SAT Math - Advanced Concepts & Edge Cases

Front

Nonlinear System with Circle and Line: How many solutions exist when a line intersects a circle?

Back

Three possibilities: 0 solutions (line misses circle entirely), 1 solution (line is tangent to circle), or 2 solutions (line is a secant passing through the circle). To determine which, substitute the linear equation into the circle equation and examine the discriminant of the resulting quadratic.

Front

Horizontal Transformations: In y = f(bx), what is the effect of 'b' on the graph?

Back

The graph is horizontally compressed by a factor of 1/b (not b). For y = f(2x), every x-coordinate is halved, compressing the graph toward the y-axis. This is counterintuitive: multiplication inside the function argument compresses, while division stretches. Common trap: confusing horizontal compression factor with the coefficient itself.

Front

Vertex Form Analysis: What does the coefficient 'a' in y = a(x - h)^2 + k reveal about the parabola?

Back

'a' determines three properties: (1) direction—opens up if a > 0, down if a < 0; (2) vertical stretch/compression—larger |a| means narrower parabola; (3) rate of change—the parabola is wider than y = x^2 when |a| < 1. Vertex is always at (h, k) regardless of 'a' value.

Front

Rational Function Hole vs. Asymptote: How do you distinguish a removable discontinuity from a vertical asymptote?

Back

Factor both numerator and denominator. If a factor cancels completely, the zero of that factor creates a hole (removable discontinuity). If a factor remains in the denominator after simplification, its zero creates a vertical asymptote. Holes occur when the same linear factor appears in both numerator and denominator.

Front

Exponential vs. Linear Growth: How can you distinguish them from a table of values?

Back

Check differences vs. ratios. Linear: constant first differences (y-values increase by same amount). Exponential: constant ratios (y-values multiply by same factor). For example, (1,2), (2,4), (3,8), (4,16) has ratios of 2, indicating y = 2^x. SAT often mixes these in data interpretation questions.