sat math mistakes
if √ab = 11 what is a²b²
(ab) squared is a²b² this equals 121
which of the following equivalent forms display the _________ as constants or coefficients?
(x-a)(x+a) represent x-intercepts as constants a(x-h)² + k represents the vertex as constants ax²+bx+c represents the y-intercept as constants or coefficients
equation of circle
(x-h)² + (y-k)² = r²
a certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. which inequality represents all possible weights in pounds for the elephant 365 days after birth?
200+2(365)<w< 200+3(365) 930 < w < 1295
perimeter of a semicircle
2πr/2 + diameter
density is
mass/volume
it is impossible for a line and a parabola to intersect at
more than 2 points 2 is maximum
(2+i)/(2-i) =
multiply both top and bottom by 2+i because it is the conjugate
exterior angle theorem
the measure of an exterior angle to A equals the sum of the interior angles B+C
whenever there is an x intercept at 0
"x" is one of the factors
if h(x) = x^a for some positive constant a, and h(27x) = 3h(x) for some positive value of x, what is the value of a?
(27x)^a = 3(x^a) 27^a * x^a = 3 * x^a since x^a is same on both sides, eliminate it 27^a = 3 a= 1/3
xiggis formula
(2d1d2)/(d1+d2)
a higher standard deviation means a lower standard deviation means the same standard deviation means
a higher spread in data a lower spread in data the same spread
i² =
-1
i⁴ =
1
properties of parallelograms
1. opposite sides are congruent 2. opposite angles are congruent 3. consecutive angles are supplementary 4. diagonals bisect each other 5. diagonals separate parallelogram into two congruent triangles
1/(1/4x + 1/x+2)
1/ (x+2+4x / 4x² + 8x)) 1/ (5x+2 / 4x² + 8x)) 4x² + 8x / 5x + 2 always flip reciprocols
if a semi circle has an area of 8π, what is its perimeter
1/2 πr² = 8π πr² = 16π r = 4 1/2 (2πr) = 1/2 (8π)= 4π remember to add the radius to the perimeter, the radius here is 4 4π + 4
t= 0.06(a-m) t= 300 a= 30000 no calculator
300 = 0.06(30000-m) 300/.6 = 30000-m 300/.6 means 5000 5000 = 30000 - m -25000 = -m m = 25000
for gym class, Shayla completed a 4-mile walking and running exercise. She ran for 7t miles and walked for 3t miles. the equation 7t+3t = 4 models this situation. what is the best interpretation of the value 7 in this equation?
4 means the amount of miles she walked and ran 7t + 3t = 4 miles 7t reveal the miles she ran 7 * t = 7 is miles/hour t is hours so that when multiplied, 7t represents the number of miles this means shayla ran at 7 miles per hour
kelly enlarged the area of a photograph to 250% of its original size. the original dimensions were 5 by 7, what is the new area?
5 * 7 = 35 250 percent of 35 is 35(1) + 35(1.5) = 87.5
π/3 = π/6 = π/2 =
60 30 90
pure beeswax has a density of 0.555 ounce per cubic inch. an online company sells pure beeswax at a price of $8 per ounce. what is the selling price for pure beeswax from this company?
8 dollars/ 1 ounce = x dollars/0.555 ounce
y= 9.8x what is the best interpretation of 9.8
9.8 is the increase in y for every 1 increase in x
when finding graphs to equations with real zeroes, remember
a zero is a factor, so if a factor is x-1 there will be an intercept at x=1
the function f is defined for all real numbers, and the graph of y=f(x) is a line with a negative slope. which of the following MUST be true? a) if a<b then f(a)>f(b) b) if a<0 then f(a)>0 c) if a>0 then f(a)<0
a) this statement is always true because a lesser value will equate to a higher output in a negative slope b) this statement is not always true, what if the equation was f(x) = -x-1000000 and x=-1. c) this statement is not always true, what if the equation was f(x) = -x+100 and x=1
each y intercept in a parabola is either
above the vertex if the function is positive below the vertex if the function is negative
x² + 6x + y² - 8y + 9 = 0 what the coordinates (h,k) of the center?
