SAT Questions I Got Wrong/What I Want To Remember
i^4
1
0<x<90 degrees quadrant 90<x<180 degrees quadrant 180<x<270 degrees quadrant 70<x<360 degrees quadrant
1 2 3 4
area of a circle
πr²
cylinder volume
πr²h
volume of cylinder
πr²h
y = 2(x+1)^2 - 8 (h,k) vertex is ___
(-1,-8)
volume of triangular prism
(1/2bh)h
(x-3)^2 + (y - 4)^2 = 25 center is ___ radius is ___
(3,4) 5
percent equation
(x%)(total) = part the % is in decimal format
circle equation center
(x-h)^2 + (y-k)^2 = r^2 (h,k)
i^2 =
-1
i^3
-i
Percent Decreases starting value 1200 35% decrease
.65(1200) = 780
ratios 1/2 = ____
1:2
left off
1:32:00
Percent Increases starting value 1200 20% increase
1.20(1200) = x 1440
area of a triangle
1/2bh
Pyramid Volume
1/3 lwh
volume of pyramid
1/3Bh
Cone Volume
1/3πr²h
volume of cone
1/3πr²h
the sum of the measures in degrees of angles of a triangle is
180
Conversion Ratios convert 2 miles to feet
2 miles / 1 * 5280 feet/ 1 mile cross out the miles = 10,560 feet
if a= 5radical2 and 2a= radical2x what is the value of x
2(5radical2) = radical2x 10radical2 = radical2x square 100 * 2 = 2x 100 = x
the number of radians of arc in a circle is
2pi
circumference of a circle
2πr
convert 3 days to minutes
3 days/1 * 24 hours/ 1 day * 60 minutes/ 1 hour = 4320 minutes
y = 3(2)^x
3 is the initial value it is growing by 2 every x (maybe hour yknow; x is the time) yint is (0, 3) (look at the initial value meow meow)
if 3/5 w = 4/3, what is the value of w? A) 9/20 B) 4/5 C) 5/4 D) 20/9
3/5 w = 4/3 multiply both by 15 9w=20 w= 20/9
the number of degrees of arc in a circle is
360
(3x^2 - 2x - 5) - (4x^2 + x + 2)
3x^2 - 2x - 5 - 4x^2 - x - 2 -x^2 - 3x - 7
Sphere Volume
4/3πr³
volume of sphere
4/3πr³
To make 4 batches of cupcakes requires 7 tablespoons of sugar. How many tablespoons is required for 5 batches?
4/7 = 5/x 8.75
special triangle
45 45 90 x x x*radical2 60 30 90 x*radical3 x 2x
45-45-90 triangle
45: x 45: x 90: xradical2
5 + i^2 = 4
5 + -1 = 4 4 = 4
60-30-90 triangle
60: xradical3 30: x 90: 2x
7 + i^8 = 8
7 + 1 = 8 8 = 8
If 7 pounds of plums makes 8 fruit rolls, how many pounds is needed to make 20 fruit rolls?
7/8 = x/20 17.5
30 is 75% of what number
75%(x) = 30 .75x = 30 x = 40
(3x^2 - 2x - 5) + (4x^2 + x + 2)
7x^2 - x - 3
quadrant 1234 All Students Take Calculus
A Q1 all positive S Q2 sin values positive T Q3 tan values positive C Q4 cos values positive sin, cos, and tan are all positive between 0 and 90
Sector area formula
A = inner angle/360 pir^2
scatterplot looks like (__ but curved the whole way what equation could this be? A)t=96 - 32r B)t= 64-16r C)t=128(0.5)^r D)t= 128(2)^r
A and B look linear / so no the line appears to be decreasing and not growing so it cant be D therefore, it could be C
In a circle with center O, central angle AOB has a measure of 5π/4 radians. The area of the sector formed by central angle AOB is what fraction of the area of the circle?
A complete rotation around a point is 360° or 2π radians. Since the central angle AOB has measure 5π/4 radians, it represents 5π/4 /2π = 5/8 of a complete rotation around point O. Therefore, the sector formed by central angle AOB has area equal to 5/8 the area of the entire circle. The correct answer is 5/8 or .625
similar triangles
AA, SAS, SSS
Standard Form
Ax+By=C where A, B, and C are real numbers and A & B are not both zero; A must be positive
If 7/9x -4/9x = 1/4 + 5/12 , what is the value of x ?
First, clear the fractions from the given equation by multiplying each side of the equation by 36. The equation becomes 28x − 16x = 9 + 15. Combining like terms on each side of the equation yields 12x = 24. Finally, dividing both sides of the equation by 12 yields x = 2. The correct answer is 2
congruent
Having the same size and shape
simplify 3x^2 - 2x - 5 / x^2 + x divided by 9-15x / 3x
KCF 3x^2 - 2x - 5 / x^2 + x * 3x / 9 - 15x (3x-5)(x+1) / x(x+1) * 3x / 3(3x - 5) cross cancel everything actually got knocked out 1
Arc Length Formula
L = inner angle/360 * pir^2
A rectangle was altered by increasing its length by 10 percent and decreasing its width by p percent. If these alterations decreased the area of the rectangle by 12 percent, what is the value of p ? A) 12 B) 15 C) 20 D) 22
Let l and w be the length and width, of the original rectangle. The area of the original rectangle is A = lw The rectangle is altered by increasing its length by 10% and decreasing its width by p% the length is altered by 1.1l and the width is altered by (1-p/100)w the alterations decrease the area by 12% so the area of the altered rectangle is (1-0.1)A = 0.88A. the altered rectangle is the product of its length and width so, 0.88A = (1.1l)(1-P/100)w solve for P .88 = (1.1)(1 - P/100) p = 20
The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth-grade students at Keisel Middle School. There are 5 times as many right-handed female students as there are left-handed female students, and there are 9 times as many right-handed male students as there are left-handed male students. If there is a total of 18 left-handed students and 122 right-handed students in the school, which of the following is closest to the probability that a right-handed student selected at random is female? (Note: Assume that none of the eighth-grade students are both right-handed and left-handed.) A) 0.410 B) 0.357 C) 0.333 D) 0.250
Let x be the number of left-handed female students and let y be the number of left-handed male students. Then the number of righthanded female students will be 5x and the number of right-handed male students will be 9y. Since the total number of left-handed students is 18 and the total number of right-handed students is 122, the system of equations below must be satisfied. 5 x + y = 18 5x + 9y = 122 Solving this system gives x = 10 and y = 8. Thus, 50 of the 122 right-handed students are female. Therefore, the probability that a right-handed student selected at random is female is 50/122, which to the nearest thousandth is 0.410 Choice A is correct.
3 + 4i / 6 - 5i
NO IMAGINARY IN THE DENOMINATOR KCF 18 + 15i + 24i + 20i^2 / 36 + 30i - 30i * 25i^2 18 + 39i - 20 / 36 + 25 - 2 + 39i / 61 yay
The sum of three numbers is 855. One of the numbers, x, is 50% more than the sum of the other two numbers. What is the value of x ? A) 570 B) 513 C) 214 D) 155
One of the three numbers is x; let the other two numbers be y and z. Since the sum of three numbers is 855, the equation x + y + z = 855 is true. The statement that x is 50% more than the sum of the other two numbers can be represented as x = 1.5(y + z) x/1.5 = y + z. Substituting x/1.5 for y + z in x + y + z = 855 gives x + x/1.5 = 855 x = 513 Choice B is correct.
3x-5y = 8 find the slope
QUICK WAY -a/b -3/-5 = 3/5 3/5 is the slope
In a circle with center O, central angle AOB has a measure of 5pi/4 radians. The area of the sector formed by central angle AOB is what fraction of the area of the circle?
REMEMBER THIS ------------------------- sector area/area of circle = central angle/ degrees or 2pi = arc length/circumference ---------------------------- 5pi/4 / 2pi = = 5/8 that's the answer
Congruent Triangles
SSS, SAS, AAS, ASA, HL (ONLY IF RIGHT TRIANGLE)
The angles shown above are acute and sin( a°) = cos( b°). If a = 4k − 22 and b = 6k − 13 , what is the value of k ? A) 4.5 B) 5.5 C) 12.5 D) 21.5
Since the angles are acute and sin(a°) = cos(b°), it follows from the complementary angle property of sines and cosines that a + b = 90. Substituting 4k − 22 for a and 6k − 13 for b gives (4k − 22) + (6k − 13) = 90 10k − 35 = 90 10k = 125 k = 12.5. Choice C is correct.
