SAT Math – Advanced Math
Polynomial and Other Nonlinear Graphs
Understanding cubic functions, rational graphs, absolute value, and other complex curves
Beyond quadratics and exponentials, the SAT tests higher-degree polynomials, rational functions, absolute value graphs, radical functions, and circles. You'll analyze end behavior, identify asymptotes, count zeros and turning points, recognize transformations, and interpret features like discontinuities and domain restrictions.
Success requires understanding how degree affects graph shape, recognizing vertical and horizontal asymptotes in rational functions, interpreting absolute value V-shapes, analyzing radical function domains, and identifying key features from equations and graphs. These functions model diverse phenomena—cubic curves describe volume-surface area relationships, rational functions represent average costs and concentration dilutions, and circles model orbits and wave patterns.
Understanding Nonlinear Function Types
Polynomial Functions
Polynomials are sums of power terms: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)
Cubic (degree 3): Can have up to 3 zeros, 2 turning points
Quartic (degree 4): Can have up to 4 zeros, 3 turning points
End behavior: Determined by leading term and degree (odd/even)
Rational Functions
Rational functions are ratios of polynomials: \(f(x) = \frac{P(x)}{Q(x)}\)
Horizontal asymptotes: End behavior limits as \(x \to \pm\infty\)
Domain: All x except where denominator = 0
Discontinuities: Breaks or jumps in the graph
Absolute Value Functions
Absolute value creates V-shaped graphs: \(f(x) = a|x - h| + k\)
Opens up: If a > 0 (V shape)
Opens down: If a < 0 (∧ shape)
Slope: Left side has slope -a, right side has slope +a
Radical and Circle Equations
Square root and circle functions have restricted domains.
Domain restriction: Radicand must be ≥ 0
Circle: \((x-h)^2 + (y-k)^2 = r^2\) centered at (h, k)
Not a function: Circles fail vertical line test
Essential Properties and Patterns
Polynomial End Behavior
Even degree, positive leading coefficient:
Both ends rise: \(x \to -\infty\), \(y \to +\infty\); \(x \to +\infty\), \(y \to +\infty\)
Odd degree, positive leading coefficient:
Falls left, rises right: \(x \to -\infty\), \(y \to -\infty\); \(x \to +\infty\), \(y \to +\infty\)
Rational Function Asymptotes
Vertical asymptote: x = a where denominator = 0
Horizontal asymptote:
• If degree(numerator) < degree(denominator): y = 0
• If degree(numerator) = degree(denominator): y = ratio of leading coefficients
• If degree(numerator) > degree(denominator): no horizontal asymptote
Absolute Value Vertex Form
\(f(x) = a|x - h| + k\)
Vertex: (h, k)
Slope magnitude: |a|
Opens up if a > 0, down if a < 0
Circle Standard Form
\((x - h)^2 + (y - k)^2 = r^2\)
Center: (h, k)
Radius: r
All points are distance r from center
Common Pitfalls & Expert Tips
❌ Confusing zeros with turning points
Degree 3 polynomial can have UP TO 3 zeros and 2 turning points—not always that many! Some may be complex.
❌ Missing domain restrictions
Rational functions can't have denominators = 0. Square roots need radicand ≥ 0. Always check domain!
❌ Thinking asymptotes can't be crossed
Graphs can cross HORIZONTAL asymptotes (just not at infinity). Only vertical asymptotes are never crossed!
❌ Sign errors in absolute value vertex
\(f(x) = |x + 3|\) has vertex at x = -3, not +3. Form is \(|x - h|\), so \(|x - (-3)|\)!
✓ Expert Tip: Use end behavior to check degree
If both ends go same direction, even degree. Opposite directions = odd degree. Quick visual check!
✓ Expert Tip: Count turning points to bound degree
Polynomial with n turning points has degree at least n+1. If you see 2 turns, degree is at least 3!
✓ Expert Tip: Vertical asymptotes from factored form
For \(\frac{1}{(x-2)(x+3)}\), vertical asymptotes at x = 2 and x = -3. Read directly from factors!
Fully Worked SAT-Style Examples
What is the end behavior of \(f(x) = -2x^4 + 5x^2 - 3\)?
Solution:
Identify key features:
Degree: 4 (even)
Leading coefficient: -2 (negative)
Apply end behavior rules:
Even degree + negative coefficient
Both ends go in SAME direction (downward)
Answer: As \(x \to -\infty\), \(y \to -\infty\); As \(x \to +\infty\), \(y \to -\infty\)
Find the vertical and horizontal asymptotes of \(f(x) = \frac{2x + 3}{x - 4}\)
Solution:
Find vertical asymptote:
Set denominator = 0: \(x - 4 = 0\)
Vertical asymptote: \(x = 4\)
Find horizontal asymptote:
Degree of numerator = 1
Degree of denominator = 1
Same degree → ratio of leading coefficients
Horizontal asymptote: \(y = \frac{2}{1} = 2\)
Answer: Vertical asymptote x = 4; Horizontal asymptote y = 2
What is the vertex of \(f(x) = 3|x + 2| - 5\)?
