sat math

cos²x+sin²x=

1

x⁰=

1

lesson 4 summary

1) convert the average to sum, where sum= average*number of terms 2) average= sum/number 3) whenever the number of terms is not given use n. the first sum will be average*n+(change in average), the second sum will be average*n and may include a change in n if the number of terms change. In the case that the number of terms change, the second sum is average(n+change).

lesson 3 summary

1) directly means, x/1 is in ratio of y/1 and is proportional. the equation is y=kx for some constant k. you have to find k first 2) inverse variation means as y goes up x goes down, the equation is y=k/x. you have to find k first. 3) whenever a new input is applied, plug it into ALL outputs that have x (ex. if f(x)= 14x²-30x+15, then f(4x)= 14(4x)²-30(14x)+15) 4) in compositions of functions work inside out 5) in a problem that uses direct variation and inverse variation, the direct goes on top and inverse goes on top and inverse on bottom

bill has cows, pigs and chickens on his farm. The number of chickens is 4 times the number of pigs, the number of pigs is three more than cows. Which of the following could be the total number of these animals? A) 28 B) 27 C) 26 D) 25

cows=x pigs= x+3 chickens= 4(x+3) x+x+3+4x+12= 6x+15, the only number that when subtracted by 15 and divided by 6 is an integer is 27, so B)

if 5x/y=10, what is the value of 8y/x

cross multiply to get 5x=10y the trick happens here: since we want to get y/x, divide both sides by 10x, the result is 1/2, multiply 8 by 1/2, the answer is 4

sin(90-x)= cos(90-x)=

cos x sin x

cos(-x) = sin(-x)= tan(-x)=

cos(x) -sin(x) -tan(x)

which of the following is equal to cos (π/5)? A) -cos (-π/5) B) -sin (π/5) C) sin (3π/10) D) -cos (3π/10)

cos(x)= sin(90-x) cos(π/5)= sin(π/2-π/5) = sin(3π/10)

cos (A-B)=

cosA cosB + sinA sinB

x^a*x^b=

x^a+b

(x/y)^a

x^a/y^a

(x^a)^b=

x^ab

(x+y)² (x-y)²

x²+2xy+y² x²-2xy+y²

general form for the equation of a circle

x²+y²+ax+by+c

if x>0 and x⁴-16=0, what is x

x⁴-16=0 x⁴=16 x= 2

fractional exponents ⁴√x³ =

x∧¾

y intercept is labelled as x intercept is labelled as

(0,a) so f(0)=a (b,0) so f(b)=0

if a²-169/a-13=b², what does a equal in terms of b

(a+13)(a-13)/a-13 = b² a+13=b² a=b²-13

area of a polygon

(n-2)*180t

if r²s>10ⁿ where n=2000, then the value of (rs+1/r)/7rs is closest to which of the following: A) 0.1 B) 0.15 C) 0.2 D) 0.25

(rs+1/r)/7rs, first multiply both top and bottom by r to get rid of the complex fraction r(rs+1)/r(7rs)= r²s+1/7r²s= r²s/7r²s + 1/7r²s= 1/7 + 1/7r²s because r²s is very large, 1/r²s is very small, so set it equal to 0 1/7+0= 0.149, so B

s²+t²=c st= c+5 what is (s+t)² in terms of c?

(s+t)²= s²+2ts+t² = s²+t²+2ts = c+ 2(c+5) c+2c+10 3c+10

if x+y=2k-1 and x²+y²=9-4k+2k² what is xy?

(x+y)²=x²+2xy+y² (2k-1)²=4k²-4k+1 4k²-4k+1=x²+2xy+y² 4k²-4k-(x²+y²)+1= 2xy 4k²-4k-(9-4k+2k²)+1=2xy 2k²-8=2xy k²-4=xy