In the xy-plane, the parabola with equation y = ( x− 11)^2 intersects the line with equation y = 25 at two points, A and B. What is the length of AB? A) 10 B) 12 C) 14 D) 16
Substitute 25 for y in the equation y = (x − 11)^2 25 = (x − 11)^2 find square root +- 5 = x-11 so the x-coordinates of the two points of intersection are x = 16 and x = 6, respectively. Since both points of intersection have a y-coordinate of 25, it follows that the two points are (16, 25) and (6, 25). Since these points lie on the horizontal line y = 25, the distance between these points is the positive difference of the x-coordinates: 16 − 6 = 10. Choice A is correct.
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N= rT . This relationship is known as Little's law. The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time. Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?
The average number of shoppers, N, in the checkout line at any time is N = rt, where r is the number of shoppers entering the checkout line per minute and T is the average number of minutes each shopper spends in the checkout line. Since 84 shoppers per hour make a purchase, 84 shoppers per hour enter the checkout line. This needs to be converted to the number of shoppers per minute. Since there are 60 minutes in one hour, the rate is 84 shoppers /60 minutes = 1.4 shoppers per minute. Using the given formula with r = 1.4 and t = 5, N = (1.4)(5) = 7. Therefore, the average number of shoppers, N, in the checkout line at any time during business hours is 7. The correct answer is 7.
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N= rT . This relationship is known as Little's law. The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time. The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? (Note: Ignore the percent symbol when entering your answer. For example, if the answer is 42.1%, enter 42.1)
The estimated average number of shoppers in the original store at any time is 45. In the new store, the manager estimates that an average of 90 shoppers per hour enter the store, which is equivalent to 1.5 shoppers per minute (90shoppers/60mins = 1.5). The manager also estimates that each shopper stays in the store for an average of 12 minutes. Thus, by Little's law, there are, on average, N = rt N = (1.5)(12) = 18 shoppers in the new store at any time. This is 45 − 18= 27/45 = .60 .60 × 100 = 60 60 percent less than the average number of shoppers in the original store at any time. The correct answer is 60
An online store receives customer satisfaction ratings between 0 and 100, inclusive. In the first 10 ratings the store received, the average (arithmetic mean) of the ratings was 75. What is the least value the store can receive for the 11th rating and still be able to have an average of at least 85 for the first 20 ratings?
The mean of a data set is the sum of the values in the data set divided by the number of values in the data set. The mean of 75 is obtained by finding the sum of the first 10 ratings and dividing by 10. Thus, the sum of the first 10 ratings was 750. In order for the mean of the first 20 ratings to be at least 85, the sum of the first 20 ratings must be at least (85)(20) = 1700. Therefore, the sum of the next 10 ratings must be at least 1700 − 750 = 950. We want to find the 11th rating so leave that out of the 10. The maximum rating is 100, so the maximum possible value of the sum of the 12th through 20th ratings is 9 × 100 = 900. Therefore, for the store to be able to have an average of at least 85 for the first 20 ratings, the least possible value for the 11th rating is 950 − 900 = 50. The correct answer is 50.
In the xy-plane, the line determined by the points (2,k) and ( k, 32) passes through the origin. Which of the following could be the value of k ? A) 0 B) 4 C) 8 D) 16
The line passes through the origin (0,0), (2, k), and (k, 32). Any two of these points can be used to find the slope of the line. Since the line passes through (0, 0) and (2, k), the slope of the line is equal to k-0/2-0 = k/2 . Similarly, since the line passes through (0, 0) and (k, 32), the slope of the line is equal to 32-0/k-0 = 32/k . Since each expression gives the slope of the same line, it must be true that k/2 = 32/k . k = 8 or k = −8. Therefore, of the given choices, only 8 could be the value of k. Choice C is correct.
y = a(x − 2)(x + 4) In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy-plane is a parabola with vertex (c,d). Which of the following is equal to d ? A) -9a B) -8a C) -5a D) -2a
The parabola with equation y = a(x − 2)(x + 4) crosses the x-axis at the points (−4, 0) and (2, 0). The x-coordinate of the vertex of the parabola is halfway between the x-coordinates of (−4, 0) and (2, 0). Thus, the x-coordinate of the vertex is −4+2 / 2 = −1. This is the value of c. To find the y-coordinate of the vertex, substitute −1 for x in y = a(x − 2)(x + 4) a(−1 − 2)(−1 + 4) = a(−3)(3) = −9a. Therefore, the value of d is −9a. Choice A is correct.
S(P) = 1/2P + 40 D(P) = 220 - p The quantity of a product supplied and the quantity of the product demanded in an economic market are functions of the price of the product. The functions above are the estimated supply and demand functions for a certain product. The function S (P) gives the quantity of the product supplied to the market when the price is P dollars, and the function D (P) gives the quantity of the product demanded by the market when the price is P dollars. How will the quantity of the product supplied to the market change if the price of the product is increased by $10 ? A) The quantity supplied will decrease by 5 units. B) The quantity supplied will increase by 5 units. C) The quantity supplied will increase by 10 units. D) The quantity supplied will increase by 50 units.
The quantity of the product supplied to the market is given by the function S(P) = 1/2P + 40. If the price P of the product increases by $10, the effect on the quantity of the product supplied can be determined by substituting P + 10 for P as the argument in the function. This gives S(P + 10) = 1/2 (P + 10) + 40 1/2 P + 45 this shows that S(P + 10) = S(P) + 5. [ Why? (+5 because it was originally 1/2p + 40 now its 1/2P + 45 so there was an increase of 5) ] Therefore, the quantity supplied to the market will increase by 5 units when the price of the product is increased by $10. Choice B is correct.
triangle sum theorem
The sum of the measures of the interior angles of a triangle is 180 degrees
In triangle ABC, the measure of ∠B is 90°, BC = 16, and AC = 20. Triangle DEF is similar to triangle ABC, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side of triangle DEF is 1 3 the length of the corresponding side of triangle ABC. What is the value of sin F ?
Triangle ABC is a right triangle with its right angle at B. By the Pythagorean theorem, AB = squarerootof20^2 − 162^2 squarerootof400 − 256 squarerootof144 = 12. Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of angle F equals the measure of angle C. Thus, sinF = sinC. From the side lengths of triangle ABC, sinC = opposite/hypotenuse AB/AC =12/20 = 3/5 Therefore, sinF = 3/5 The correct answer is 3/5 or .6
g(x)=ax^2+24 For the function g defined above, a is a constant and g(4)=8. What is the value of g(-4) ? A)8 B)0 C)-1 D)-8
We can substitute 4 for x and 8 for g(x) into the equation to calculate a, then evaluate g(-4). x= 4 g(x) = 8 8=a(4)^2+24 -1=a now we can do g(-4)= -1(-4)^16+24 =8 A
exponential functions can show growth or decay growth y = ___ b > ___ percentages say it grows by 5% every year ___ decay ___ < b < ___ percentages say it decays/decreases by 5% every year ___
a(b)^x 1 b = 1.05 0; 1 b= .95
Proportions where a fraction equals a fraction
a/b = c/d a:b = c:d bc=ad
right triangle formula
a^2+b^2=c^2
Find the linear equation (equation of the line)
always do y-y1 = m(x-x1)
complements supplements
any two angles added together to equal 90 degrees any two angles added together to equal 180 degrees
distance =
average speed x time
Pythagorean Theorem
a²+b²=c²
X^2 - 4x + 3 find the product of the solutions
c/a 3/1 = 3
arc measure is equivalent to ___ and double the ___
central; inscribed (point from circle not center)
rate of change formula
change in value/time
with radical equations, you always gotta
check for extraneous answers whatever your x's equal, plug them in to make sure they work
heres an example for the i things Which of the following complex numbers is equal to (5 + 12i) - (9i^2 - 6i); for i = the square root of -1 ? A) -14 - 18i B) -4 - 6i C) 4 + 6i D) 14 + 18i
combine like terms and remember that i^2 = -1 5 + 12i + 9 + 6i 14 + 18i
8^2/3
cuberootof 8^2 = x cuberootof 64 = x 4 = x OR (cuberoot of 8)^2 = x 2^2 = x 4 = x
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Laura can swim 2 laps in the local swimming pool every 1.5 minutes. At this rate, how many minutes does Laura need to swim 12 laps?
