SAT Math – Advanced Math
Nonlinear Functions
Analyzing quadratics, exponentials, and other curved relationships
Nonlinear functions are functions whose graphs are not straight lines—they curve, bend, or change direction. On the SAT, you'll analyze quadratic parabolas, exponential growth/decay, absolute value V-shapes, and other nonlinear relationships, identifying key features like vertices, axes of symmetry, zeros, and transformations.
Success requires understanding function transformations, recognizing standard forms, finding maximum and minimum values, analyzing rates of change, and connecting algebraic equations to graphical representations. These functions aren't just mathematical abstractions—they model projectile motion, population growth, radioactive decay, profit optimization, and countless real-world phenomena where relationships involve acceleration, curvature, or exponential change.
Understanding Nonlinear Functions
Quadratic Functions (Parabolas)
Quadratic functions have degree 2 and create U-shaped or inverted U-shaped graphs.
Vertex form: \(f(x) = a(x - h)^2 + k\)
Vertex: Point (h, k) where parabola turns
Axis of symmetry: \(x = h\) (vertical line through vertex)
Opens: Up if \(a > 0\), down if \(a < 0\)
Exponential Functions
Exponential functions involve variable exponents and show rapid growth or decay.
Growth: \(b > 1\) (increases rapidly)
Decay: \(0 < b < 1\) (decreases toward zero)
Horizontal asymptote: y = 0 (never touches x-axis)
Applications: Population, compound interest, radioactive decay
Function Transformations
Transformations shift, stretch, compress, or reflect functions.
Horizontal shift: \(f(x - h)\) moves right h units
Vertical stretch: \(a \cdot f(x)\) where \(|a| > 1\)
Vertical compression: \(a \cdot f(x)\) where \(0 < |a| < 1\)
Reflection: \(-f(x)\) reflects over x-axis
Key Features of Nonlinear Functions
Important characteristics to identify and analyze.
Vertex/turning point: Maximum or minimum value
Domain: All possible x-values
Range: All possible y-values
Increasing/decreasing: Where function rises or falls
Essential Formulas and Properties
Quadratic Vertex Formula
For \(f(x) = ax^2 + bx + c\):
Vertex x-coordinate: \(x = -\frac{b}{2a}\)
Then substitute to find y-coordinate: \(y = f\left(-\frac{b}{2a}\right)\)
Quadratic Formula (Finding Zeros)
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Discriminant: \(b^2 - 4ac\)
• If positive: two real solutions
• If zero: one real solution
• If negative: no real solutions
Exponential Growth/Decay
\(f(x) = a(1 + r)^x\) (growth if r > 0)
\(f(x) = a(1 - r)^x\) (decay if 0 < r < 1)
a = initial value, r = rate of change
Vertex Form of Quadratic
\(f(x) = a(x - h)^2 + k\)
Vertex is at point (h, k)
Maximum if \(a < 0\), minimum if \(a > 0\)
Axis of symmetry: \(x = h\)
Common Pitfalls & Expert Tips
❌ Confusing vertex formula signs
Vertex form is \(a(x - h)^2 + k\), so if you see \((x + 3)^2\), the h-value is -3, not +3! Opposite sign inside parentheses.
❌ Mixing up maximum and minimum
If \(a > 0\), parabola opens UP → vertex is MINIMUM. If \(a < 0\), opens DOWN → vertex is MAXIMUM. Check the sign of a!
❌ Forgetting exponential never reaches zero
Exponential functions approach but never touch the x-axis. The range is \(y > 0\) (for standard form), not all real numbers!
❌ Transformation direction errors
\(f(x - 3)\) shifts RIGHT 3 units, not left. Inside transformations do the opposite of what you expect!
✓ Expert Tip: Use vertex form for transformations
Converting to vertex form \(a(x - h)^2 + k\) immediately reveals the vertex location and direction. Complete the square if needed!
✓ Expert Tip: Check end behavior
For large |x| values, how does the function behave? Parabolas go to infinity, exponentials grow/decay, understanding limits helps.
✓ Expert Tip: Use symmetry
Parabolas are symmetric about their axis. If you know one point, you know its mirror image across the axis of symmetry!
Fully Worked SAT-Style Examples
Find the vertex of \(f(x) = 2x^2 - 8x + 3\)
Solution:
Step 1: Identify a, b, c
\(a = 2\), \(b = -8\), \(c = 3\)
Step 2: Find x-coordinate of vertex
\(x = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2\)
Step 3: Find y-coordinate by substituting
\(f(2) = 2(2)^2 - 8(2) + 3\)
\(= 2(4) - 16 + 3 = 8 - 16 + 3 = -5\)
Interpretation:
Since \(a = 2 > 0\), parabola opens upward
Vertex (2, -5) is the minimum point
Answer: Vertex is (2, -5)
The function \(f(x) = -3(x + 4)^2 + 7\) has a maximum value at what point?