Standard form equation for circle

(x-h)²+(y-k)²=r² center is (h,k), both must be subtracted in order for it to be negative

how to pick a number

1) don't pick a 0 or 1, go for 2 2) don't pick numbers that appear in the problem 3) picking numbers is only a strategy for eliminating answer choices, do not choose the first answer choice that comes out to the correct answer, if multiple come out then start again 4) if there are fractions then choose the least common denominator or a multiple of it 5) in percent problems use the number 100

lesson 2 summary

1) if a problem uses a, plug in 2 2) if a line intersects another line at its midpoint, the y is half what it normally would be 3) perpendicular= opposite recipricol 4) the only thing perpendicular to a horizontal line is a vertical line

lesson 5 summary

1) if only partial variables are given, multiply them together and find the square root to get the answer 2) one method of finding a variable is setting them one on top of another and taking out the one you need by subtraction or addition 3) dividing exponents is the same as subtracting them

lesson 1 summary

1) only positive square roots 2) multiply to get rid of fractions 3) create variables in terms of x, 2x, 2x+3 in word problems

as x gets very large

1/x becomes very small

14²

196

if factoring the function by number 2 gets a result that is 0

2 is a factor of the function in the form x-2

an elephant traveled 7 miles at an average rate of 4 miles per hour and then traveled the next 7 miles at an average rate of 1 miles per hour. what is the average speed?

2(4)(1)/(4+1)= 8/5

conjugate of imaginary numbers

2-3i= 2+3i -5+i= -5-i

if y=7^x, what is equivalent to 49^x-7^(x+2)

49^x= (7²)^x 7^x+2= 7^x*49 = y²*y49

if 2^9x=27z^3 and 8^x=nz, what is n

2^9x= (2³)x= (8)^x= 27z³ nz= (3z)³ n=3

if 2x=7-3y and 5y=5-3x, what is the value of x?

2x=7-3y 5y=5-3x, first set the two equations on the same side 2x=7-3y 3x=5-5y, now use elimination by getting rid of the y by multiplying the top by -3 and bottom by 5 -9x=-15 10x=35 x= 20

a circle has

360 degrees or 2π radians

volume of a sphere

4/3 πr³

percent change=

change/original * 100

exponential functions

A= initial value(increase in percent)^time

surface area of a rectangular solid

A=2lw+2lh+2wh

type of problem: fractional if 3z=(y-5)/2, and z=5, what is the value of y A) 20 B) 25 C) 30 D) 35

knowing z=5, 3z=15 cross multiply to get 30=y-5 y=35

k is

a constant

If h=a+b+c+d+e+f+g, what is the average of a+b+c+d+e+f+g and h?

a+b+c+d+e+f+g=h h+h/8 2h/8 h/4

surface area of a cube

a=6s²

if (x^a)^b= x^c^2/x^d^2 and ab=c-d, what is the value of c+d

ab=c²-d² ab=(c+d)(c-d) ab=c-d so ab/c-d= c+d 1=c+d

in percent problems

always use the number 100

angle/360 =

arc/circumference

angle=

arc/radius

the x-intercepts of a function

are also called zeroes, roots, or solutions

xiggi's formula

average speed= 2(speed1)(speed2)/speed 1 + speed 2 can only be used when two individual speeds for the same distance are known

general form of equation

ax+by=c slope= -a/b

let r and s be the roots of the quadratic equation x²+bx+c=0 then

b=-(r+s) and c=rs

P(x)= 20x/98-x The function P above models the monthly profit, in thousands of dollars, for a company that sells x percent of their inventory for the month. If $90,000 is earned in profit during the month of April, what percent of April's inventory, to the nearest whole percent, has been sold?

basically the problem asks for the solution to x when p=90 90= 20x/98-x 8820-90x= 20x 8820=110x 80.18=x

if f(0)=7 f(-3)=7/27 f(3)=189 in function g(x)=ab^x, what is the value of b?

because ab^x= 7 and not 1, this means that a=7 so 7b^-3=7/27 b=3, because 3^-3=1/27 and multiplying that by 7 equals 7/27

if a graph f(x) has a x-intercept at point (5,0). what is the intercept of graph f(x-1)?

because f(x-1) means shifting one unit to the right, the new x intercept will be (6,0)

A group of students take a test and the average score is 72. One more student takes the test and receives a score of 88 increasing the average score of the group to 76. How many students were in the initial group?