d=rt 2 = r(1.5) r = 4/3 d=rt 12 = 4/3(x) 9 OR!!! 2/1.5 = 12/x 2x = 18 x = 9
Rates
distance = rate * time
What is the ratio of a:b? 3a=7b
divide by 3 a = 7b/3 divide by b a/b = 7/3
i^9 = i^10 = i^8 =
divide by 4; 4 goes in two times with a remainder of one. go down one. i 4 goes in twice with a remainder of two. go down two. -1 anything divisible by 4 is 1
if x>3, which of the following is equivalent to 1 / 1/x+2 + 1/x+3 (end of fraction)
first give the fractions below the 1 common denominators (mult one by x+3 and the other by x+2) 1 / x+3/x^2+5x+6 + x+2/x^2+5x+6 add them 1 / 2x+5 / x^2+5x+6 when you have a fraction in the denominator, you can multiply the numerator by the fraction's reciprocal so 1 * x^2+5x+6 / 2x+5 which would be answer B but I didnt want to write out all of the choices but yea
What is the solution set of the equation above? square root of 2x + 6 (end square root) + 4 = x + 3 A) -1 B) 5 C) -1, 5 D) 0, -1, 5
first make equation look like square root of 2x + 6 = x-1 then square both sides 2x+6 = x^2 -2x +1 make equal to zero to factor then x^2 -4x -5 = 0 (x-5)(x+1) = 0 x= 5, x= -1 CHECK FOR EXTRANEOUS GO BACK TO ORIGINAL EQUATION -1 doesn't work. the answer is just 5.
2x+6/(x+2)^2 - 2/x+2 the expression above is equivalent to a/(x+2)^2, where a is a positive constant and x does not equal -2. What is the value of a?
first we find common denominators multiply by (x+2) to the 2/x+2 now we have 2x+6/(x+2)^2 - 2x+4/(x+2)^2 we can combine the fractions now since they have the same denominator we have 2/(x+2)^2 look back at the equation of a/(x+2)^2 we are looking for a. therefore, a=2
When you divide or multiply both sides by a negative, you must ___
flip the inequality symbol
slope positive ___ negative ___ slope that is flat like ---- is ___ slope that is straight up like | is ____
increasing decreasing slope of 0 no slope or undefined
yint is the ____ value xint is the point of ____ vertex is either the ___ or ___ point
initial no value min; max
linear and quadratic systems y = x^2 - 6x + 3 y = -2x + 3 the solutions are the points of ____ find where they interect
intersections - 2 x + 3 = x^2 - 6x + 3 x^2 - 4x = 0 x(x-4) = 0 x= 0; plug in to find y; 3 x = 4; plug in to find y; -5
graph y less than or equal to x + 1
line's yint is at 1 with a slope of 1; / ; the line is solid NOT DOTTED LINE; since y is less than, we shade everything below the line
angle of rectangle
lw
rectangular prism volume
lwh
volume of rectangular prism
lwh
h=3a+28.6 A pediatrician uses the model above to estimate the height h of a boy, in inches, in terms of the boy's age a, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, of a boy's height each year? A)3 B)5.7 C)9.5 D)14.3
make a = 1 the height increases by 3 or make a= 2 and then a= 3 a 2yr old's height would be 34.6 and the 3yr old's height would be 37.6 find the difference. the difference is 3 every year, the height increases by 3 A
vertex the ___ value (__,___)
max or min (depending on whether or not the parabola is upside down or rightside up) h; k
data inferences margin of error 72 with a margin of error of 5 confidence level is the ____ margin of error gets bigger, Confidence level gets ___ margin of error gets smaller, Confidence level gets ___
means that the true value is within 5 of 72 67 to 77 confidence that your answer is within the margin of error bigger smaller
left off
minute 13
rationalize the denominator 3 / radical5
multiply both sides by radical5 3 * radical5 / 5
rationalize the denominator 3 / 4+radical5
multiply both sides by the conjugate (flip the sign) so multiply both sides by 4 - radical5 12 - 3 * radical5 / 16 - 5 (you had to foil the denominator with the 4 - radical5 but the middle parts cancelled out which is why we got 16 - 5) 12 - 3 * radical5 / 11
converting degrees to radians
multiply the degree by π/180
converting radians to degrees
multiply the radian by 180/π
Area of a circle: Circumference of a circle:
pi r^2 2 pi r
x-1/2a = 0 If x = 1 in the equation above, what is the value of a?
plug in 1 for x 1 - 1/2a = 0 1= 1/2a 2 = a the answer is 2
x+1 = 2/sqrt of 2 in the equation above, which of the following is a possible value of x+1? A) 1- sqrt of 2 B) sqrt of 2 C) 2 D) 4
plug in numbers sqrt2 = 2/sqrt2 remember power over root? 2^1/2 = 2^1/2^1/2 we can subtract exponents 2^1/2 = 2^1/2 (because 1 - 1/2 = 1/2) so B is the correct answer
sin(120) sin(195) tan(220) tan(330) cos(335) cos(135)
positive negative positive negative positive negative
similar means
proportionate same shape, different size
2000:862 2010:846 The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population and year is linear, which of the following functions P models the population of Greenleaf t years after 2000? A) P(t) = 862-1.6t B) P(t) = 862 - 16t C) P(t) = 862+16(t-2000) D) P(t) = 862-1.6(t-2000)
rate of change = change in population/change in years 2010-2000= 10 = t 846-862/10 -16/10 =-1.6 P(t) = 862-1.6t good luck lol here's my thoughts: you already calculated the 2000 so why would you add that back into the equation like D and C idk though but there's an explanation
Mr. Kohl has a beaker containing n milliliters of solution to distribute to the students in his chemistry class. If he gives each student 3 milliliters of solution, he will have 5 milliliters left over. In order to give each student 4 milliliters of solution, he will need an additional 21 milliliters. How many students are in the class? A) 16 B) 21 C) 23 D) 26
say x is the number of students n = 3x + 5 n = 4x - 21 x = 26 the answer is D
sin(x) = cos(90-x)
sin(10) = cos(80) sin(30)=cos(60) they all add up to equal 90
traignle shit why
sin(x) = cos(90-x) sin(30)=cos(60) sin(45)=cos(45) sin(0)=cos(90)
SOHCAHTOA
sin=opp/hyp cos=adj/hyp tan=opp/adj
graph y > -2x
since its > and not an "or equal to", we use a dotted line line goes through origin, with a slope of -2; \; since y is greater than, we shade everything above the line
the mean score of 8 players in a basketball game was 14.5 points. If the highest individual score is removed, the mean score of the remaining 7 players becomes 12 points. What was the highest score? A) 20 B) 24 C) 32 D) 36
so, we need to know what the mean is mean = sum of all scores / # of players we can use this formula to find the total score for 8 players and the total score for 7 players; and when we subtract those totals, all that will be left will be the highest score that was removed according to the story. 14.5 = x/8 x= 116 12 = x/7 x = 84 116-84 = 32 the answer is C: 32
sqrt of 32x^3y^6z^9 simplify
sqrt of (2)(16)(x)(x^2)(y^6)(z)(z^8) 4xy^3z^4 sqrtof 2xz
i =
square root of -1
yint is the ____ point at (____, y) throwing a ball in the air, the yint would be ____
starting 0 your starting point; how high you were above the ground when you threwthe ball
In the xy-plane, the point (p, r) lies on the line with equation y = x + b where b is a constant. The point with coordinates (2p, 5r) lies on the line with equation y = 2x + b. If p does not equal 0, what is the value of r/p? A) 2/5 B) 3/4 C) 4/3 D) 5/2
substitute (p,r) into the y = x + b r=p+b substitute (2p, 5r) into y = 2x+b 5r=4p + b We can subtract each side of the equation r=p+b from that of 5r=4p+b 5r - r = (4p + b) - (p+b) 4r=3p divide both sides by p (we wanna get r/p) 4r/p = 3p/p 4 * r/p = 3 r/p = 3/4 the answer is 3/4
Two lines; see where they intersect ex) y = 2x + 1 y = -2/3x - 7
substitution or elimination work 2x + 1 = -2/3x - 7 6x + 3 = -x -21 8x = -24 x= -3 y = 2(-3) + 1 y = - 5 (-3, -5)
linear vs exponential x 1 2 3 4 5 6 y 3 6 9 12 15 18 x 1 2 3 4 5 6 y 3 6 12 24 48 96
the first one is linear -as x goes up by 1, y goes up by 3 the second one is exponential -as x goes up by 1, y goes up by doubling
A photocopy machine is initially loaded with 5000 sheets of paper. The machine starts a large job and copies at a constant rate. After 20 minutes, it has used 30 percent of the paper. Which of the following equations models the number of sheets of paper, p, remaining in the machine m minutes after the machine started printing? A) p = 5000 -20m B) p = 5000-75m C) p = 5000(0.3)^m/20 D) p = 5000(0.7)^m/20
the machine used 30% of 5000 or 1500 sheets in 20 minutes. 5000-1500 = 3500 3500 sheets remained at 20 minutes. now plug in and test. lets try B. 3500 = 5000 - 75(20) 3500=3500 yay the answer is B
In a right triangle, one angle measures x degrees, where sinx =4/5. What is cos(90-x)?
