Interpreting Nonlinear Functions Flashcards

The function f(x) = -2(x + 3)² + 8 is graphed in the xy-plane. Which of the following statements about f is true?

Answer: The function has a maximum value of 8 at x = -3.

f(x) = -2(x + 3)² + 8 is in vertex form f(x) = a(x - h)² + k, where the vertex is at (h, k). Here h = -3 and k = 8, so the vertex is (-3, 8). Because a = -2 < 0, the parabola opens downward, meaning the vertex is a maximum. Thus the maximum value is 8, occurring at x = -3.

The exponential function g is defined by g(x) = 80 · (0.5)^(x/12). What does the value 12 represent in the context of this function?

Answer: The number of units of x required for g to decrease to half its current value.

When x increases by 12, the exponent x/12 increases by 1, so g is multiplied by 0.5. This means every 12 units, the output is halved. In context, 12 is the half-life — the number of units of x needed for g to decrease to half its current value. It does not represent growth, an initial value, or a per-unit rate.

The graph of y = f(x) passes through (0, 5) and (3, 40). If f(x) = a · bˣ for constants a and b, what is the value of b?

Answer: 2

At x = 0: f(0) = a · b⁰ = a = 5. At x = 3: f(3) = 5 · b³ = 40, so b³ = 8, giving b = ∛8 = 2. The base b = 2, not ∛8 (which is just the intermediate step before simplifying).

A function h is defined by h(x) = x² - 6x + k, where k is a constant. If the equation h(x) = 0 has exactly one real solution, what must be true?

Answer: k = 9, and the graph of h is tangent to the x-axis at x = 3.

For exactly one real solution, the discriminant must equal zero: b² - 4ac = 0. Here a = 1, b = -6, c = k, so 36 - 4k = 0, giving k = 9. Completing the square: h(x) = (x - 3)² + (k - 9). With k = 9, h(x) = (x - 3)², which is tangent to the x-axis at x = 3 (the vertex). x = -3 is incorrect — be careful about the vertex location from the completed square.

In the xy-plane, the graph of the function p(x) = -(x - 1)(x + 5) intersects the x-axis at two points. What is the x-coordinate of the midpoint of these two intersection points, and what does this value represent on the graph?

Answer: x = -2; it is the x-coordinate of the vertex, where the parabola reaches its maximum value.

The zeros occur at x = 1 and x = -5. The midpoint of the zeros is (1 + (-5))/2 = -4/2 = -2. The axis of symmetry of a parabola passes through the midpoint of the zeros, so x = -2 is the axis of symmetry and the x-coordinate of the vertex. Since the leading coefficient is negative (the parabola opens downward), the vertex is a maximum. The y-intercept is found by setting x = 0, which gives a different point entirely.

The function f(x) = 3ˣ and the function g(x) = 3^(x+2) are both graphed in the xy-plane. Which of the following best describes the relationship between their graphs?

Answer: The graph of g is the graph of f shifted 2 units to the left, which is equivalent to a vertical stretch by a factor of 9.

g(x) = 3^(x+2) = 3^x · 3² = 9 · 3^x = 9f(x). This means the graph of g is a horizontal shift of f by 2 units to the LEFT (not right), which is algebraically equivalent to multiplying f by 9 — a vertical stretch by a factor of 9. These two transformations produce identical graphs. A rightward shift would be 3^(x-2). Adding 2 outside the exponent (f(x) + 2) would be a vertical shift.