Maths with Lemon

Key Points

• A rational function has the form \[ f(x)=\frac{p(x)}{q(x)} \] where \(p(x)\) and \(q(x)\) are polynomials.
• The denominator cannot equal zero.
• Rational functions often have asymptotes.

Key Points

• Vertical asymptotes occur when the denominator equals zero.
• Example: \[ f(x)=\frac{1}{x-3} \] Vertical asymptote: \[ x=3 \]
• For \[ f(x)=\frac{a}{x-h}+k \] the horizontal asymptote is \[ y=k \]

Key Points

• The parent function is \[ y=\frac1x \]
• General form: \[ y=\frac{a}{x-h}+k \]
• \(h\) shifts the graph horizontally.
• \(k\) shifts the graph vertically.
• \(a\) controls reflection and stretch.

Key Points

• Start with: \[ y=\frac{a}{x-h}+k \]
• Determine \(h\) from the vertical asymptote.
• Determine \(k\) from the horizontal asymptote.
• Substitute a known point to find \(a\).

Key Points

• Parent function: \[ y=\frac1x \]
• General form: \[ y=\frac{a}{x-h}+k \]
• Vertical asymptote: \[ x=h \]
• Horizontal asymptote: \[ y=k \]
• Domain: \[ x\neq h \]
• Range: \[ y\neq k \]