this has to do with sig sinx = cos(90-x) the answer is 4/5
Function Notation f(x) = 3x+7 g(x) = 5x h(x) = x^2 + 3 (g * h )(-1) = ____ (h o g)(-2)
this means that -1 is going in for g and h, and we are multiplying both of those values (-5)(4) -20 is the answer this means h(g(-2)) first, plug -2 for g and then plug g for h h(-10) 103 is the answer
x^7 * x^4 (SAME BASE; xs are the same #) = ____
x^ 7+4 x^11
x^7 / x^4 (again same base obviously)
x^ 7-4 x^3
In the xy-plane, the line determined by points (2,k) and (k, 32) passes through the origin. Which of the following could be the value of k? A) 0 B) 4 C) 8 D) 16
we know that the line passes through the origin so we have (0,0) we can find the slopes m= y2-y1/x2-x1 m= 32-0/k-0 m= 32/k m= k-0/2-0 m= k/2 theyre of the same line so the slopes would be the same therefore, k/2 = 32/k k^2 = 64 k= +- 8 therefore, the answer is C
Structure in expressions a+b/c-d - a+b/d-c simplify
we wanna find common denominators factor out a negative one from d-c to make the denominators the same a+b/c-d - -a-b/c-d 2a + 2b / c-d
x^2 + 4x + y^2 - 8y = -11 find center and radius
we wanna get it to the circle formula we have to complete the square x^2 + 4 x + 4 + y^2 - 8y + 16 = - 11 + 4 + 16 (x+2)^2 + (y-4)^2 = 9 center (-2,4) radius 3
h = -4.9t^2 + 25t the equation above expresses the approximate height h, in meters, of a ball t seconds after it is launched vertically of 25 meters per second. After approximately how many seconds will the ball hit the ground? A) 3.5 B) 4.0 C) 4.5 D) 5.0
well its like a parabola like a bouncing ball animation one hump we can calculate the distance between the two zeroes to find how many seconds we want h = 0 0 = -49t^2 + 25t t(-4.9t) + 25 = 0 one solution is that t outside of the () =0 because then 0 = 0 another is if the t inside the () = 0 because again, anything times 0 = 0. so, find what t inside () is if the () = 0 -4.9t + 25 = 0 -4.9t = - 25 t= 5.10 the answer would be D QUICK WAY 25/ -4.9 if theyre asking the same question lol
x^2 - 6x + 8 complete the square to solve for x
x^2 - 6x = -8 take half of 6 and then square it, and add to both sides x^2 - 6x + 9 = -8 + 9 now, its factorable (x-3)(x-3) = 1 (x - 3)^2 = 1 x - 3 = 1 x - 3 = - 1 x= 4; 2 (obviously you should have factored but thats how you complete the square)
Which of the following is an equivalent form of the equation of the graph shown in the xy-plane above, from which the coordinates of vertex A can be identified as constants in the equation? the equation is y=x^2 - 2x - 15 A) y = (x+3)(x-5) B) y= (x-3)(x+5) C) y= x(x-2)-15 D) y= (x-1)^2 - 16
when we want to find the vertex of a parabola, we complete the square however theyre just asking for an equation here and the answer is D because thats the only one in vertex form vertex form: y= a(x-h)^2 + k vertex: (h,k) looking at D, we know that k is -16 and h is 1 (1, -16) a is 1 because theres nothing outside the parenthesis in D so yeah
cuberootof x^3 = cuberootof x^6 =
x 2 (you basically just divide the exponent under the radical by 3)
2x^2 - 4x - 16 = 0 solve for x
x = -b +- sqrt b^2 - 4ac/ all over 2a 4 +- sqrt 16+128/4 4+- sqrt 144 / 4 4 + 12 / 4 = 4 4 - 12 / 4 = - 2 x = 4; -2
How to find the vertex when not in vertex form? 2x^2 + 4x - 6 find the vertex
x = -b/2a -4/2(2) x = - 1 plug -1 in to find y 2(-1)^2 + 4(-1) - 6 2 - 4 - 6 = - 8 (-1 , -8)
9 is what percent of 45?
x%(45) = 9 =.20 =20%
what is 12% of 50?
x/50 = 12/100 x=6 OR using percent equation .12(50) = x x = 6
Radicals sqrtof x^6 =
x^3 (you basically just divide the exponent under the radical by 2)
x^3 + b^3 = (x+b)(x^2 - xb + 9) find b
x^3 + b^3 = x^3 - x^2b + 9x + x^2b - b^2x + 9b x^3 + b^3 = x^3 + 9x * b^2x + 9b b^3 = 9b they are both the constants b^2 = 9 b = 3
write y = -2/3x + 7 in standard form
y = -2/3x + 7 2/3x + y = 7 2x+ 3y = 21
y = 2x^2 + 4x - 6 in intercept form is
y = 2(x+3)(x-1) the xints of the parabola are (-3,0) and (1,0)
exponential functions standard form a is ___ b is ___
y = a(b)^x initial value growth factor x is the time (ex: every hour)
standard form highest degree is ___
y = ax^2 + bx + c 2
point slope form? Example problem: example) y - 3 = 2/5(x-2) slope? what is the point?
y-y1=m(x-x1) m= slope (x1,y1)= any point on this line example) y - 3 = 2/5(x-2) slope= 2/5 (2, 3) coordinate
Vertex form
y=a(x-h)^2+k
y = 2x^2 + 4x - 6 the -6 is the ___
yint (0, - 6 )
y lessthanequalto -15x +3000 y lessthanequalto 5x in the xy-plane, if a point with coordinates (a,b) lies in the solution set of the system of inequalities above, what is the maximum possible value of b?
you can graph them but youll find that its the intersection of these two lines we can set them equal, solve for x, and then solve for y -15x + 3000 = 5x 3000 = 20x x = 150 plug in x y lessthanequalto 750 and 750 for both equations the answer is 750
If a^b/4 = 16 for positive integers a and b, what is one possible value of b?
you literally have to guess what to a power could equal 16? 4^2 could. what in b/4 = 2? 8 8 is a possible value for b
xint is the point of ____ (x,____) throwing a ball in the air, the xint would be ___
zero value 0 when it hits the ground/gets to the height of zero
100x^2 - 25
(10x+5)(10x-5) just be on the lookout itll help you go faster
a^2 - b^2 =
(a+b)(a-b) just remember this itll help you go faster
If a^-1/2 = x, where a>0, what is a in terms of x? A) radical x B )-radicalx C) 1/x^2 D) -1/x^2
(a^m)^n = a^m*n a^-n = 1/a^n a^m/n = nradical a^m first raise both sides by the reciprical (-2) (a^-1/2)*^-2 = x^-2 a = 1/x^2 the answer is C
The glass pictured to the left can hold a maximum volume of 473 cubic centimeters, which is approximately 16 fluid ounces. Jenny has a pitcher that contains 1 gallon of water. How many times could Jenny completely fill the glass with 1 gallon of water? (1 gallon = 128 fluid ounces). A) 16 B) 8 C) 4 D) 3
128/16 = 8 the answer is B
If 16+4x is 10 more than 14, what is the value of 8x? A)2 B)6 C)16 D)80
16+4x=24 4x=8 x=2 plug into 8x 8(2) 16 the answer is c
a worker uses a forklift to move boxes that weigh either 40 pounds or 65 pounds each. Let x be the number of 40 pound boxes and y the number of 65 pound boxes. The forklift can carry up to either 45 boxes or a weight of 2400 pounds. Which of the following system of inequalities represents the relationship?
