Nonlinear functions
The function, f, is defined by f(x) = x^2 - 6x + 14. What is the minimum value of f(x)?
5
The quadratic equation 2x^2 - 11x + 5 = 0 has roots r_1 and r_2. What is the value of r_1 + r_2?
5.5
If x^2 - 2x - 15 = 0 and x > 0, what is the value of x?
5
If 3/(x-2) =5/(x+4), what is the value of x?
11
If |4x-4|=112, what is the positive value of x-1?
28
A population of bacteria starts at 500 and increases by 12% every hour. Which of the following functions models the population, P(t), after t hours?
A) P(t) = 500 + 1.12t
B) P(t) = 500(0.12)^t
C) P(t) = 500(1.12)^t
D) P(t) = 500 + 12t
C
The graph of y = f(x) is transformed to create the graph of g(x) = f(x - 3) + 2. Which statement correctly describes the transformation from the graph of f to the graph of g?
A) Shifted 3 units to the left and 2 units up.
B) Shifted 3 units to the right and 2 units up.
C) Shifted 3 units to the left and 2 units down.
D) Shifted 3 units to the right and 2 units down.
B) Shifted 3 units to the right and 2 units up.
If square root of {2x + 7} - 3 = 2, what is the value of x?
9
The equation x^2 - 8x + 4 = 0 has solutions x = a + square root of b and x = a - square root of b , where a and b are positive integers. What is the value of a + b?
P(t) = 24.8(1.036)^t
The function, P, gives the predicted population, in millions, of a certain country for the period from 1984 to 2018, where t is the number of years after . According to the model, what is the best interpretation of the statement “P(8) is approximately equal to 32.91”?
A) In 1984, the predicted population of this country was approximately 8 million.
B) In 1984, the predicted population of this country was approximately 32.91 million.
C) 8 years after 1984, the predicted population of this country was approximately 32.91 million.
D) 32.91 years after 1984, the predicted population of this country was approximately 8 million.
C) 8 years after 1984, the predicted population of this country was approximately 32.91 million.
The graph of the quadratic function
,f(x) = -2x^2 + bx - 12,
has its vertex at x = 3. What is the maximum value of the function?
6
In the quadratic equation 3x^2 + kx + 12 = 0, k is a positive constant. If the equation has exactly one distinct real solution, what is the value of k?
12
If (x, y) is the solution to the system of equations above, what is the value of x + y?
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Equivalent expressions bonus:
(x-11y)(2x-3y)-12y(-2x+3y)
Which of the following is equivalent to the expression above?
A) x-23y
B) 2x^2-xy-3y^2
C) 2x^2+24xy+36y^2
D) 2x^2-49xy+69y^2
B) 2x^2-xy-3y^2
The value of a certain antique item grows exponentially over time. At time t = 0 years, the item is worth $500. At time t = 2 years, the item is worth $720. What is the value of the item at t = 3 years?
A) $830
B) $864
C) $930
D) $1,080
B) $864
The graph of the function f(x) has a vertex at the point (2, -5). If the function g is defined as g(x) = -3f(x + 4) - 1, what are the coordinates of the vertex of the graph of g(x)?
A) (-2, 14)
B) (-2, 16)
C) (6, 14)
D) (-6, -16)
A) (-2, 14)
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What is the complete solution set to the equation square root of {2x + 6} = x - 1$?
A) {-1}
B) {5}
C) {-1, 5}
D) The equation has no real solutions.
y= -1.5
y= x^2+8x+a
In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a?
14.5, 29/2
A system of equations consists of a line and a parabola:
If the system has no real solutions, which of the following could be the value of b?
A) -5
B) -2
C) 0
D) 3
A) -5
The roots of the quadratic equation x^2 - kx + 24 = 0 are two distinct positive integers. What is the sum of all possible values of the constant k?
60
Which of the following contains all real solutions to the equation x/(x-3) - 3/(x+3) = 18/(x^2-9)
A) -3
B) 3
C) {-3, 3}
D) The equation has no real solutions.
D) The equation has no real solutions.
A system of two linear equations is graphed in the coordinate xy-plane. The first line passes through the coordinate points (0, 4) and (2, 8). The second line is represented algebraically by the equation ax - 3y = 9, where a is a constant. If the system has no solutions, what is the value of a?
A) -6
B) 2
C) 3
D) 6
D) 6
Function f is defined by f(x) = -a^x+b where a and b are constants. In the xy-plane, the graph of y = f(x)-12 has a y-intercept at (0, -75/7). The product of a and b is 320/7. What is the value of a?
20