Free SAT Interpreting Nonlinear Functions Questions and Answers 2 — Questions and Answers
Question 1: The function h(t) = -4.9t^2 + bt + c models the height, in meters, of a projectile t seconds after it is launched. The projectile reaches its maximum height of 25 meters at t = 2. Which of the following defines the function?
- h(t) = -4.9(t - 2)^2 + 25 (Correct answer)
- h(t) = -4.9(t + 2)^2 + 25
- h(t) = 4.9(t - 2)^2 + 25
- h(t) = -4.9(t - 25)^2 + 2
Correct answer: h(t) = -4.9(t - 2)^2 + 25
The vertex form of a parabola, y = a(x - h)^2 + k, reveals the vertex at (h, k). In this problem, the maximum height is the y-coordinate of the vertex, and the time it takes to reach that height is the x-coordinate. Therefore, the vertex is (2, 25). Plugging h=2 and k=25 into the vertex form gives y = a(t - 2)^2 + 25. The coefficient 'a' is given as -4.9, which determines the parabola's vertical stretch and direction (opening downwards, which is correct for a projectile's path).
Question 2: A scientist models the decay of a radioactive substance with the function A(t) = A₀(1/2)^(t/h), where A(t) is the amount remaining at time t, A₀ is the initial amount, and h is the half-life. If the half-life of the substance is 200 years, and the initial amount is 500 grams, how much of the substance remains after 800 years?
- 125 grams
- 62.5 grams
- 31.25 grams (Correct answer)
- 15.625 grams
Correct answer: 31.25 grams
First, substitute the given values into the function: A₀ = 500 and h = 200. The function becomes A(t) = 500(1/2)^(t/200). We need to find the amount remaining after t = 800 years. Plug in t = 800: A(800) = 500(1/2)^(800/200). Simplify the exponent: 800/200 = 4. The equation is now A(800) = 500(1/2)^4. Calculate (1/2)^4 = 1/16. Finally, calculate A(800) = 500 * (1/16) = 500/16 = 31.25 grams.
Question 3: The graph of y = f(x) is a parabola that opens upward with a vertex at (3, -5). The graph of y = g(x) is a line with a positive slope. If the system of equations y = f(x) and y = g(x) has exactly one solution, what must be true about the line g(x)?
- The line is tangent to the parabola at a point where x < 3.
- The line must pass through the origin.
- The line is tangent to the parabola at its vertex. (Correct answer)
- The line is perpendicular to the parabola's axis of symmetry.
Correct answer: The line is tangent to the parabola at its vertex.
A system of a parabola and a line has exactly one solution when the line is tangent to the parabola. This means the line touches the parabola at exactly one point. For a parabola with a vertex at (3, -5), the line must touch it at this single point, the vertex, to satisfy the condition of having only one solution. Any other line would either miss the parabola entirely (zero solutions) or intersect it at two points.
Question 4: The value of a car, V, in dollars, is modeled by the function V(t) = 25000(0.88)^t, where t is the number of years since its purchase. Which of the following statements best interprets the number 0.88 in this context?
- The car's value increases by 88% each year.
- The car's value decreases by 88% each year.
- The car retains 88% of its value from the previous year. (Correct answer)
- The car's value decreases by $0.88 each year.
Correct answer: The car retains 88% of its value from the previous year.
This is an exponential decay function of the form y = a(b)^x, where 'a' is the initial amount and 'b' is the decay factor. Here, b = 0.88. This factor represents the percentage of the value that remains each year. So, the car's value each year is 88% of its value from the year before. This corresponds to a 12% annual decrease (1 - 0.88 = 0.12).
Question 5: A function is defined by f(x) = a(x-3)(x+1), where 'a' is a positive constant. A second function is g(x) = f(x) + k. For what value of k will the graph of g(x) have its vertex on the x-axis?
- k = 2a
- k = -4a
- k = 4a (Correct answer)
- k = -a
Correct answer: k = 4a
The x-intercepts of f(x) are at x=3 and x=-1. The x-coordinate of the vertex of a parabola is halfway between its x-intercepts, so the vertex is at x = (3 + (-1))/2 = 1. To find the y-coordinate of the vertex of f(x), we evaluate f(1): f(1) = a(1-3)(1+1) = a(-2)(2) = -4a. The vertex of f(x) is at (1, -4a). The function g(x) = f(x) + k represents a vertical shift of f(x) by k units. To move the vertex onto the x-axis, we need to shift the graph up by the absolute value of the vertex's y-coordinate. Therefore, we must add 4a. So, k = 4a.
Question 6: The number of users of a new app is modeled by the function N(d) = 100(b)^d, where d is the number of days after launch. If the number of users triples every 5 days, which of the following is the closest value for b?
- 1.25 (Correct answer)
- 3
- 1.5
- 1.20
Correct answer: 1.25
We are given that the number of users triples every 5 days. This means that when d=5, N(5) should be 3 times the initial amount, N(0). The initial amount is N(0) = 100(b)^0 = 100. So, N(5) = 3 * 100 = 300. We can set up the equation: 300 = 100(b)^5. Divide by 100: 3 = b^5. To solve for b, we take the fifth root of 3: b = 3^(1/5). Using a calculator, 3^(1/5) is approximately 1.2457. The closest answer choice is 1.25.
Free SAT Interpreting Nonlinear Functions Questions and Answers 2
The function h(t) = -4.9t^2 + bt + c models the height, in meters, of a projectile t seconds after it is launched.
The projectile reaches its maximum height of 25 meters at t = 2.
Which of the following defines the function?