Math Mixed Practice Test 6

1. Point-slope form: y - y_1 = m(x - x_1).
2. From the point (-2, 4): x_1 = -2 and y_1 = 4. The slope is m = -3.
3. Substitute: y - 4 = -3\big(x - (-2)\big).
4. Subtracting a negative is adding: x - (-2) = x + 2. y - 4 = -3(x + 2)

Answer: y - 4 = -3(x + 2)

Check: Put in the point (-2, 4): left side 4 - 4 = 0; right side -3(-2 + 2) = -3(0) = 0 ✓.

Tip: In point-slope form the signs inside the equation are the opposite of the signs in the point. The point (-2, 4) shows up as (x + 2) and (y - 4).

1. Start with V = \tfrac{1}{3}\pi r^2 h.
2. Multiply both sides by 3 to undo the \tfrac{1}{3}: 3V = \pi r^2 h.
3. Divide both sides by \pi r^2: \dfrac{3V}{\pi r^2} = h.

The answer is h = \dfrac{3V}{\pi r^2}.

Check: With numbers, a cone with r = 3 and h = 4 has V = \tfrac{1}{3}\pi(9)(4) = 12\pi. Then \dfrac{3(12\pi)}{\pi(9)} = \dfrac{36\pi}{9\pi} = 4 ✓

Tip: To undo multiplying by \tfrac{1}{3}, multiply by 3 — don't divide by 3. That slip gives \dfrac{V}{3\pi r^2}, which is 9 times too small.

1. In y = 0.6x + 18, the number multiplied by x is the slope: the change in y for each increase of 1 in x.
2. One more year (x goes up by 1) makes y go up by 0.6.
3. But y is measured in thousands of people, so an increase of 0.6 means 0.6 \times 1{,}000 = 600 \text{ people}
4. So the population grows by about 600 people per year.

The best interpretation is: the town's population increases by about 600 people each year.

Check: Compare 2010 (x = 0, y = 18, so 18,000 people) and 2011 (x = 1, y = 18.6, so 18,600 people): 18{,}600 - 18{,}000 = 600 ✓

Tip: Always attach units to the slope: (units of y) per (unit of x). When a variable is "in thousands" or "in millions," convert at the end, or your answer will be off by a factor of 1,000.

1. Step 1 substitutes -3 for both x's with parentheses. Correct.
2. Step 2 claims (-3)^2 = -9. But (-3)^2 = (-3)(-3) = +9. This is the first error.
3. Correct Step 2: -(9) + 4(-3) = -9 + 4(-3).
4. Correct Step 3: 4(-3) = -12, so -9 - 12.
5. Correct Step 4: -9 - 12 = -21.

Answer: The first error is in Step 2; the correct value is f(-3) = -21.

Check: Factor: f(x) = -x(x - 4), so f(-3) = -(-3)(-3 - 4) = 3 \cdot (-7) = -21 ✓

Tip: In -x^2, the square happens first and the negative sign is applied after. So -x^2 is always negative (or zero), whatever you put in for x.

1. The legs are a = 9 and b = 12; c is across from the right angle.
2. Pythagorean theorem: c^2 = a^2 + b^2 = 9^2 + 12^2 = 81 + 144 = 225.
3. Square root: c = \sqrt{225} = 15.

c = 15 cm.

Check: 9 = 3 \times 3 and 12 = 4 \times 3, so this is the 3-4-5 triangle times 3, and 5 \times 3 = 15 ✓. Also 15 is longer than each leg but shorter than 9 + 12 = 21 ✓.

Tip: Memorize the common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples (6-8-10, 9-12-15). They save time and are a fast check.

1. When w = 0 (no payments yet), he owes 1{,}200 - 0 = 1{,}200 dollars. So he borrowed $1,200.
2. Each extra payment lowers the amount by 85 dollars. So each weekly payment is $85.
3. Check the other ideas: he does not pay $1,200 a week, and the loan is paid off when 1{,}200 - 85w = 0, which is about 14.1 weeks, not 85 weeks.

