate vertices and intercepts precisely.

Calculator Support

• GC TRACE verifies intercepts and intersections quickly—record the coordinates used.
• dy/dx feature confirms the location of stationary points.
• For parametric plots, use GC parametric mode and state the viewing window chosen.

Exam Watch Points

• Always annotate asymptotes with equations and show arrows approaching them.
• Use exact values for key points (fractions, radicals) rather than decimals when possible.
• For modulus and reciprocal graphs, indicate excluded points clearly.
• Double-check final sketches against derivative sign charts to avoid contradicting monotonicity.

Quick Revision Checklist

• [ ] Execute full sketching workflow: intercepts, asymptotes, stationary points, symmetry.
• [ ] Apply transformation chains accurately with correct order and orientation.
• [ ] Translate between parametric/polar and Cartesian forms fluently.
• [ ] Explain graph features (turning points, asymptotes) in written solutions, not just diagrams.

Next steps: Dive into 1.3 to master analytic and graphical techniques for equations and inequalities.