add (b/2)² of each factor x²+6x+9+y²-8y+16 = 16 (x+3)² + (y-4)² = 4² (-3,4)
when finding area of a parallelogram
always do bh dont get confused by slant height
inscribed angle
angle on the far back of a circle that has half the angle measure of the arc it intercepts
height= 3 width= h-3 length= h+4 area of base is 60 what is volume
area of base= lw (h-3)(h+4)= 60 h²+h-12 = 60 h² + h - 72 = 0 h = 8 volume =480
relationship between number of people and margin of error
as the number of people increase, the margin of error decreases
12 = ax² + 16x + 8 which of the following inequalities represents all values of a for which the equation has two distinct real solutions?
ax² + 16x - 4 two solutions means b²-4ac > 0 256 - 4(a)(-4) > 0 256 + 16a > 0 256 > -16a -16 < a
the vertex is equidistant to
both the roots
survey questions with a margin of error
correct answer choices will have plausible language, like "it is possible" or "it could be suggested" and will account for all percentages in the margin of error- so if the percentage is 20% and the margin of error is 3% the answer would say "between 17 to 23 percent"
100 percent increase 200 percent increase
double triple
absolute value function
everything is above x axis
at what second does max height occur
find -b/2a
when finding the distance on a velocity time graph
find the area
"if c is a constant such that there are four values of x that f(x) = c what is the value of c"?
find the value of x and draw a horizontal (left to right) line. choose the one that meets the graph four times
how many seconds after launch does the object hit the ground?
find the x-intercepts by using quadratic formula
to find vertex of parabola
find x, it is -b/2a plug in -b/2a into the function
0.5(x-k) = kx in order for the equation to have no solution, what is the value of k?
first multiply the whole equation by 2, it makes the solving easier x-k = 2kx now see each part of the equation as a separate linear graph, x-k and 2kx, remember that if two lines have the same slope they have no solution y= x-k y= 2kx the first line has a slope of 1, for the second line to have a slope of 1 k must equal 1/2
0.5x - 0.25y = 5 ax - 3y = 20 in the system of linear equations above, a is a constant. if the system has no solution, what is the value of a?
first, set the equation equal to y 0.5x - 5 = 0.25y 2x - 20 = y y = 2x - 20 3y = ax- 20 y = (a/3)x - (20/3) for the system to have no solution, both slopes have to be equivalent to each other. a/3 = 2 a= 6
bouncing x intercept
has a zero of x²
discriminant values
if b²-4ac = 0, the function has only one solution negative, the function has no solutions, 1 or higher, the function has two solutions
ax²+8x+2 if the function has only one real solution, what is the value of a?