The true statement is Kofi pays $85 each week.

Check: after 2 payments he owes 1{,}200 - 170 = 1{,}030 dollars, which is $1,200 minus two payments of $85 ✓

Tip: to interpret an expression, plug in 0 for the variable (that gives the starting value) and then 1 (that shows the change per step).

1. To decide, look at the inputs in the x row: -3, -1, 0, 1, 3.
2. Every input appears once, so each input has exactly one output.
3. The outputs 9 and 1 each appear twice, but that is allowed: different inputs may share an output. (-3 and 3 both give 9 — this table follows y = x^2.)
4. The order of the outputs does not matter either.

Answer: Yes, because each x-value is paired with exactly one y-value.

Check: Name the output for every input without hesitation: -3 \to 9, -1 \to 1, 0 \to 0, 1 \to 1, 3 \to 9 — one answer each ✓.

Tip: For a table, scan only the input row for repeats. If no input repeats, the table is a function — no matter what the outputs do.

1. Turn every result into a decimal:
   • A: \dfrac{9}{40} = 9 \div 40 = 0.225
   • B: 0.2 = 0.200
   • C: \dfrac{7}{32} = 7 \div 32 = 0.21875
   • D: \dfrac{11}{50} = 11 \div 50 = 0.22
2. Write with 4 places: 0.2250, 0.2000, 0.2188 (rounded), 0.2200.
3. Order from highest: 0.2250 > 0.2200 > 0.2188 > 0.2000.

Answer: Line A has the highest defect rate.

Check: \tfrac{9}{40} = \tfrac{22.5}{100} and \tfrac{11}{50} = \tfrac{22}{100}, so A is half a percentage point higher than D. ✓

Tip: The line with the most defective items (D, with 11) is not automatically the worst — it also checked more items. Always divide before you compare.

1. Fill the restricted seat first. Rosa must be in the middle chair: 1 choice.
2. The other 4 coworkers fill the remaining 4 chairs in any order: 4! = 4 \times 3 \times 2 \times 1 = 24
3. Multiply: 1 \times 24 = 24.
4. There are 24 arrangements.

Check: Without the rule there are 5! = 120 arrangements. Rosa is in each of the 5 chairs equally often, so \tfrac{1}{5} of them have her in the middle: 120 \div 5 = 24 ✓

Tip: A person or item locked into one spot has just 1 choice. Place it first, then arrange everything else.

1. Pyramid formula (on the formula sheet): V = \tfrac{1}{3}Bh.
2. The base is a rectangle: B = 12 \times 5 = 60\text{ ft}^2.
3. Multiply by the height: 60 \times 4 = 240.
4. Take one-third: V = \tfrac{1}{3}(240) = 80.

Answer: 80 ft³

Check: Work backwards: h = \dfrac{3V}{B} = \dfrac{3(80)}{60} = 4 ft, which matches the height in the problem ✓.

Tip: The pyramid formula V = \tfrac{1}{3}Bh works for any base shape. Only the way you find B changes: square, rectangle, triangle, and so on.

1. List the points: (-3, 2), (-1, 4), (0, -1), (2, 3), (2, -2), (4, 1).
2. Look for a repeated x-value: x = 2 appears twice, with y = 3 and y = -2.
3. That pair makes the relation fail: a vertical line at x = 2 hits two points.
4. Removing either (2, 3) or (2, -2) fixes it. Among the choices, only (2, -2) is one of these points.

Answer: (2, -2)

Check: Without (2, -2) the x-values are -3, -1, 0, 2, 4 — all different, so every vertical line meets at most one point ✓.

Tip: For a graph of separate points, the vertical line test still works: two dots stacked in the same column mean "not a function."

