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Graphing Techniques
A-Level H2 Math
Conics
$\text{Circle: } (x-a)^2 + (y-b)^2 = r^2$
$\text{Ellipse: } \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$
$\text{Hyperbola: } \dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$
$\text{Oblique asymptotes for hyperbolas: } y - k = \pm\dfrac{b}{a}(x-h)$
Rational Graphs
FormVertical AsymptoteHorizontal / Oblique
$y = \dfrac{ax+b}{cx+d}$$x = -\dfrac{d}{c}$Horizontal: $y = \dfrac{a}{c}$
$y = \dfrac{ax^2+bx+c}{dx+e}$$x = -\dfrac{e}{d}$Oblique: $y = \dfrac{a}{d}x + f + \dfrac{\text{remainder}}{dx+e}$
(perform long division)
Translation
$y \to y + a$
Translate in negative $y$-direction by $a$ units
$y \to y - a$
Translate in positive $y$-direction by $a$ units
$x \to x + a$
Translate in negative $x$-direction by $a$ units
$x \to x - a$
Translate in positive $x$-direction by $a$ units
Scaling
$y \to ay$
Scale along $y$-axis by factor $\dfrac{1}{a}$
$y \to \dfrac{1}{a}y$
Scale along $y$-axis by factor $a$
$x \to ax$
Scale along $x$-axis by factor $\dfrac{1}{a}$
$x \to \dfrac{1}{a}x$
Scale along $x$-axis by factor $a$
Reflection
$y \to -y$
Reflect about $x$-axis
$x \to -x$
Reflect about $y$-axis
Special Transformations
$y = |f(x)|$
  • Reflect bottom part up (below $x$-axis → above)
$y = f(|x|)$
  • 1. Abandon left half
  • 2. Reflect right half to left
$y = f'(x)$
  • 1. Vertical asymptote remains
  • 2. $x$-coordinates of turning points become $x$-intercepts
  • 3. Trace the gradient
$y = \dfrac{1}{f(x)}$
  • 1. $x$-intercept → asymptote
  • 2. Asymptote → $x$-intercept
  • 3. Max → min, min → max
  • 4. All $y$ values become $\dfrac{1}{y}$
  • 5. Observe behavior near asymptote

Formulas for the Singapore A-Level H2 Mathematics syllabus