The Correct Answer is (D) — If we subtract x from both sides, we get 6 > x + 4; subtracting 4 from both sides gives us 2 > x. So 2 is not a possible value for x.
The Correct Answer is (D) — The graph shows a negative (downward-opening) quadratic function. The equations in (A) and (B) are linear functions, since the variable p has an exponent of 1 (if the exponent is 1 it is not written), so we can eliminate those options. The general form of a quadratic function is , where a, b, and c are constants. (It is not necessary to expand the equations to solve this question, but if we do, we see that is the same as and is the same as , so both equations fit the general form). If a is negative, we call the function a negative quadratic function, and the graph will open downward. In equation (D), a = -1, so (D) is the correct answer.
The Correct Answer is (B) — To find the next number in the sequence, add 3: -2 + 3 = 1; 1 + 3 = 4, and so on.
The Correct Answer is (B) — If we rearrange the second equation into the form and simplify, we get . We can rearrange the first equation to get . Plugging this in for x in the first equation, we get . Solving for y gives us y = 1. If we plug this value of y into the first equation, we find that x = 4. (4, 1) is therefore the point of intersection, and 4 > 1, so a > b is the correct answer.
The Correct Answer is (A) — The slope of this line is -2, since a rise of -1 occurs over a run of ½. The y-intercept is 1. So .
The Correct Answer is (A) — Using the FOIL method, we can find that .
The Correct Answer is (A) — The possible ordered pairs are (1, 1) and (1, 2) (zero is neither negative nor positive).
The Correct Answer is (B) — If we plug in 5 for x, we get , which reduces to 3 = 3. If we plug in -6 for x, we get , which reduces to .
The Correct Answer is (C) — Squaring both sides and rearranging the equation gives us . Factoring gives , so the two possible values for y are 3 and 4. The product of 3 and 4 is 12.
The Correct Answer is (D) — The sum of 1 and 1 is 2, which is greater than 0, the absolute value of their difference.
The Correct Answer is (D) — g(x) is translated by 2 in the positive x direction and 5 in the positive y direction. This means that h must be 2 and k must be 5, since f(x – 2) will translate f(x) in the positive x direction and f(x) + 5 will translate f(x) in the positive y direction. So hk = 2 × 5 = 10.
The Correct Answer is (B) — Setting the functions equal to each other gives . Rearranging gives , and factoring gives . This means that the two points of intersection occur when x = -1 and x = 3. Since x = -1 is not in the first quadrant, we must use x = 3. Plugging 3 in for x in either equation yields y = 29.
The Correct Answer is (D) — If we look at the values in the chart, we see that while b increases by one each time, h increases by 10, then 14, then 18. This implies that the equation that would represent the relationship is not linear (if it were, h would increase by a constant amount each time), so we can eliminate (A) and (B) since they are linear equations. Now, checking the values for b and h in equation (C), we can see that they do not fit in this function (for example, 29 ≠ 42 + 1). That leaves equation (D), which, if we plug in for b and h, satisfies all values in the chart.
The Correct Answer is (6) — If 2f(p) = 8, f(p) = 4. Plugging this into the original equation, we get 4 = 2 – p, so p = -2. Plugging -2 into f(2p) gives f(-4) = 2 + 4 = 6.
The Correct Answer is (1/3) — To find the smallest value of k, we must find the smallest value for –x + 2, meaning we must find the largest value for x (since x becomes negative in the equation). The largest value for x, given x ≤ 1, is 1. Plugging 1 in for x and solving for k gives k ≥ 1/3. The minimum value of k, then, is 1/3.
The Correct Answer is (5) — Multiplying each side by the denominators gives -6x = x2 – 9x – 10; simplifying gives 0 = x2 – 3x – 10. Factoring gives 0 = (x – 5)(x + 2). The solutions are therefore 5 and -2; 5 is the positive solution.
The Correct Answer is (29) — The length of each side can be found using the Pythagorean Theorem: x2 + y2 = s2, where s is the length of the side of the square. Taking the bottom edge as an example, we can see that x = 5 since the difference between the two x-coordinates is 5. Similarly, y = 2, so we have 52 + 22 = s2. Since the area of a square is s2, solving for this term yields 25 + 4 = 29.