because we don't know the number of people, let it=n sum 1= average*number sum 1= 72n because to get to the second sum, there is an additional 88 added, this gets refined to sum1= 72n+88 the second sum has an increase in one person, so sum2= 76(n+1) setting them equal to each other: 76n+76=72n+88 4n=12 n=3

a ladder rests against the side of a wall and reaches a point that is h meters above the ground. The angle formed by the ladder and the ground is θ. A point on the ladder is k meters from the wall. What is the vertical distance, in meters, from this point on the ladder to the ground? A) (h-k)tan θ B) (h-k)cos θ C) h-k sinθ D) h-k tanθ

by drawing a picture, we see that tan θ = x/y and tan θ= h/y+k y tanθ= x (y+k) tanθ= h x+ ktanθ= h x= h-ktanθ

x³-3x²+5x-15=0 for what value of x is the equation untrue?

by factoring by grouping, we get x²(x-3)*5(x-3)= (x²+5)(x-3) x=3 is the only real solution

discriminant

b²-4ac

the average of x,y,z and w is 15 and the average of z and w is 11. What is the average of x and y?

change averages to sums, the sum of x,y,z,w is 15*4= 60 and the sum of z and w= 22, if x+y+w+z=60 and z+w=22, then x+y=38, the average is 19

(x+y)/x= 2/9, which of the following must be true A) x/y = 9/11 B) x/y= -9/7 C) x-y/x= 11/9 D) x-y/x= -9/7

do cross multiplication to get 9x+9y= 2x -9y=7x knowing that we must get x/y, divide both sides by 7y, which equals -9/7

to find area of sector

do x/360 = a/(radius*pi)² where x is the angle measure and radius*pi=

if a question asks for how many values of x does a graph equal constant b

draw a line between b and the graph and count how many times the graph touches that point

5a+2y+3z=23 5a+y+2z=15 what is y+z

easy way to solve this is to subtract the two terms y+z=8

exponential growth and decay

f(t)= a(1+r)^ct a= initial amount set what is inside parentheses to equal 1+r, then times r*100 to get the percentage change ct= the amount of time in years, set t=1 year to find amount after how much time the change happens

simple probability principle

favorable outcomes/total outcomes

y= ax²+bx+c

find the x point of the vertex by -b/2a plug that x point in to find the y point

when doing average problems

first convert to sum by multiplying average * number of terms

the x-coordinate for the vertex of function g(x)=3(a-h)²+k is

h

effect on the mean

if a mean changes, for example if a new number is added or one is subtracted, then this is reflected in the mean by doing so, set the previous mean equal to the new mean with the effect on the old mean

when is the hypotenuse greater than the squares of the other two sides

if it is a obtuse triangle

when is the hypotenuse less than the squares of the other two sides

if it is an acute triangle

cutting open a cylinder to get a rectangle

if opening a cylinder along the top, we get that the top of triangle represents the circumference of the circle and side represents the height we can use this to find the height

how many solid wood cubes each with a total surface area of 294 square centimeters can be cut from a solid wood cube with a total surface area of 2646 square centimeters

if the total surface area is 294, the side is 7 if the total surface area is 2646, the side is 21 7³=343 and 21³=9261, 343/9261= 27

lines with infinite solutions

if there are two lines: ax+by=c dx+ey=f lines with infinite solutions will be such that a/d = b/e = c/f

to get from cos of an angle to sin of an angle

if there is cos(x), get to sin by doing sin(90-x)a

the median of a set of consecutive integers

if x is the least integer in a list of n+1 consecutive integers, then the median is x+n/2

a function satisfies f(3)=7, and f(7)=1, g(7)=3, g(1)=4 what is f(g(7))

in compositions of functions, always work inner out f(g(7))= f(3)= 7

lines that are parallel or have no solution

in two lines ax+by=c dx+ey=f parallel lines will be such that a/d = b/e

to reduce the margin of error

increase the number of subjects

if a family of guppies double from an initial population of 2 each month, how would this be expressed as an exponential function

initial value=2 time= 12t since it is doubled, 1+r=2. r=1 so 2(2)^12t 2^12t+1

y varies inversely as x

is saying that while one increases, the other decreases like a hyperbola some other ways of saying this is: 1) y is inversely proportional to x 2) y=k/x for some constant k 3) xy is a constant