if the function has only one real solution,
f(x) = 4x²+px+p if x+3 is a factor, what is the value of p
if x+3 is a factor, then at x=-3 f(x)=0 plug in x=-3 into the equation 4(-3)²-3p+p = 0 36-3p+p = 0 p = 18
utility curve For her job, Natasha spent a total of n minutes processing ID card applications and driver's license applications. It takes Natasha 15 minutes to process an ID card and 20 minutes to process a driver's license application
in a utility curve, each point when multiplied with the coefficient equals the same number in this problem each number in the vertical axis is multiplied by 20 and each in the horizontal 15; that means point (a,b) will output an utility of 15a+20b of a certain number
which function indicates the x-intercepts of y=4x²+4x-24 a) 4(x-2)(x+3) b) (2x-4)(2x+6)
in order to find x intercepts of b), you have to do algebra a) reveals the x-intercepts clearly the answer is a
triangle proportionality
in/out small version/ whole
proportions with trapezoids
in/out = in/out
in a linear function, the c value y=-3.9x + 44
is the initial value, or y-intercept of a function
in a quadratic written in vertex form, the c value y=-4.9(x-3)²+44
is the maximum value
adding complex numbers
is the same as normally adding numbers
if a line passes through the center of a circle, it means
it divides the circle into two
whenever a question asks for the maximum value or the minimum value of a function
it is asking for the maximum y value or the minimum y value
three circles are tangent at points A,B,C the radius of each circle is 9 what is the length of arc AB
joining the three points at the intersection of the three circles results in an equilateral triangle this is a radian of π/3 times radius 9 this results in 3π
to reflect a function across x axis when the function is already beneath the x axis
just flip it across
formula for conversion of arc to angle
length of arc/circumference = degree measure of intercepting central angle/ 360
minato drove 390 miles. part of that distance was on local roads with an average speed of 20 mph, the rest was on a highway with an average speed of 60 mph. the drive was 8 hours. what is the distance of minato's drive on local roads
lt = local time ht= highway time l + h = 390 d = rt l = 20 * lt h = 60 * ht divide both l and h by 20 and 60 respectively l/20 = lt h/60 = ht l/20 + h/60 = 8 hours 60(l/20 + h/60 = 8 hours) 3l + h = 480 l + h = 390 solve the systems of equations process: 1. add the two different types of roads to equal total miles 2. set one equation equal to local miles = speed * time and another highway miles = speed * time the time will not be known 3. get l and h by itself and use the original systems of equations
area of right rectangular prism
lwh
complex fractions
multiply the numerator and denominator of the fraction of the conjugate of denominator a complex number "a+bi" has conjugate "a-bi"
a function such as y= a(0.5)^x when raised to power 0
only multiplies the fraction by the exponent, the coefficient stays
ax²+8x+2 if the equation has only one real solution, what is the value of a?
only one real solution means b²-4ac = 0 64-4(a)(2) = 0 64-8a = 0 a=8
y= x²-6x-16 which form includes the x and y coordinates as constants? A) y= (x-3)² - 25 B) y= x(x-6) -16
only the vertex form can display x and y as constants vertex form is a(x-h)² + k this is A) note that B) does not square the first term, therefore is incorrect
what is the max height of object
plug in -b/2a
arc separate formula
radius * radian
-f(x) means f(-x) means
reflect across x axis reflect across y axis
y= bx(x-a)(x-a)(x+b)(x-b) how many distinct x-intercepts does the function have?
remember that a coefficient in front of a factored polynomial means the function has an x intercept at 0 the other intercepts are a, b and -b there are 4 total zeroes
to convert degrees to radians
remember that π equals 180 degrees
parabola D in the xy-plane has the equation x-2y²-8y-11 = 0 which equation shoes the x-intercepts as constants or coefficients? A) x= 2y²+8y+11 B) x= 2(y+2)² + 3 C) x=3 = 2(y+2)²
see that the x-intercept happens when y=0 plug in 0 for y, x=11 A) is the right answer
the total d, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function defined in terms of t, where t is time in seconds. at a time of 10 seconds, the total distance traveled is 50 meters. at a time of 20 seconds, the total distance traveled was 200 meters. if the object was at a distance of 0 meters when t=0, what is the distance traveled at 30 seconds?
see that this function is a quadratic, which means it is a variation of y=x² the points (10,50) and (20,200) show that the function is half of the square of x, meaning it is y=1/2x² at x=30, the value will be 450
best interpretation questions asking about slope
slope is the increase in y for increase in x
in a science classroom, when labs are performed, students are seated at lab tables. if the teacher assigns 2 students in each lab table, 4 additional lab tables are needed. if the teacher assigns 4 students in each lab table, 4 lab tables will not be used. how many students are in the science class
t= tables s= students 2t+2(4) = s 2t + 8 = s 4t - 4(4) = s 4t - 16 = s 2t + 8 = 4t - 16 8 = 2t - 16 24 = 2t t = 12
(-1,a) and (3,b) are the endpoints of a diameter of a circle. if the center is (h,k) and a+b = 4 what is the value of h+k?
