[Video: Introduction to Nonlinear Graphs]
Learn about polynomial, absolute value, and other nonlinear graphs and their key features that appear on the SAT.
Polynomial Graphs
Basic Characteristics
General form: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
- Degree: Highest power of x (determines basic shape)
- Roots: x-intercepts (real zeros)
- End behavior: Determined by leading term
- Turning points: At most n-1 for degree n
[Graph of cubic polynomial]
Figure 1: Cubic polynomial (degree 3)
[Graph of quartic polynomial]
Figure 2: Quartic polynomial (degree 4)
Other Nonlinear Graphs
Absolute Value Graphs
V-shaped graphs with a vertex at (h,k):
f(x) = a|x - h| + k
- Vertex: (h, k)
- Slopes: ±a on either side
- Symmetrical about x = h
Square Root Graphs
Curved graphs starting at (h,k):
f(x) = a√(x - h) + k
- Starting point: (h, k)
- Domain: x ≥ h
- Increasing if a > 0, decreasing if a < 0
Piecewise Functions
Graphs composed of different functions over different intervals.
Example:
f(x) = { x² if x ≤ 1
{ 2x - 1 if x > 1
• Parabola left of x=1, line right of x=1
Key Features to Identify
Common SAT Focus Points
- Intercepts (x and y)
- End behavior (as x → ±∞)
- Symmetry
- Maximum/minimum points
- Points of discontinuity or sharp turns
- Intervals of increase/decrease
Key Takeaways
- Recognize polynomial graphs by their smooth, continuous curves
- Identify absolute value graphs by their V-shape
- Spot square root graphs by their starting point and gradual curve
- For piecewise functions, examine each interval separately
- On SAT, focus on interpreting key features in context