the last term of the parabola

is the x intercept

when a question asks "how many boxes" "how many ounces of flour"

it is asking for volume of a cube

which of the following is the equation of a line in the xy-plane that is perpendicular to the line with equation y=3? A) y=-3 B) y=-1/3 C) x=-2 D) y=-3x

it is easy to go by without noticing the slyness of this question, the equation y=-3 is a horizontal line on the xy-plane; the only thing perpendicular to it is a vertical line, so C is the answer

if a function varies directly with another function

it is like x/1 = z/1

if a function varies inversely with another function

it is like x= 1/z

if a function z varies inversely with function a, but directly with function b

it is like z= b*k/a

the y-coordinate for the vertex of function g(x)=3(a-h)²+k is

k

the length of a rectangle was increased by r percent and decreased in width by 20%. If the area of the rectangle was increased by 4%, what is the value of r? A) 20 B)30 C)40 D)50

lets make the width equal to 10 and length equal to 10, the area is then 10, now lets try each option A) (12*8)= 96, this is a reduction in 4 percent B) (13*8)= 104, this is an increase in 4 percent C) (14*8)= 112, this is an increase in 12 percent B is the answer

there are m bricks that need to be stacked. After n of them have been stacked, then in terms of m and n, what percent have not been stacked? A) m/100(m-n) % B) 100(m-n)/m % C) 100m/n % D) 100n/m %

lets say m=100 and n=20, there would need to be 80% of bricks that need to be stacked the only option that results in 80% is B

three points on a graph are labeled (0,0), (18,3), and (15,k) the distance between (0,0) and (18,3) and (18,3) and (15,k) is the same what is the value of k?

making a right triangle, the hypotenuse between 0 and (18,3) is √333 since the distance is the same between (18,3) and (15,k), the distance between those two points is also √333 since there is a 3 point reduction between 18 to 15, the opposite value is 3, making the adjacent value 18 because the graph is shifted 3 up, the value of k is 21 (see number 11 on page 75 for a better description)

conditional probability

measures the probability of an event given that another event has occurred

when dividing imaginary numbers

multiply both numerator and denominator by the conjugate of the imaginary number

to get a variable by itself

multiply by opposite reciprocol

when there is a graph with average

multiply x*y to get the total frequency and divide by total numbers

if a question what is the probability that a random selected 9-11 year old is overweight

name the events as they follow: first they want they probability of selecting a 9-11 year old then the probability that it is overweight so if there are 743 total people, 426 of whom are 9-11 and 317 are 6-8, and 194 total overweight people and 163 overweighters 9-11 while 31 are 6-8, the numbers divided would be 163/426 this is because

The mean length of a pop song released in the 80's was 4 minutes 8 seconds. The mean length of a pop song released in the 1990's was 4 minutes 14 seconds. Which of the following must be true about the mean length of a pop song released between 1980 and 1999 A) mean length is equal to 4 minutes 11 seconds B) mean length is less than 4 minutes 11 seconds C) mean length is greater than 4 minute 11 seconds D) mean length is between 4 minute 8 seconds and 4 minute 14 seconds

notice that without even using any numbers, the answer must be D.

Dana has pennies, nickels and dimes in her pocket. The number of dimes she has is three times the number of nickels, and the number of nickels she has is 2 more than the number of pennies. Which of the following could be the total number of coins in Dana's pocket? A) 15 B) 16 C) 17 D) 18

number of pennies= x number of nickels= x+2 number of dimes= 3(x+2) x+x+2+3x+6= 5x+8, the only number that is divisible is 18, so D

p(1)=-7, q(1)=0, r(1)=-5 p(0)=5, q(0)=-1, r(0)= -6 p(-1)=2, q(-1)=1, r(-1)= 2 p(-2)=3, q(-2)=4, r(-2)=-3 what is the value of p(-2)+q(0)-r(1)

p(-2)=3, q(0)= -1, r(1)=-5 3-1-(-5) 3-1+5= 7

what is the slope of the line in the x-y plane that passes through the points (a²,a⁴) and (a³,a⁶)