40x + 65y lessthanequalto 2400 x+y lessthanequalto 45
Maria plans to rent a boat. The boat rental costs $60 per hour, and she will also have to pay for a water safety course that costs $10. Maria wants to spend no more than $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Maria can rent the boat?
60x + 10 lessthanorequalto 280 60x lessthanorequalto 270 x lessthanorequal to 270/60 4.5 whole number so we gotta go down to 4. the answer is 4`
A quality control manager at a factory selects 7 lightbulbs at random for inspection out of every 400 lightbulbs produced. At this rate, how many lightbulbs will be inspected if the factory produces 20000 lightbulbs? A) 300 B) 350 C) 400 D) 450
7/400 = x/20000 x= 350 B These next ones are all from a video showing what little over 50% of the math SAT section is on. Even the easy ones like this will be included in my quizlet why am i explaining this i cant curse anymore in quizlet why cant i
A customer's monthly water bill was $75.74. Due to a rate increase, her monthly bill is now $79.86. To the nearest tenth of a percent, by what percent did the amount of the customer's water bill increase? A) 4.1% B)5.1% C) 5.2% D)5.4%
79.86 - 75.74 = 4.12 4.12 was the increase in price so if we divide 4.12 by the old bill of 75.74, we'd get 5.4% D is our answer
Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 dollars per gallon, which equation can Alan use to determine how many fewer average miles m he should drive each week? A) 25/4m = 95 B) 25/4m=5 C) 4/25m=95 D)4/25m=5
A bunch of conversion mumbo jumbo. No clue tbh. Im assuming so if you dont wanna take a gamble leave. Im assuming that the units you want should be on the top of the fraction. here we go. we know that he wants to go something miles to get 5 dollars. we know that the end of the equation should be m=5 so A and C are cancelled. now we need to figure out what goes on top. no clue. im going with my assumption so... 4/25m = 5 4 dollars to 5 dollars yknow? thats why im asusming the 4 is on top. D is the correct answer
isosceles triangles
A triangle with two sides of the same length and two angles of the same measure.
4(80+n)=(3k)n In the equation above, k is a constant. For what value of k are there no solutions to the equation?
There will be no solution if the n-terms on the left cancel with the n-terms on the right, leaving a false statement such as 1=0 This can only happen if 4n=(3k)n; that is, if 4 = 3k solve for k k = 4/3
In the figure above, what is the value of x? (the figure is four sided; one angle is 45 degrees while the others are all labeled x)
In a 4-sided figure, all of the interior angles will add up to 360. so, subtract 360 from 45 and then divide by 3 105=x
Which of the following is an equation of a circle in the xy-plane with center (0,4) and a radius with endpoint (4/3, 5)? A) x^2 + (y-4)^2 = 25/9 B) x^2 + (y+4)^2 = 25/9 C) x^2 + (y-4)^2 = 5/3 D) x^2 + (y+4)^2 = 3/5
In the standard form equation of a circle in the xy-plane, (x-h)^2+(y-k)^2=r^2 are the coordinates of the center and r is the radius. we need to find r and square it distance formula: d= square root of (x2-x1)^2 + (Y2-y1)^2 r= square root of (4/3 - 0)^2 + (5-4)^2 r= square root of 25/9 r= 5/3 but remember, we need to square it to fit back into the standard form equation of a circle so the answer is 25/9 The answer is A x^2 + (y-4)^2 = 25/9
x^2 + 10x +31 what are the sums of the solutions to the equation?
QUICK WAY -b/a -10/1 = -10 -10 is the sum
The graph above shows the positions of Paul and Mark during a race. Paul and Mark each ran at a constant rate, and Mark was given a head start to shorten the distance he needed to run. Paul finished the race in 6 seconds, and Mark finished the race in 10 seconds. According to the graph, Mark was given a head start of how many yards? (Mark's yint is at 18 on the graph and Paul's is at 0) A)6 B) 12 C) 18 D) 24
Since mark's yint is at 18 and paul's is at zero, mark was given an 18 yd headstart because mark's start position (18) - paul's start position (0) = 18 so the answer is 18
the graph of a line in the xy-plane passes through the point (1,4) and crosses the x-axis at the point (2,0). The line crosses the y-axis at the point (0, b). What is the value of b?
So we have three points (2,0) (1,4) (0,b) the distance from 2,0 to 1,4 is (-1, +4) the distance from 1,4 to (0, b) is the same b=8
(x - 6)^2 + (y + 5)^2 = 16 In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates (10, -5). If line PQ is a diameter of the circle, what are the coordinates of point Q? A) (2, -5) B) (6, -1) C) (6, -5) D) (6, -9)
So, we have a circle. Point P is somewhere on the circle at (10, -5). We want to find point Q. We know PQ together is the circle's diameter. So, the line of PQ would have to pass through the center of the circle somewhere. Looking at the equation above, we know the center of the circle's coords. The x center would be 6 and the y center would be -5. Therefore, the center of the circle is (6, -5). We need to find point Q now. From point P (10, -5) to the center (6, -5) the difference is (4, 0). If the center of the circle is an even distance from point P as it is to Q (which it is), we'd have (2, -5) I dont know the technical terms lol but it looks like you just subtract the center (6, -5) by the difference (4, 0) to get (2, -5). The answer is A), (2, -5)
exterior angle theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
A granite block in the shape of a right rectangular prism has dimensions 30 centimeters by 40 centimeters by 50 centimeters. The block has a density of 2.8 grams per cubic centimeter. What is the mass of the block, in grams? (Density is mass per unit volume.) A) 336 B) 3360 C) 16800 D) 168000
Volume = lwh (30)(60)(40) volume = 60000 mass = density * volume 2.8 * 60000 = 168000 the answer is D
I just wanna remember how to do this... A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents? A)30 B)20 C)19 D)18
We wanna set up two equations. One for the total of people and one for the total of tents. 2x + 4y = 202 x + y = 60 We wanna find how many two-person tents so for me, I need to find x. Therefore, in order to find x, I need to kick y's a5s out. Multiply x + y = 60 all by -4 to cancel the y's. 2x + 4y = 202 -4x - 4y = -240 -2x = -38 x = 19 C) x = 19 is the correct answer
g(x) = ax^2 + 24 For the function g defined above, a is a constant and g(4) =8. What is the value of g(-4)? A) 8 B) 0 C) -1 D) -8
What I did was solve for a since I know g(4)=8 I did this 8 = a(4)^2 + 24 a= -1 then I solved for g(-4) g(-4) = -1(4)^2 + 24 g(-4) = 8 the answer is 8 quick way: this is an even function so if g(4) = 8, g(-4) = 8 also an even function always has an even power of x (ex x^2) an odd function would only have odd powers of x (ex x^3) just do my way just in case it took like one minute please
Which of the following is a value of x for which the expression -3/x^2 +3x -10 is undefined? A) -3 B) -2 C) 0 D) 2
a fraction is undefined when its denominator is 0 plug answers in the denom and solve to see if any equal 0 2^2 + 3(2) - 10 = 0 x=2 D is the correct answer
the volume of right circular cylinder A is 22 cubic cm. what is the volume in cubic cm of a right circular cylinder with twice the radius and half the height of cylinder A? A) 11 B) 22 C) 44 D) 66
amanda's sexy shortcut. if its twice the radius and half the height (or 2r halfh), just double the volume. 22x2 = 44 Actual way we should learn i guess... volume= pir^2h 22=pir^2h we want twice the radius and half the height so 22= pi(2r)^2 * (1/2 h) square the 2 in (2r)^2 22= pi (4r^2) (1/2 h) simplify 22= 2pir^2h look at original volume formula. its pir^2h. so we just multiply the original volume by the 2 in front of our new equation. so 22x2 = 44 the answer is c
for circle problems always remember
arc length/circumference = inner angle/360 circumference = 2pir area = pi(r^2)
On April 18, 1775, Paul Revere set off on his midnight ride from Charlestown to Lexington. If he had ridden straight to Lexington without stopping, he would have traveled 11 miles in 26 minutes. In such a ride, what would the average speed of his horse have been, to the nearest tenth of a mile per hour?