Solution:
Recognize vertex form: \(a|x - h| + k\)
Rewrite: \(3|x - (-2)| + (-5)\)
Identify vertex:
\(h = -2\), \(k = -5\)
Vertex: (-2, -5)
Additional info:
Since a = 3 > 0, opens upward (V shape)
Vertex is minimum point at y = -5
Answer: (-2, -5)
Find the center and radius of the circle \((x - 3)^2 + (y + 1)^2 = 25\)
Solution:
Standard form: \((x - h)^2 + (y - k)^2 = r^2\)
Rewrite: \((x - 3)^2 + (y - (-1))^2 = 5^2\)
Identify components:
\(h = 3\), \(k = -1\)
Center: (3, -1)
\(r^2 = 25\) → \(r = 5\)
Answer: Center (3, -1); Radius 5
What is the domain of \(g(x) = \frac{x + 5}{x^2 - 9}\)?
Solution:
Find where denominator = 0:
\(x^2 - 9 = 0\)
\((x - 3)(x + 3) = 0\)
\(x = 3\) or \(x = -3\)
Domain:
All real numbers except x = 3 and x = -3
Answer: All real numbers except \(x = \pm 3\) or \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\)
How many real zeros does \(f(x) = x^3 - 4x\) have?
Solution:
Set function equal to zero:
\(x^3 - 4x = 0\)
Factor:
\(x(x^2 - 4) = 0\)
\(x(x - 2)(x + 2) = 0\)
Solutions:
\(x = 0\), \(x = 2\), \(x = -2\)
Three real zeros
Answer: 3 real zeros
What is the domain of \(h(x) = \sqrt{2x - 6}\)?
Solution:
Radicand must be ≥ 0:
\(2x - 6 \geq 0\)
\(2x \geq 6\)
\(x \geq 3\)
Answer: \(x \geq 3\) or \([3, \infty)\)
Function Type Quick Reference
Polynomial Clues
• Smooth, continuous curves
• No breaks or asymptotes
• Turning points = degree - 1 max
• All real number domain
Rational Clues
• Vertical asymptotes (breaks)
• Horizontal/oblique asymptotes
• Fraction with polynomials
• Restricted domain
Beyond Quadratics: Exploring Complex Function Behavior
Higher-degree polynomials, rational functions, absolute value graphs, and other nonlinear curves extend mathematical modeling far beyond parabolas and exponentials, capturing relationships too complex for simple functions. The SAT tests these advanced functions because they represent mathematical sophistication essential for calculus, engineering, economics, and sciences—cubic polynomials model volume relationships where surface area and dimensions interact nonlinearly, quartic functions describe certain physics phenomena, and rational functions represent average costs where fixed expenses distribute across variable production. Understanding that degree determines maximum turning points and zeros (degree n allows up to n zeros and n-1 turns) connects algebra to geometry, explaining why cubic graphs can change direction twice while quadratics turn only once. End behavior depends fundamentally on degree parity: even-degree polynomials have ends moving together (both up or both down), while odd-degree polynomials have opposite-direction ends, patterns visible immediately from graphs. Rational functions introduce discontinuities through vertical asymptotes where denominators vanish, modeling real-world breaks like division by zero in rate calculations or undefined values in physical laws, while horizontal asymptotes reveal long-term behavior limits. Absolute value functions create sharp V-shaped turns representing situations where direction matters—distance from origin, error magnitude, or tolerance bounds—while radical functions model constrained growth with domain restrictions reflecting physical impossibilities like negative area or imaginary lengths. Circle equations represent all points equidistant from centers, fundamental to orbits, waves, and rotational symmetry. Master the analytical skills for extracting information: factoring denominators reveals vertical asymptotes instantly, comparing numerator and denominator degrees determines horizontal asymptotes systematically, and recognizing vertex form in absolute value or completing squares in circles provides immediate geometric insights. These functions transcend test preparation, equipping you to analyze pharmaceutical concentration decay, optimize manufacturing costs with rational profit functions, model constrained biological growth, understand resonance frequencies involving higher polynomials, and recognize when simple linear or quadratic models fail to capture essential complexity—mathematical maturity that distinguishes quantitative literacy from formula memorization.