plug in 2 for a the points are (4,16) and (8,64) 64-16/8-4= 48/4= 12 the slope is 12

whenever there is an x in the parentheses

plug in that x for all variables

when a question asks if a function is divisible by 3x+1,

plug in x=-1/3 into both functions to see if they add up to 0

circumference is

radius*2pi or pi*diameter

distance=

rate*time

-f(x)

reflect in x axis

f(-x)

reflect in y axis

when doing ab^x

remember to raise to exponent first, then multiply

x+k<y m-x>y in the xy-plane (0,0) is a solution to the system of inequalities above. Which of the following relationships k and m must be true? A) k=-m B) k>m C) k<m D) |k|<|m|

replace (0,0) in both equations, 0+k<0 m-0>0 so k<0 and m>0, so m>k or k<m which is C

if rs=4, st=7, rt=63, what is rst

rs=4 st=7 rt=63 (rs)(st)(rt)= 4*7*63 r²s²t²=1764 rst=42

if there are two graphs shows, and the question asks "for which of the following values of x is f(x)-g(x)<0", how would you solve this

see that f(x)-g(x)<0 that means that f(x)<g(x) that means for which values is the first graph lower than the second graph

a cube with volume 343 cubic inches is inscribed in a sphere so that each vertex of the cube touches the sphere. what is the length of the radius, in inches, of the sphere?

see that this is just a cute way of using the generalized pythagorean theorem. a cube with volume 343 means that each side is 7 inches. so d²=7²+7²+7² d²= 3*7² d= 7√3 since this is the diameter, you can find the radius by dividing it in 2 r= (7√3)/2

to find the area of a sector

set angle x/360= sector area/ circle area

if the hypotenuse of a triangle is the radius of the circle

set the other sides squared equal to hypotenuse squared and find the radius

6z+8+2t+3+9=52 2z+8+w+3+9=34 find 4z+2t-w

set the two equations below one another 6z+8+2t+3+9=52 2z+8+w+3+9=34 by subtracting the two equations, you get 4z+(2t-w)=18

f(x-1)= f(x+1)= f(x)-1= f(x)+1=

shift right one unit shift left one unit shift down one unit shift up one unit

see figure on page 149

side AB= 10-(10-2c) AE= 10-10+2c= 2c the area of the square is 1/2(2c)² = 2c² because the probability of landing in the square is 100, the probability of landing in the triangle is 2c²/100, which is given to be 2/25, meaning that c=2

volume of cube

side cubed

if a function g is defined by g(x)=p(x-h)²+k, where p,h,k are all positive what cannot be true? A) g(7)=-h B) g(7)=2 C) g(0)=-2 D) g(0)=2

since k is positive, g(0) cannot be a negative number therefore C is the answer

y≥-12x+600 y≥3x if a point with coordinates (a,b) lies in the solution set of the system of inequalities above, what is the minimum possible value of b?

since the value of y is given in the second equation, you can replace it in the first equation 3x=-12x+600 15x=600 x=120

tanx

sinx/cosx

x-3= √x+3 what is the solution set of the equation above?

start with option B, which says the answer is 6 6-3 = √6+3 3= √9 3=3 6 works, but that leaves the possibility of it also being option c, which is 6 or 1, using 1: 1-3= √1+3 -2= √4 -2=2 remember that on the SAT, negative roots are not valid. so √100 is not -10 or 10, it is just 10.

the average of 11 numbers is j. If one of the numbers is k, what is the average of the remaining 10 numbers in terms of j and k?

sum=average*number sum=11*j sum=11j the effect is subtracting k sum= 11j-k/10 because there are 10 numbers

The average age of the people in a certain group was 35 years before one of the members left the group and was replaced by someone 12 years older. The average is now 37 years, how many people are in the group?

