What are Radicals and Rationals?
- Radical expressions contain roots: \(\sqrt{x}\) (square root), \(\sqrt[3]{x}\) (cube root), and n-th roots.
- Even roots (\(\sqrt{\phantom{x}}\), 4th root, ...) require the expression inside the root \(\geq 0\) for real values.
- Odd roots (cube root, 5th root, ...) accept all real values inside the root.
- Rational functions are quotients of polynomials: R(x) = P(x)/Q(x).
- Domain excludes \(x\) where \(Q(x) = 0\) (non-permissible values), which cause vertical asymptotes or holes.
Graphing Radical Functions
- y = \(\sqrt{x}\): Starts at \((0,0)\), increases gently to the right. Range \([0, ∞)\).
- y = \(\sqrt[3]{x}\): Passes through origin, increases for all x. Domain and range: all reals.
- Identify domain (expression inside the root \(\geq 0\) for even roots). Find left-most endpoint if any.
- Make a table of easy \(x\)-values (e.g., \(0\),\(1\),\(4\),\(9\) for sqrt) and compute \(y\)-values.
- Plot points and connect smoothly; keep correct end-behaviour.
- Label domain and range clearly on the graph.
Graphing Rational Functions
- Factor numerator and denominator completely. Identify domain and non-permissible values.
- Simplify if possible to reveal holes (cancelled factors). Mark hole coordinates exactly.
- Find vertical asymptotes: remaining zeros of denominator after cancellation.
- Determine horizontal/slant asymptote using degree rules and polynomial division if needed.
- Find \(x\)-intercepts (zeros of numerator after cancellation) and y-intercept (evaluate \(R(0)\) if permissible).
- Do sign analysis or use test points in each interval between vertical asymptotes and holes.
- Sketch using asymptotes as guides; plot a few additional points to ensure correct curvature. Label holes and asymptotes.
Transformations: Master Form & Mapping
- Use y = a f(b(x + c)) + d for all transformations (\(f =\) parent: sqrt, cuberoot, \(1/x\), etc).
- Mapping: If \((x, y)\) is on \(y = f(x)\), then on transformed graph:
$$(x_{new}, y_{new}) = (x / b - c,\ a y + d)$$ - a: vertical stretch/compression/reflection; b: horizontal stretch/compression/reflection; c: horizontal shift; d: vertical shift.
- For radicals: domain for \(y = a \sqrt{b(x + c)}\) + d is b(x + c) \(\geq 0\) (solve for \(x\), flip if \(b < 0\)).
- For rationals: domain excludes zeros of denominator; holes if factors cancel.
Describing Transformations in Words
- "Vertical stretch by 2, up 3" → \(y = 2 f(x) + 3\)
- "Compressed horizontally by 1/3, left 4" → \(y = f(3(x + 4))\)
- "Reflected in x-axis, right 2, up 1" → \(y = -f(x - 2) + 1\)
- For radicals: "Standard sqrt curve shifted right \(h\), up \(k\), stretched by \(a\)" → \(y = a \sqrt{x - h} + k\) (watch sign inside!)
Finding Equations from Domain & Range
- Identify parent (sqrt or cuberoot) from shape and domain.
- Translate domain/endpoint to find c (horizontal shift).
- Translate range starting value to find d (vertical shift).
- Use known points to solve for a and b if needed.
- Check domain requirement for even roots: b(x + c) \(\geq 0\).
Example: Domain \([2, \infty)\), range \([3, \infty)\), shape of \(\sqrt{x}\):
Candidate: \(y = \sqrt{x - 2} + 3\). If steeper, \(y = 2\sqrt{x - 2} + 3\). Verify with a point.
Candidate: \(y = \sqrt{x - 2} + 3\). If steeper, \(y = 2\sqrt{x - 2} + 3\). Verify with a point.
Making \(y = \sqrt{f(x)}\) from \(f(x)\)
- Domain: only where \(f(x) \geq 0\).
- For x where \(f(x) = 0\), y = 0. For x where \(f(x) > 0\), \(y = \sqrt{f(x)}\).
- Remove parts where \(f(x) < 0\). Plot transformed points and connect smoothly.
- Range: [0, \(\sqrt{\max(f(x))}\)] if \(f(x) \geq 0\).
- Invariant points: where \(f(x) = 0\) or \(f(x) = 1\) (since \(\sqrt{0} = 0\), \(\sqrt{1} = 1\)).
Solving Radical Equations
- Isolate the radical.
- Raise both sides to the necessary power.
- Solve the resulting equation.
- Check all solutions in the original (discard extraneous roots).
Example: Solve \(\sqrt{x + 3} = x - 1\).
Domain: \(x - 1 \geq 0 \implies x \geq 1\).
Square: \(x + 3 = (x - 1)^2 = x^2 - 2x + 1\).
Rearrange: \(0 = x^2 - 3x - 2\). Solve: \(x = \frac{3 \pm \sqrt{17}}{2}\).
Check which satisfy original and domain.
Domain: \(x - 1 \geq 0 \implies x \geq 1\).
Square: \(x + 3 = (x - 1)^2 = x^2 - 2x + 1\).
Rearrange: \(0 = x^2 - 3x - 2\). Solve: \(x = \frac{3 \pm \sqrt{17}}{2}\).
Check which satisfy original and domain.
Rational Functions: Asymptotes, Holes, and Domain
- Non-permissible values: x where denominator = 0.
- Vertical asymptote: denominator zero, numerator nonzero (uncancelled factor).
- Hole: denominator and numerator zero, factor cancels. Find y by plugging x into simplified function.
- Horizontal asymptote: degree rules:
\(n < m\): \(y = 0\)
\(n = m\): \(y =\) leading coeffs
\(n = m + 1\): slant asymptote (use division)
\(n > m + 1\): no horizontal asymptote, use division for end behavior.