1. Maximum of f: the values rise to 9 at x = 1 and fall again symmetrically (8, 8 and 5, 5). So the maximum of f is 9.
2. Maximum of g: a = -1, b = 6. x = -\frac{6}{2(-1)} = 3, \qquad g(3) = -(3)^2 + 6(3) + 1 = -9 + 18 + 1 = 10 So the maximum of g is 10.
3. Compare: 10 > 9.

Answer: The maximum of g (10) is greater than the maximum of f (9).

Check: g(2) = -4 + 12 + 1 = 9 and g(4) = -16 + 24 + 1 = 9, both below 10 ✓

Tip: Compare the same feature on both sides: maximum value against maximum value. The 1 in g(x) is its y-intercept, and 3 is where its maximum occurs — neither is the maximum value.

1. Isolate the squared group. Divide both sides by 4: (x - 1)^2 = 9
2. Take the square root of both sides with \pm: x - 1 = \pm 3
3. Solve each equation by adding 1: x = 1 + 3 = 4 \quad \text{or} \quad x = 1 - 3 = -2
4. The solutions are x = 4 or x = -2.

Check: 4(4 - 1)^2 = 4(9) = 36 ✓ and 4(-2 - 1)^2 = 4(9) = 36 ✓

Tip: Undo operations in reverse order: the 4 multiplies the square, so divide by 4 first, then take the root, then move the 1.

1. Group: (x^3 - 5x^2) + (3x - 15).
2. Factor the first group: x^3 - 5x^2 = x^2(x - 5).
3. Factor the second group: 3x - 15 = 3(x - 5).
4. Now both parts share (x - 5): x^2(x - 5) + 3(x - 5).
5. Pull out the shared binomial: (x - 5)(x^2 + 3).

Answer: (x - 5)(x^2 + 3)

Check: (x - 5)(x^2 + 3) = x^3 + 3x - 5x^2 - 15 = x^3 - 5x^2 + 3x - 15 ✓

Tip: After you factor each group, the parentheses must match exactly. If they don't, try changing the sign you pull out of the second group.

1. Start with 3x - 5y = 15.
2. Subtract 3x from both sides: -5y = -3x + 15
3. Divide every term by -5: y = \frac{-3}{-5}x + \frac{15}{-5} = \frac{3}{5}x - 3
4. Compare with y = mx + b: m = \tfrac{3}{5}.

The slope is \tfrac{3}{5}.

Check: Two points on the line: x = 0 \Rightarrow -5y = 15 \Rightarrow y = -3; x = 5 \Rightarrow 15 - 5y = 15 \Rightarrow y = 0. Slope = \frac{0 - (-3)}{5 - 0} = \frac{3}{5} ✓

Tip: Shortcut for Ax + By = C: slope = -\dfrac{A}{B}. Here -\dfrac{3}{-5} = \dfrac{3}{5}. A negative divided by a negative is positive.

1. Consecutive integers differ by 1, so the three integers are n, n + 1, and n + 2.
2. Sum: n + (n + 1) + (n + 2).
3. Combine like terms: n + n + n = 3n and 1 + 2 = 3.
4. Sum = 3n + 3.

Answer: 3n + 3

Check: With n = 7: 7 + 8 + 9 = 24, and 3(7) + 3 = 24 ✓.

Tip: Use +1 steps for consecutive integers, and +2 steps for consecutive even or odd integers. "Sum" means add; "product" means multiply.

1. Perfect squares around 150: 12^2 = 144 and 13^2 = 169.
2. Since 144 < 150 < 169, we have 12 < \sqrt{150} < 13 (about 12.25).
3. Taking the opposite flips the order: -13 < -\sqrt{150} < -12.
4. So -\sqrt{150} \approx -12.25, which lies between -13 and -12.

Answer: between -13 and -12

Check: (-12.25) is to the left of -12 and to the right of -13 on a number line ✓. And 12.25^2 = 150.06 \approx 150 ✓.

Tip: The negative sign is outside the root, so -\sqrt{150} is a real number (just negative). Find the root of the positive number first, then attach the sign.