sum=average*number of terms say the number of terms are n sum= 35n then the sum=37n 37n=35n+12 2n=12 n=6

in the equation y=12/x, y varies inversely as x

that would mean that y(x)=12 so some appropriate values for this function would be (1,12), (2,6), (3,4), (4,3)

in the xy-plane, the line determined by the points (c,5) and (10,2c) passes through the origin. Which of the following could be the value of c? A) 0 B) 5 C) 10 D) 25

the answer cannot be A, as it would mean the line passes through point (0,5) if the answer were to be B, the points would be (5,5) and (10,10) which would indicate a slope of y=x if the answer were to be C, the points would be (10,5) and (10,20), this equation cannot pass through the origin because it has NO slope if the answer were to be D, the points would be (25,5) and (10,50), which would indicate that the graph reaches the horizontal axis after the x point 25, which doesn't meet the criteria

the negative sum of all zeros is

the b value

the multiplied zeros are

the c value

when unfolding a cylinder, the bottom is

the circumference for the lateral area to get the whole area, add the lateral area+two circles

whenever two numbers with same bases are multiplied with exponents

the exponents are added

see page 147 what is the probability that a randomly selected overweight child aged 6-11 is less than 9 years old?

the first part of the question always clarifies exactly which row/column to look into here you would look into the overweight column, and divide 31/194

suppose that z varies directly as x² and inversely as y³, when x=3 and y=2 z=9, what is y when z=4.5 and x=6

the function is z=k*x²/y³, 9= k*3²/8 9= 9k/8 72= 9k k= 8 4.5= 8*6²/y³ 4.5= 8*36/y³ y³= 8*36/4.5 y³= 8*8/1 y³= 64 y=4

generalized pythagorean theorem

the length d of a long diagonal of a rectangular solid is d²=a²+b²+c² where a,b,c are the length width and height of a rectangle

triangle rule

the length of the third side is between the sum and the sum and the difference of the other two sides if a triangle has sides length 2,5,x, x will be between 3 and 7

if a circle is tangent to another circle

the line between them is perpendicular

which of the following is an equation of the line in the xy-plane that passes through the point (4,-2) and is perpendicular to the line y=-4x+7 a) y=-4x-3 b) y=-4x+3 c) y= 0.25x-3 d) y= 0.25x+6

the line perpendicular to y=-4x+7 is y=1/4x+b plug in point (4,-2) -2= 1/4(4) +b -2= 1+b -3=b the line is y=1/4x-3, so C

if the sum is changed by adding one more person or subtracting one less person

the new sum undergoes an average change of n+1 or n-1

fence post formula

the number of integers from a to b are b-a+1

if p(-5)=6

the result of p(x) when factored by (x+5)=6

the last number of a polynomial is

the result of the multiplied factors

the points (5,e) and (f,7) are on a line perpendicular to y=-1/5x+12 what is e in terms of f? A) 5f+32 B) -5f+32 C) -1/5f+32 D) 1/5f+32

the slope of the line is y=5x, the points are (7-e)/(f-5)=5/1 so e=2, and f=6 A) e=64 B) e=-30+32=2 C) e=-6/5+32= 31.2 D) e= 6/5+32= 32.2 B is the correct answer

if the two lines intersect at a single point

the slopes will not be the same

how many spherical snowballs with radius of 4 centimeters can be formed with the amount of snow in a spherical snowball of radius 8 centimeters

the volume of sphere with radius r is given by: 4/3πr³ the amount of snow with 8 centimeter radius is 2048π/3 the amount of snow with 4 centimeter radius is 256π/3 when dividing the amount of snow from the first to the second, the result is 8

if on a graph point (3,4) is present

then f(3)=4

if the graph is above the x-axis

then f(x)>0, if it is below the x-axis then f(x)<0

if discriminant is negative

there are no solutions

if discriminant is positive

there are two solutions

if discriminant is 0

there is 1 solution

if given the length of the arc that encapsulates a circle

times that angle to get to 360 then multiply that same number to the length of the arc

it is given that cos x= k, where is the radian measure of an angle and is between 0 and 3π/2. If cos z=-k, which of the following could not be the value of z? A) x-π B) π-x C) 2π-x D) 3π-x

unanswered

Marco drove from home to work at an average speed of 50 miles per hour and returned home along the same route at an average speed of 46 miles per hour. If his total driving time for the trip was 4 hours, how many minutes did it take Marco to drive from work from home?

