FREE SAT Interpreting Nonlinear Functions Questions and Answers 3

Question 1

The function f is a rational function defined by f(x) = (x^2 - 4) / (x - 2). The graph of y = f(x) in the xy-plane has a point of discontinuity (a hole). What are the coordinates of this hole?

Accepted Answer

(2, 4)

Answer

To find the hole, factor the numerator: f(x) = ((x - 2)(x + 2)) / (x - 2). The term (x - 2) cancels out, which indicates a hole at x = 2. To find the y-coordinate, substitute x = 2 into the simplified function, which is f(x) = x + 2. So, y = 2 + 2 = 4. Therefore, the hole is at (2, 4).

Question 2

The graph of a nonlinear piecewise function is shown, consisting of a parabola and a square root function. Which of the following defines the function for x ≥ 0?

Accepted Answer

f(x) = { -x^2 + 4, if 0 ≤ x < 2; √(x-2), if x ≥ 2 }

Answer

For the interval 0 ≤ x < 2, the graph is a downward-opening parabola with a y-intercept at (0, 4) and a root at x=2. The equation is y = -x^2 + 4. For x ≥ 2, the graph is a square root function starting at (2, 0) and passing through (6, 2). The equation is y = √(x-2). Combining these gives the correct piecewise definition.

Question 3

The function g(x) = a(x - h)^2 + k represents a parabola. The graph of y = g(x) is a parabola that opens downward and has a vertex at (-3, 5). Which of the following MUST be true?

Accepted Answer

a < 0 and k = 5

Answer

In the vertex form of a parabola, y = a(x - h)^2 + k, the vertex is at (h, k). The sign of 'a' determines the direction the parabola opens. Since the parabola opens downward, 'a' must be negative (a < 0). The vertex is given as (-3, 5), so h = -3 and k = 5. Therefore, the statement that must be true is a < 0 and k = 5.

Question 4

A scientist models the decay of a radioactive substance with the function N(t) = N₀e^(-λt), where N(t) is the amount of the substance at time t, N₀ is the initial amount, and λ is the decay constant. If the half-life of the substance is 100 years, which equation correctly represents the decay constant λ?

Accepted Answer

λ = ln(2) / 100

Answer

The half-life is the time it takes for the substance to decay to half its initial amount. So, N(100) = N₀/2. We set up the equation: N₀/2 = N₀e^(-λ*100). Dividing by N₀ gives 1/2 = e^(-100λ). Taking the natural logarithm of both sides gives ln(1/2) = -100λ. Since ln(1/2) = -ln(2), we have -ln(2) = -100λ. Solving for λ gives λ = ln(2) / 100.

Question 5

The graph of y = f(x) is shown, which has a horizontal asymptote at y = 2 and a vertical asymptote at x = -1. Which of the following could be the equation for f(x)?

Accepted Answer

f(x) = (2x - 3) / (x + 1)

Answer

A vertical asymptote occurs where the denominator is zero. For the vertical asymptote to be at x = -1, the denominator must be (x + 1). A horizontal asymptote for a rational function where the degree of the numerator and denominator are equal is the ratio of the leading coefficients. For the horizontal asymptote to be at y = 2, the ratio of the leading coefficients must be 2. The function f(x) = (2x - 3) / (x + 1) meets both criteria.

Bluebook SAT Practice Test Practice Test

Bluebook SAT Practice Test Practice Test

FREE SAT Interpreting Nonlinear Functions Questions and Answers 3