avg speed = distance/time 1 hour = 60 mins convert 11 miles over 26 minutes to mph avgspeed= 11miles/26mins * 60mins/1hr the minutes cancel out = 660miles/26hr = 25.38 mph round to nearest tenth = 25.4
oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000? A) f(t) = 21/130 t +4 B) f(t) = 19/130 t +4 C) f(t) = -21/130 t +4 D) f(t) = -19/130 t +4
constant rate means a line if a line is in decrease, the slope is negative. A and B are canceled. (remember slope is rise/run) lets find the slop. m= y2-y1/x2-x1 m= 1.9 - 4/2013-2000 (WHY? we started with 4 mil barrels and ended with 1.9; so y1=4 and y2=1.9; same thing with the years but with x) m= -2.1/13 decimals in fractions are gross so multiply to get rid of them. we can multiply by 10/10. m= 21/130 that matches C C is the correct answer
A motor powers a model car so that after starting from rest, the car travels s inches in t seconds, where s= 16t radical t. Which of the following gives the average speed of the car, in inches per second, over the first t seconds after it starts?
distance = average speed x time distance/time = average speed 16t radical t all over (/)t = average speed simplify 16 * radical t = average speed
The distance traveled by Earth in one orbit around the Sun is about 580,000,000 miles. Earth makes one complete orbit around the sun in one year. Of the following, which is closest to the average speed of Earth, in miles per hour, as it orbits the Sun? A) 66,000 B) 93,000 C) 210,000 D) 420,000
distance = avg speed * time avg speed = distance/time x= 580,000,000/365 then /24 x= 66,210.04566 x= 66,210 A is the correct answer
A line in the xy-plane passes through the origin and has a slope of 1/7 . Which of the following points lies on the line? A) 0,7 B) 1,7 C) 7,7 D) 14,2
do slope intercept form with the two points (0,0) and one of the four multiple choice options to find which one has a slope of 1/7 y2-y1/x2-x1 try (14,2) with (0,0) 2-0/14-0 = 1/7 D is the correct answer
which of the following is an equation of a circle in the xy-plane with center (0,4) and a radius with endpoint (4/5,5)? A) x^2 + (y-4) = 25/9 B) x^2 + (y+4) = 25/9 C) x^2 + (y-4) = 5/3 D) x^2 + (y+4) = 5/3
equation of a circle (x-h)^2 + (y-k)^2 = r^2 center: (h,k), radius: r (x-0)^2 + (y-4)^2 = r^2 x^2 + (y-4)^2 = r^2 we need to find the radius now. use distance formula d= sqrtof (x2 - x1)^2 + (y2-y1)^2 (4/3-0)^2 + (5-4)^2 16/9 + 1 16/9 + 9/9 radius = sqrt25/9 r= 5/3 but the formula needs it to be squared so the answer is x^2 + (y-4)^2 = 25/9 the answer is choice A
f(x) = 2^x + 1 the function f is defined by the equation above. which of the following is the graph of y = - f(x) in the xy-plane?
f(x) = 2^x + 1 or y = 2^x + 1 lets find y so plug in 0 for x y= 2^0 + 1 y= 2 (remember #^0 = 1) lets look at the y= - f(x) they put a negative in front of the equation so it'd be flipped and going downward y would now equal -2 so find the image where it is flipped over the x axis (going downward) and whose yint is -2.
The expression x^-2 * y^1/2 / x^1/3 * y^-1 ,where x>1 and y>1, is equivalent to which of the following? A) squareroot of y / cuberoot of x^2 B) y * sqrt of y / cuberoot of x^2 C) y * sqrt of y / x * sqrt of x D) y * sqrt of y / x^2 * cuberoot of x
first we fix up exponents. turn the negatives to positives. they are losing in a volleyball game and want to switch sides. so.. y^1 * y^1/2 / x^2 * x^1/3 now we take care of the fractional exponents. remember its power/root. (for example, x^1/5 would be 5sqrt of x^1) we'd have y * sqrt of y / x^2 * cuberoot of x the answer is D
g(x)=2x−1 h(x)=1−g(x) The functions g and h are defined above. What is the value of h(0) ?
first we need to find g(0) because we need to subtract it from h 2(0) - 1 -1 then we plug that into the second equation 1 - - 1 = 2 the answer is 2
if x>3, which of the following is equivalent to 1 / 1/x+2 + 1/x+3 ? A) 2x+5/x^2 +5x +6 B) x^2+5x+6/2x+5 C) 2x+5 D) x^2+5x+6
first we want common denominators with the x+2 and x+3 so what we do is multiply them by each other now we have 1 / x+3+x+2 / x^2 + 5x + 6 then we have 1 / 2x+5 / x^2+5x+6 now we do KCF 1 / 2x+5 * x^2+5x+6 = x^2+5x+6/2x+5 the answer would be B
If 2a/b = 1/2. What is the value of b/a? A) 1/8 B) 1/4 C) 2 D) 4
first write 2a/b = 1/2 we need to flip this to get b on top so we gotta flip both. b/2a = 2 we want b/a so we need to get rid of that 2 next to the a. multiply both sides by 2 b/a = 4 D is the correct answer you could also write 2a/b = 1/2 and cross multiply to get 4a=b 4=b/a 4
4∣6+2s∣−27≤−3 Which of the following best describes the solutions to the inequality shown above? A) -24 less than or equal to s less than or equal to 0 B) -6 less than or equal to s less than or equal to 0 C) no solution D)s lessthan or equal to 0 or s lessthan or equal to 3
get the absolute value alone |6+2s| less than or equal to 6 now put into two equations 6+2s less than or equal to 6 to get s less than or equal to 0 and then 6+2s greater than or equal to -6 to get s greater than or equal to -6 the answer is B: -6 less than or equal to s less than or equal to 0
VERY IMPORTANT REMEMBER; EASY POINTS i, i^2, i^3, i^4
i -1 -i 1
M = 1800(1.02)^t The equation above models the number of members, M, of a gym t years after the gym opens. Of the following, which equation models the number of members of the gym q quarter years after the gym opens? A) M = 1800(1.02)^q/4 B) M = 1800(1.02)^4q C) M = 1800(1.005)^4q D) M = 1800(1.082)^q
in one year there are 4 quarter years, so the number of quarter years, q, is 4 times the number of years, t, which gives q=4t. This is equivalent to t=q/4 and substituting this into the expression gives us M = 1800(1.02)^q/4 A is the correct answer
the line y = kx + 4, where k is a constant, is graphed in the xy-plane. If the line contains the point (c,d), where c does not = 0 and d does not = 0, what is the slope of the line in terms of c and d? A) d-4/c B) c-4/d C) 4-d/c D) 4-c/d
k is our slope. plug in c and d for x and y d=kc + 4 now solve for k k= d-4/c the answer is A
The function f is defined by a polynomial. Some values of x and f(x) are shown in the table above. Which of the following must be a factor of f(x). x y 0 3 2 1 4 0 5 -2 A) x-2 B) x-3 C) x-4 D) x-5
look for where y = 0 that would be when x = 4 so (x-4) is the answer or C
the expression 1/3x^2 - 2 can be rewritten as 1/3(x-k)(x+k), where k is a positive constant. What is the value of k? A) 2 B) 6 C) sqrt of 2 D) sqrt of 6
looking at the second equation, we see that the 1/3 is pulled out. lets do the same for the first. 1/3(x^2 - 2(3)) (if it were a 3 instead of 1/3, when factoring out it'd be 2/3. but since it is 1/3, a fraction, when dividing by a fraction we multiply by the reciprocal so 2/1 * 3/1 = 6) 1/3(x^2 - 6) lets factor this so it looks more like the second equation. 1/3 (x+ sqrt6)(x- sqrtr6) (remember x^2 - 4 is (x+2)(x-2); its actually (x+ sqrt4)(x-sqrt4) but 4 is a perfect square so we write 2. we can't do that for 6 so we just leave it like that) it matches the equation so now look where k is. sqrt6 = k; D is the correct answer
Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation P = 108 - 23d where P is the number of phones left and d is the number of days she has worked that week. What is the meaning of the value 108 in this equation? A) Kathy will complete the repairs in 108 days B)Kathy starts each week with 108 phones to fix C) Kathy repairs phones at a rate of 108 per hour D)Kathy repairs phones at a rate of 108 per day
make d = 0 P = 108 that means Kathy starts each week with 108 phones and d must be like she repairs 23 phones per hour or day or somethin because when 0 phones are fixed, 108 are left which means Kathy starts with 108 phones so B is the correct answer if that makes sense :)
-3x+4y=20 6x+3y=15 if (x,y) is the solution to the system of equations above, what is the value of x?