use a chart: from home to work the distance was d, the rate was 50, and thus the time was d/50 from work to home the distance was d, the rate was 46, and thus the time was d/46 we know the total time was 4, so set d/50 and d/46 equal to 4 multiply 4 by 50 and 46 to get the two d's by themselves 46d+50d=4*46*50 96d=4*46*50 d= (4*46*50)/96

if teds weight increased by 36 percent and jessica's weight decreased by 22 percent during a certain year, the ratio of ted's weight to jessica's weight is what at the end of the year?

use the number 100 for both ted and jessica at the end of the year ted's weight is 136 pounds at the end of the year jessica's weight is 78 pounds 136/78= 1.74

y=cx²-k y=5 for what values will c and k have no real solutions? a) c=-2 k=-6 b) c=2 k=-6 c) c=2 k=-4 d) c=2 k=4

using substitution, we get 5=cx²-k 5+k=cx² 5+k/c = x² √5+k/c =x we can plug in answers to see which one doesn't fit

isosceles triangle with base 20 and two side lengths 26 what is the area of the triangle?

using the base, we can make this a right triangle by forming an altitude. then the base becomes 10 and hypotenuse 26, which means that the length of the altitude is 24. knowing the height is 24, we can multiply 1/2b*h to get 1/2(20)(24)= 240

if a square has a side of length x+5 and a diagonal of length x+10, what is the value of x A) 5 B) 10 C) 20 D) 5√2

using the pythagorean formula= 2(x+5)²= (x+10)² 2(x²+10x+25)= (x²+20x+100) 2x²+20x+50= x²+20x+100 x²=50 x=5√2

y=-5(x-3)²+2 a line goes through the vertex of this parabola and the point (-1,5) what is the line's slope

vertex of a parabola is (h,k) so the vertex of this parabola is (3,2) (5-2)/(-1-3)= -3/4

if we know the angle x

we can set x/360 equal to arc s/circumference to find length of arc x/360 = arc/circumference

if we know the radius r

we know that d=2r c=2πr a=πr²

in g(x)=-3x-7, what is g(-4x) equal to

whenever a new input is put in place, you put it into the function -3(-4x)-7= 12x-7

if h(x)=(x-3)(x+7), which of the following is an equivalent form of the function h above in which the minimum value of h appears as a coefficient or constant

whenever a question asks for the minimum value of the function to appear, it is asking for the vertex the function (x-3)(x+7)= x²+4x-21. the x intercept of this function= -b/2a= -4/2= -2 the y-coordinate= (-2-3)(-2+7)= (-5)(5)= -25 this function can be rewritten as (x+2)²-25k

n

whenever the number of people aren't given, replace it with n. because the sum=average*num, the sum=average*n. if the number of people change throughout the scenario, you will have to multiply the first average *(n+difference). more than likely there will be an addition or subtraction in the first sum because of an increase or decrease in the average

what is the greatest integer x that satisfies the inequality 2+(x/5)<7 A) 20 B) 22 C) 24 D) 25

whenever the problem starts with greatest, go for the option with the largest number 2+(25/5)<7 2+5<7 7<7, not true. since this equation is just BARELY true, it has to be C)24

f(x)= (1/b⁵)^x what is equivalent to f(3x)

whenever there is something in parentheses, plug it in (1/b⁵)^x= (1^x)/b^5x (1/b⁵)^3x= (1^3x)/b^15x this is f(x)³

-2x²+bx+5 the graph of the equation above assumes its maximum value at x=2, what is the value of B?

x coordinate of vertex of a parabola in standard form is= -b/2a -b/-4= 2 -b=-8 b=8

if y varies directly to x

y increases as x increases some other ways of saying this is: 1) y varies directly as x 2) y is directly proportional to x 3) y=kx for some constant k 4) y/x is a constant

system inequalities

y<2x+2 y>-3x-3 the line y=2x+2 has a slope of 2>0, and therefore moves upwards from left to right. plugging in (0,0) makes the inequality true, as 0<2, so it covers everything to the right. the line y>-3x-3 has a slope of -3<0, and moves downwards as drawn from left to right. (0,0) makes the inequality true, as 0>-3, so the graph covers everything to the right.

standard form for a quadratic function

y= a(x-h)²+k vertex at (h,k) opens upwards if a>0 opens down if a<0