Unit 3: Radicals & Rationals

Radical Functions

• Square root: \(y = \sqrt{x}\)   |   Domain: \(x \geq 0\), Range: \(y \geq 0\)
• Cube root: \(y = \sqrt[3]{x}\)   |   Domain: all real, Range: all real

• \(y = a \sqrt{b(x + c)} + d\) or \(y = a \sqrt[3]{b(x + c)} + d\)
• a: vertical stretch/compression, reflect over \(x\)-axis if negative
• b: horizontal stretch/compression, reflect over \(y\)-axis if negative
• c: horizontal shift, left if \(+c\), right if \(-c\)
• d: vertical shift, up if \(+d\), down if \(-d\)

Square Root of a Function

• \(y = \sqrt{f(x)}\)
• Domain: x such that \(f(x) \geq 0\)
• Range: \(\sqrt{\text{y-values of } f(x)}\) where y \(\geq 0\)
• Invariant points: solve \(f(x) = \sqrt{f(x)}\) \(\to\) \(f(x) = 0\) or \(f(x) = 1\)

Radical Equations

1. Isolate the radical
2. Raise both sides to the appropriate power
3. Solve the resulting equation
4. Check all solutions in the original, discard extraneous

• Plot \(y = \sqrt{f(x)}\) and the other side, find \(x\)-intercepts

Rational Functions

• \(R(x) = \frac{P(x)}{Q(x)}\), P and Q polynomials
• Domain: x such that \(Q(x) \neq 0\)

• \(x\)-values making denominator \(0\)
• Causes vertical asymptotes or holes
• Vertical asymptote if factor in denominator not cancelled
• Hole if factor cancels with numerator, coordinate from simplified function

• Compare degrees: numerator \(n\), denominator \(m\)
• \(n < m\): HA \(y = 0\)
• \(n = m\): HA \(y =\) leading coeff of \(P\) / leading coeff of \(Q\)
• \(n = m + 1\): slant asymptote (use division)
• \(n > m + 1\): no HA, end behavior follows polynomial division

Transformations of Rational Functions

• \(y = a \frac{P(b(x + c))}{Q(b(x + c))} + d\)

1. Factor numerator and denominator completely
2. Identify domain and non-permissible values
3. Cancel common factors to reveal holes
4. Find vertical asymptotes from remaining denominator zeros
5. Determine horizontal or slant asymptotes
6. Find x-intercepts (zeros of numerator) and y-intercept (evaluate at x=0 if allowed)
7. Plot key points including invariant points, sketch curve approaching asymptotes

Common Patterns

• \(\sqrt{x}\): Domain \(x \geq 0\), Range \(y \geq 0\), points \((0,0)\), \((1,1)\), \((4,2)\)
• \(\sqrt[3]{x}\): Domain all real, Range all real, points \((0,0)\), \((1,1)\), \((−1,−1)\)
• \(\frac{1}{x}\): Domain \(x \neq 0\), Range \(y \neq 0\), points \((1,1)\), \((−1,−1)\), VA \(x=0\), HA \(y=0\)
• \(a\sqrt{x-c}+d\): Domain \(x \geq c\), Range \(y \geq d\), end point \((c,d)\)
• \(\frac{P(x)}{Q(x)}\): Domain \(x\) ≠ non-permissible, zeros from numerator, use asymptotes to guide sketch

Transformations Summary

• Vertical multiply \(a\) → stretch/compress, reflect \(x\)-axis if negative
• Horizontal multiply \(b\) → stretch/compress, reflect \(y\)-axis if negative
• Horizontal shift \(c\) → left \(+c\), right \(-c\)
• Vertical shift \(d\) → up \(+d\), down \(-d\)
• Map \(2–3\) key points using \((x / b - c, a y + d)\) to check