mult top by -3 mult bottom by 4 9x-12y=-60 24x+12y=60 33x=0 x=0
f(x) = x^3 - 9x g(x) = x^2 - 2x - 3 Which of the following expressions is equivalent to f(x)/g(x), for x>3 A) 1/x+1 B) x+3/x+1 C) x(x-3)/x+1 D) x(x+3)/x+1
plug in equations x^3 - 9x/x^2 - 2x - 3 now, simplify x(x^2 - 9)/(x-3)(x+3) simplify again x(x+3)(x-3)/(x-3)(x+3) cancel out x(x+3)/(x+1) the answer is D
2x^2 - 4x = t in the equation above, t is a constant. if the equation has no real solutions, which of the following could be the value of t? A) -3 B) -1 C) 1 D) 3
plug in the values one by one lets start with -3 2x^2 - 4x = -3 make equal to zero 2x^2 - 4x +3 = 0 lets do the sqrt of b^2 4ac because remember, if we get a negative, that means we have no real solutions sqrt of 16 - 4(2)(3) = sqrt of -8 theres a negaive therefore, -3 could be the value of t, making A correct
In the xy-plane, the graph of y = 3x^2 -14x intersects the graph of y=x at the points (0,0) and (a,a). What is the value of a?
plug in x for y since y=x x=3x^2-14x 3x^2-15x = 0 x^2 - 5x = 0 x(x-5) = 0 x = 0 x=5 since (0,0) is already a solution, 5 is the other solution that we were missing (a,a) therefore, the value of a is 5.
circle problem example: GIVEN: say arc is pi/3 we need to find the inner angle radius is 1
radius is 1 so circumference is 2pi so pi/3 / 2pi = x/360 multiply 2pi by the 3 above it pi/6pi cancel out the pi we have 1/6 = x/360 x= 60 the inner angle is 60 degrees
Which of the following is equivalent to 9^3/4 ? A) 3radical9 B) 4radical9 C) radical3 D) 3radical3
remember power over root 4radical9^3 we know that it aint A or B lets try to get a 3 in the radical instead as it is shown in both C and D (3^2)3/4 (remember 3^2 is still 9) =3^3/2 sqrt of 3^3 sqrt of 27 sqrt of 3 * sqrt of 9 3sqrtof3 so the answer must be D
y > 2x-1 2x > 5 Which of the following consists of the y coordinates of all the points that satisfy the system of inequalities above? y > 6 y > 4 y > 5/2 y > 3/2
see the 2x on the bottom? we wanna make that into 2x-1 like the one above. so we subtract 1 on both sides of that equation. 2x - 1 > 4. see how the 2x - 1 matches the one in the equation above? now, just tack on the y. y > 2x - 1 > 4 y > 4 as you can see, so the answer is b or y>4
b=2.35+0.25x c=1.75+0.40x In the equations above, b and c represent the price per pound, in dollars, of beef and chicken, respectively, x weeks after July 111 during last summer. What was the price per pound of beef when the price per pound of beef was equal to the price per pound of chicken? A) 2.60 B)2.85 C)2.95 D)3.35
set b and c equal to each other 2.35+0.25x = 1.75+0.40x x=4 we want to find the price of beef so plug in x for beef's equation b= 2.35 + .25(4) b= 3.35 D is the correct answer
how many liters of 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?
set it up like this its the percents of the solutions times the liters = the sum of the liters times the percent we want (15%) x(.25) + 3(.10) = (x+3)(.15) x= 1.5 liters
in the circle above, point A is the center and the length of the arc BC is 2/5 of the circumference of the circle. what is the value of x? (x is angle A and BC is the arc that is being 'eaten' by angle a)
since that arc (which is 2/5 of the circumference) is with the angle we want to find, they'd have the same ratio therefore 2/5 = x/360 cross multiply x=144
x^2 + 20x + y^2 + 16y = -20 the equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle? A) -20,-16 B) -10, -8 C) 10, 8 D) 20, 16
the equation of a circle is (x-h)^2 + (y-k)^2 = r^2 we gotta transform that equation into this one so we have to complete the square start with x^2 + 20x (x + 10)^2 - 100 (you divide that 20 by 2 to get 10 and then you square the 10 to get 100; if the equation was x^2 + 20x + 2, you'd have your answer look like this instead: (x+10)^2 - 100 + 2) then we do the y^2 + 16y (y+8)^2 - 64 write out the whole equation (x + 10)^2 - 100 + (y+8)^2 - 64 = -20 the center coordinates are (-10, -8); B is the correct answer QUICK WAY -b/2a x^2 + 20x + y^2 + 16y = -20 use x^2 + 20x -20/ 2(1) = -10 use y^2 + 16y -16/2(1)= -8 (-10, -8)
The vertex of the parabola in the xy-plane above is (0, c). Which of the following is true about the parabola with the equation y = -a(x - b^2) + c?
the vertex is (b,c) and the graph opens downward. how do I know? a is negative so its downward if it was -a(x+b^2)+c , the vertex would probably be (-b,c) im assuming no idea.
A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% percent are parents of students, 45% percent are teachers from the current high school, 25% percent are school and district administrators, and the remaining 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?
well it says the remaining 6 individuals are students. lets find the remaining percentage. 100% - 15% - 45% - 25% = 15% 15% of everyone invited = 6 so .15x = 6 x=40 therefore, 40 total people were invited. we wanna compare teacher to administrators so lets find them. .45 * 40 = 18 teachers .25 x 40 = 10 administrators 8 more teachers were invited than administrators.
c=5/9 (f-32) The equation above shows how a temperature F, measured in degrees Fahrenheit, relate to a temperature C measured in degrees Celsius. Based on this equation, which of the following must be true? 1) a temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 5/9 degree Celsius. 2) a temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit. 3) a temperature increase of 5/9 degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius. A) I only B) II only C) III only D) I and II only
tip: plug and chug 1) plug in some numbers to see c= 5/9(32-32) c=0 now lets increase the temperature as it says in 1) so lets see what c = when we plug in 33 (we're increases the temperature of Fahrenheit by 1 degree) c= 5/9(33-32) 1) is correct since 1 is correct, B and C can be cancelled out. We need to still see if 2) is correct though because we don't know if the answer is A or D. 2) c=5/9(f-32) isolate by multiplying by 9/5 which is actually 1.8. 1.8c=F-32 1.8c+32=F so if c=0, F=32 if we increase by a degree of celsius, make c=1 when c=1, F=33.8 2) is correct therefore, D is the correct answer
What is the sum of the solutions to (x - 6)(x + 0.7) = 0 ? A) -6.7 B) -5.3 C) 5.3 D) 6.7
turn it into x^2 - 5.3x - 4.2 = 0 we can do our shortcut -b/a --5.3/1 = 5.3/1 = 5.3 5.3 is our answer so C is correct
x 0 2 6 y -2 4 16 Some values of the linear function f are shown in the table above. What is the value of f(3)? A)6 B) 7 C) 8 D) 9
we can find the rate of change y2 - y1/x2- x1 lets choose two points (0, -2) and (2, 4) 4 -- 2/ 2 - 0 = 6/2 =3 the rate of change is 3 so (0, -2), (1, 1), (2, 4), (3, 7) when x = 3, y = 7 so the answer is B, 7
if 3x-y = 12, what is the value of 8^x/2^y?