Solution:
Recognize vertex form:
\(f(x) = a(x - h)^2 + k\)
Vertex is at (h, k)
Rewrite to match form:
\(f(x) = -3(x - (-4))^2 + 7\)
So \(h = -4\), \(k = 7\)
Determine maximum or minimum:
\(a = -3 < 0\) → parabola opens downward
Vertex is a MAXIMUM
Answer: Maximum at point (-4, 7)
Find the zeros of \(f(x) = x^2 + 3x - 10\)
Solution:
Method 1: Factoring
Find factors of -10 that add to 3
5 and -2 work: \(5 \times (-2) = -10\), \(5 + (-2) = 3\)
\((x + 5)(x - 2) = 0\)
\(x = -5\) or \(x = 2\)
Method 2: Quadratic Formula (verification)
\(a = 1\), \(b = 3\), \(c = -10\)
\(x = \frac{-3 \pm \sqrt{9 - 4(1)(-10)}}{2(1)}\)
\(= \frac{-3 \pm \sqrt{9 + 40}}{2} = \frac{-3 \pm \sqrt{49}}{2} = \frac{-3 \pm 7}{2}\)
\(x = \frac{-3 + 7}{2} = 2\) or \(x = \frac{-3 - 7}{2} = -5\)
Answer: Zeros are \(x = -5\) and \(x = 2\)
The graph of \(g(x) = (x - 3)^2 + 2\) is the graph of \(f(x) = x^2\) shifted in what way?
Solution:
Analyze horizontal shift:
\((x - 3)^2\) means replace x with (x - 3)
Shifts RIGHT 3 units
Analyze vertical shift:
+2 outside the squared term
Shifts UP 2 units
Transformation Summary:
Original vertex: (0, 0)
New vertex: (3, 2)
Shape unchanged (same width and direction)
Answer: Right 3 units and up 2 units
A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 9 hours?
Solution:
Set up exponential model:
Doubles every 3 hours → growth factor = 2
Number of 3-hour periods in 9 hours: \(\frac{9}{3} = 3\)
\(P(t) = 500 \cdot 2^{t/3}\)
Calculate for t = 9:
\(P(9) = 500 \cdot 2^{9/3} = 500 \cdot 2^3 = 500 \cdot 8 = 4{,}000\)
Answer: 4,000 bacteria
What is the range of \(f(x) = -2(x - 1)^2 + 5\)?
Solution:
Identify vertex:
Vertex form: \(a(x - h)^2 + k\)
Vertex: (1, 5)
Determine direction:
\(a = -2 < 0\) → parabola opens DOWNWARD
Vertex (1, 5) is the MAXIMUM point
Determine range:
Maximum y-value is 5
Parabola extends downward to negative infinity
Range: \(y \leq 5\) or \((-\infty, 5]\)
Answer: \(y \leq 5\) or \((-\infty, 5]\)
A parabola has x-intercepts at x = -2 and x = 6. What is its axis of symmetry?
Solution:
Use symmetry property:
Parabola is symmetric about vertical line through vertex
Axis of symmetry is midway between x-intercepts
Calculate midpoint:
\(x = \frac{-2 + 6}{2} = \frac{4}{2} = 2\)
Alternative Method:
If intercepts are at -2 and 6, factored form is \(a(x + 2)(x - 6)\)
Use \(x = -\frac{b}{2a}\) after expanding, or use symmetry
Answer: \(x = 2\)
Function A: \(f(x) = 100 + 50x\) (linear). Function B: \(g(x) = 100 \cdot 1.5^x\) (exponential). Which eventually grows faster?
Solution:
Test some values:
At \(x = 0\): \(f(0) = 100\), \(g(0) = 100\)
At \(x = 5\): \(f(5) = 350\), \(g(5) = 100 \cdot 7.59 \approx 759\)
At \(x = 10\): \(f(10) = 600\), \(g(10) = 100 \cdot 57.67 \approx 5{,}767\)
Key Insight:
Linear grows by constant amount (50 per step)
Exponential multiplies by constant (1.5 per step)
Exponential always overtakes linear eventually
Answer: Function B (exponential) eventually grows faster
Nonlinear Function Types
Quadratic
\(f(x) = ax^2 + bx + c\)
U-shaped parabola
Exponential
\(f(x) = a \cdot b^x\)
Rapid growth/decay
Absolute Value
\(f(x) = |x|\)
V-shaped
Square Root
\(f(x) = \sqrt{x}\)
Half parabola sideways
Nonlinear Functions: Modeling Curved Relationships
Nonlinear functions capture the complexity of real-world phenomena that linear models cannot—acceleration, deceleration, growth that compounds, paths that curve, and relationships that change direction. The SAT tests these functions because they're fundamental to every quantitative field: quadratic functions model projectile trajectories and profit optimization, exponential functions describe population growth and radioactive decay, and transformations reveal how functions shift and scale. Master the vertex form of quadratics for instant identification of maximum and minimum values—critical for optimization problems. Understand that exponential growth eventually dominates any polynomial, explaining why compound interest outpaces linear savings and why pandemics accelerate. Recognize transformations systematically: inside changes (x - h) affect horizontal position while outside changes (+k) affect vertical position, and the counterintuitive direction of inside transformations trips up countless students. These functions aren't just equations—they're the mathematical language of change itself, describing how quantities evolve when acceleration matters, when growth feeds on itself, or when constraints create boundaries. The fluency you develop with nonlinear functions—identifying vertices instantly, determining domains and ranges, applying transformations confidently—distinguishes students who merely calculate from those who truly understand mathematical relationships and their profound connections to the physical world.