SAT Math Practice Test: Advanced Math
- Advanced Math
- 11 questions
- 15 minutes
These questions cover SAT Advanced Math: equivalent expressions, quadratic and exponential functions, polynomials, radical equations and graphs of nonlinear functions. After squaring both sides of an equation, check every solution in the original equation.
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Question 1
Which expression is equivalent to (2x − 3)2 − (x + 1)(x − 1)?
Correct answer: A
(2x − 3)2 = 4x2 − 12x + 9 and (x + 1)(x − 1) = x2 − 1. Subtracting gives 3x2 − 12x + 10; the choice ending in + 8 forgets that subtracting −1 adds 1.
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Question 2
The function f is defined by f(x) = (x − 1)(x − 7). What is the minimum value of f(x)?
Correct answer: A
The parabola opens upward, and its vertex lies halfway between the zeros 1 and 7, at x = 4. The minimum value is f(4) = (3)(−3) = −9; the choice 4 is the x-coordinate of the vertex, not the minimum value.
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Question 3
What is the positive solution to the equation 2x2 + 5x − 12 = 0?
Correct answer: B
Factoring gives (2x − 3)(x + 4) = 0, so x = 32 or x = −4. The positive solution is 32; the choice 4 comes from reading the factor x + 4 with the wrong sign.
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Question 4
The number of bacteria in a culture t hours after an experiment begins is modeled by P(t) = 500(1.08)t. Which statement is the best interpretation of the number 1.08 in this model?
Correct answer: B
Multiplying by 1.08 every hour means each hour adds 8% of the current amount. A 108% increase would mean multiplying by 2.08, and an increase of exactly 8 bacteria per hour would be a linear model.
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Question 5
A radioactive sample has an initial mass of 80 grams. Its half-life is 6 days, which means its mass is halved every 6 days. Which function gives the mass M, in grams, of the sample after d days?
Correct answer: D
The mass is multiplied by 0.5 once every 6 days, so the exponent counts the number of 6-day periods, d/6. Check: when d = 6, M = 80(0.5)1 = 40. The exponent 6d would halve the mass six times every day.
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Question 6
p(x) = x3 − 4x2 + kx − 6
In the polynomial shown, k is a constant. If x − 3 is a factor of p(x), what is the value of k?
Correct answer: C
If x − 3 is a factor, then p(3) = 0: 27 − 36 + 3k − 6 = 0, so 3k = 15 and k = 5. The choice 15 is the value of 3k, not k.
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Question 7
What are all the solutions to the equation x + 7 = x − 5?
Correct answer: B
Squaring gives x + 7 = x2 − 10x + 25, so x2 − 11x + 18 = 0 and x = 2 or x = 9. For x = 9 both sides equal 4, but for x = 2 the left side is 3 and the right side is −3, so 2 is an extraneous solution.
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Question 8
Which expression is equivalent to (8x6)2/3, where x > 0?
Correct answer: D
Raise each factor to the 2/3 power: 82/3 = (cube root of 8)2 = 22 = 4, and (x6)2/3 = x4. The choice 4x9 multiplies the exponent 6 by 3/2 instead of 2/3.
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Question 9
y = x2 − 2x + 1
y = x + 5The graphs of the equations shown intersect at two points in the xy-plane. If (a, b) is the intersection point with a > 0, what is the value of b?
Correct answer: C
Setting x2 − 2x + 1 = x + 5 gives x2 − 3x − 4 = 0, so x = 4 or x = −1. With a = 4, b = 4 + 5 = 9; the choice 4 is the value of a (and the y-value of the other point).
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Question 10
In the equation kx2 − 6x + 3 = 0, k is a constant. For which of the following values of k does the equation have no real solutions?
Correct answer: D
There are no real solutions when the discriminant is negative: (−6)2 − 4(k)(3) = 36 − 12k < 0, so k > 3. Only 4 works; k = 3 makes the discriminant 0, which gives exactly one solution.
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Question 11
The function f is defined by f(x) = 2(x − 3)2 + 5, and the function g is defined by g(x) = f(x + 4) − 1. What are the coordinates of the vertex of the graph of y = g(x) in the xy-plane?
Correct answer: A
Replacing x with x + 4 shifts the graph 4 units left, and subtracting 1 shifts it 1 unit down, so the vertex (3, 5) of f moves to (−1, 4). The choice (7, 4) shifts the graph right instead of left.
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