we can rewrite 8^x to be 2^3x because the denominator is 2y x^a/x^b = x^a-b so its be 2^3x-y since 3x-y= 12 we know the answer is 2^12
Which of the following is equivalent to 4x^2+6x/4x+2 A)x B) x+4 C) x - 2/4x+2 D) x+1 - 2/4x+2
we gotta do long division for this one 4x^2+6x divided by 4x+2 4x goes into 4x^2 x times so put x on top. below 4x^2+6x put 4x^2 and 2x subtract youll have 4x. 4x goes into 4x one time so put a one on top (we have x+1 on top rn) then put a 4x on the bottom and a 2 subtract we have a -2. 4x cannot go into that so -2 is our remainder. we have x+1 - 2/4x+2 (remainder/divisor; divisor is what you divided by) D is the correct answer
2x^2 - 4x = t In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ? A) -3 B) -1 C) 1 D) 3
we gotta do the sqrt of b^2 - 4ac we want one with no solutions! remember if you get 0, there is 1 solution a negative, no solutions or any random non zero #, 2 solutions if you plug in -3 for t you'd get 2x^2 - 4x +3 = 0 do the sqrt b^2 - 4ac you'll get a negative under the radical which means that there are no solutions therefore -3 could be the value of t if there are no solutions! so A is the correct answer
y= x^2 + 3x - 7 y - 5x + 8 = 0 How many solutions are there to the system of equations above? A) 4 B) 2 C) 1 D) 0
we have a parabola and a line. how many solutions = where they intercept a line and parabola can intercept at most 2 times so A is canceled. we should solve for y and then set them equal to each other. y= x^2 + 3x - 7 y = 5x - 8 x^2 + 3x - 7 = 5x - 8 x^2 - 2x + 1 = 0 this is factorable. (x-1)(x-1) x=1; there is one solution. C is correct. ALTERNATIVE to factoring. you can do the sqrt of b^2 - 4ac if you get 0, there is 1 solution a negative, no solutions or any random non zero #, 2 solutions
which of the following could be the equation of the graph above? (the graph has like a parabola with some humps. it touches the xaxis 3 times)(theres a zero at -3, 0, and 2) A) y = x(x-2)(x+3) B) y= x^2 (x-2)(x+3) C) y= x(x+2)(x-3) D) y= x^2 (x+2)(x-3)
we know it has 3 zeros if it touches the xaxis 3 times. so y= (x+3)x(x-2) what about multiplicity? if the x had an odd exponent, it'd go through the x axis. it'd have an odd exponent if there is a sign change. but its touching the x axis; if there's no sign change it'd also have an even exponent apparantly so its gotta have an even exponent which is why we'd go with x^2 rather than x. the question is asking what COULD not IS. so the answer would be B) since there's no sign change for the x so it'd have an even exponent.
the angles shown above are acute and sin(a) = cos(b). If a = 4k-22 and b= 6k -13, what is the value of k? A) 4.5 B) 5.5 C) 12.5 D) 21.5
we know sin(30) = cos(60) sin(45)=cos(45) sin(0)=cos(90) in all of these examples, they all add up to equal 90 so, a + b = 90 therefore we can write 4k - 22 + 6k - 13 = 90 10k = 125 k= 12.5 the answer is C
if f(x)= 5x^2 - 3 and f(x+a) = 5x^2+30x+42, what is the value of a? A) -30 B) -3 C) 3 D) 30
we wanna add a to f(x) to make it also f(x+a) so whenever we see an x in that equation, we also add +a 5(x+a)^2 - 3 f(x+a) = 5x^2 + 10ax + 5a^2 - 3 now since both are f(x+a), we can make them equal to eachother 5x^2 + 10ax + 5a^2 - 3 = 5x^2+30x+42 10ax + 5a^2 - 45 = 5x^2 + 30x see that 30x at the end? and see that 10ax. that 10ax was originally 30x. so, what is 30/10? 3 the answer is 3; C idk please don't kill the messager
kx-3y=4 4x-5y=7 In the system of equations above, k is a constant and x and y are variables. For what value of k will the system of equations have no solution? A) 12/5 B) 16/7 C) -16/7 D) -12/5
we want no solution a solution is the intersection of two lines. therefore, when lines are parallel, they dont intersect; and if they dont intersect, there is no solution lines are parallel when they have the same slope however, if the lines are RIGHT on top of each other, theyd intersect at EVERY point and that is INFINITE SOLUTIONS (they have the same slope AND yint) therefore, we just want our lines to have the same slope and different yints. turn them into y=mx+b y = k/3 x - 4/3 y= 4/5 x - 7/5 they have different yints so yay. we want the slopes to be the same though so k/3 must = 4/5 k/3=4/5 5k = 12 k = 12/5 the answer is A
Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p? A) 0.88p B) p/0.88 C) (0.8)(1.08)p D) p/(0.8)(1.08)
we're saying a 20 percent DISCOUNT, not 20 percent off!!!! so its not (.20)x its actually (.8)x we're also adding on an 8 percent sales tax (if we were taking off 8 percent, we'd write .92) but we're adding an 8 percent sales tax, so we'd write 1.08. so (1.08)(.8)x = p p/(1.08)(.8) = x the answer is D
y=a(x-2)(x+4) in the quadratic equation above, a is a nonzero constant. the graph of the equation in the xy-plane is a parabola with the vertex (c,d). which of the following is equal to d? A) -9a B) -8a C) -5a D) -2a
well we know that the zeros are -4 and 2 we can find the vertex with those a parabola is symmetrical so the vertex is in the middle so if we average the -4 and 2 -4+2 = -2/2 = -1 we know the the x coordinate of our vertex is -1; c = -1 we want to find d though so plug in -1 to the equation y = a(x-2)(x+4) y= a(-1-2)(-1+4) y = a(-3)(3) y= -9a therefore, d = -9a the answer is A
x 1 2 3 y 11/4 25/4 39/4 Which of the following equations relates y to x for the values in the table above? A) y = 1/2 * (5/2)^x B) y= 2 * (3/4)^x C) y= 3/4x + 2 D) y = 7/2x - 3/4
well, as x increases by 1, y increases by 14/4 or 7/2 D is the only one with 14/4 or 7/2 you can also plug x and y into the equation to see if it works y = 7/2(1) - 3/4 y= 14/4(1) - 3/4 y = 14/4 - 3/4 y = 11/4 when 1 = x, y = 11/4 so D is the correct answer
A customer paid $53 for a jacket after a 6% sales tax was added. What was the price of the jacket before the sales tax was added? A) 47.60 B) 50.00 C) 52.60 D) 52.84
x(1.06) = 53 why 1.06? Its like 100% of the price plus the sales tax. when we divided 53 by that, we'd find the price before the sales tax. 50; the answer is B
Between 1497 and 1500, Amerigo Vespucci embarked on two voyages to the New World. According to Vespucci's letters, the first voyage lasted 43 days longer than the second voyage, and the two voyages combined lasted a total of 1003 days. How many days did the second voyage last? A) 460 B) 480 C) 520 D) 540
x+y = 1003 y+43 = x make them both solve for x and then set equal to one another x= 1003 - y x= y + 43 1003-y = y + 43 1003 = 2y + 43 960 = 2y 480 = y
In the xy-plane, the parabola with equation y=(x-11)^2 intersects the line with equation y=25 at two points, A and B. What is the length of line AB? A) 10 B) 12 C) 14 D) 16
y=(x-11)^2 the parabola would be off to the right by 11. (unrelated but x^2 + 3 moves the parabola up 3) line ab intersects this parabola at point a and at point b and we want the length of point a to point b or line ab. if we find the x coordinates of the intersection, we can just subtract them to find the distance to find the intersections, set the equations equal to each other. (x-11)^2 = 25 x-11 = +- 5 x= 16 x= 6 16-6 = 10 therefore the answer is A
vertex form
y=a(x-h)^2+k and vertex is (h,k) say you wanna find which of the following equations defines the parabola. look for the vertex first. eliminate any that don have it right off the bat. now would it be this answer or this one for example: f(x) = 4(x-3)^2 + 1 or f(x) = (x-3)^2 + 1 if you know another point on that parabola, say (2,5) for example, plug in to see if a is 1 or 4 given those are the two examples you are stuck on above 5 = a(2-3)^2+1 5= a + 1 a = 4 therefore, the equation f(x) = 4(x-3)^2 + 1 would be the correct answer
a start-up company opened with 8 employees. the company's growth plan assumes that 2 new employees will be hired each quarter (every 3 months) for the first 5 years. If an equation is written in the form y=ax+b to represent the number of employees, y, employed by the company x quarters after the company opened, what is the value of b?
y=ax+b b in the y-intercept the yint is the initial amount. so, they started with 8 employees b=8
The line y=kx+4, where k is a constant, is graphed in the xy-plane. If the line contains the point (c,d), where c does not = 0 and d does not = 0, what is the slope of the line in terms of c and d? A)d-4/c B) c-4/d C) 4-d/c D) 4-c/d
y=mx+b y=kx+4 d=kc +4 k is the slope. we want to find the slope so we solve for k. d-4/c = k the answer is A
4x^2 + 6x all over (/) 4x+2
you do the long division thing 4x * x = 4x^2 and then 2 * x =2x we subtract we have 4x 4x * 1 = 4x and 2 * 1 = 2 we subtract we have -2 4x cannot go into -2 that is our remainder we are left with x+1 - 2/4x+2 and that is our answer!