the center of a circle is the midpoint of the endpoints (-1 + 3)/2 = 1 h=1 (a+b)/2 = 2 k-2 h+k = 3
which is true of the circle with equation (x-0.2)² + (y+0.2)² = 0.2 and line y= -x A) the line and the circle do not intersect B) the line is tangent to the circle C) the line separates the circle into two arcs of equal lengths D) the line separates the circle into two arcs of unequal lengths
the center of the circle is (0.2,-0.2) and the slope is y=-x this means that the slope goes through the center of the circle because plugging in (0.2,-0.2) into the linear equation results the same value if a line passes through the center of a circle, it means it divides the circle into two
in circles with central arcs and bisected arcs, remember
the central angle is always the same as the arc it intercepts the angle that is far back at the end of the circle is half the angle it intercepts
how to read a boxplot
the first line to the start of the box: minimum to 25% start of box to middle of box: 25-50% middle of box to end of box: 50-75% end of box to last line: 75%-100%
if the value decreases by a percent,
the function is exponential, NOT QUADRATIC
w(l) = 0.04(1.22)^L the function w gives the estimated weight W(L) in pounds of a rainbrow trout based on its length L. What is the best interpretation of the number 1.22 in this context?
the number inside parentheses describes the change in percent 1.22 equates to an increase in 22 percent the variable that is being changed is L or length. and the variable being measured is weight. this means that a one length increase results in a 22 percent increase in weight.
whenever circular graphs are given and a function (sine,cosine,tangent) is asked, remember
the quadrant each function is positive in A- all positive in q1 S- sine positive in q2 T- tan positive in q3 C- cos positive in q4
the mean and median of x consecutive integers is always
the same
in the xy-plane, the graph of a linear equation of the form y=mx+b and the graph of an exponential equation y=ab^x both contain points (1,3) and (2,4). the line contains (r,s) and the quadratic (r,t) where 0<r<4 and s>t which of the following must be true? a) 0<r<1 b) 1<r<2 c) 2<r<3 d) 3<r<4
the slope of the line is 1 the y intercept is 2 y= x+ 2 y= ab^x has points (1,3) and (2,4) 3= ab 4= ab^2 a = 3/b 4= (3/b)*b^2 4= 3/b b= 4/3 3/ (4/3) = 9/4 = a the quadratic eq is y=(9/4)(4/3)^x this problem is asking for a certain value at which the output of the line is greater than the quadtatic if you plug in values, this value occurs between intervals 1 and 2
h(x) = (x-5)(x+5) for what value of x does the function reach its minimum value?
the two roots are at 5 and -5 the vertex is equidistant to both, it is at 0
an x intercept always has 0 as
the y coordinate
whenever a linear graph is shown and the x values do not start at 0
the y intercept is not what it appears on the graph, it is likely lower
the equation 9x+5 = a(x+b) has no solutions. which of the following must be true? a) a= 9 b) b = 5 c) b does not equal 5/9
think of both of these as two lines y= 9x+5 and y= ax+ab in order for there to be no solutions, the slopes have to be the same and the y-intercepts different therefore a=9 and b cannot equal 5/9 see that b is a true statement, but it is not always true. b does not have to equal 5, it can equal any number that is not 5/9
(x-1)²-12(x-1)+36 = 0 what is x
this is not the problem to expand and find x to, that just wastes time and is difficult because of the lack of a calculator see that, similar to a quadratic, (x-1) acts as a coefficient replace (x-1) with p p²-12p+36 = 0 (p-6)(p-6) = 0 p= 6 (x-1) = 6 x = 7
finding f(x) or g(x) from a table of values
to find f(5), find the f(x) value under 5 as x to find f(g(5)), find the g(x) value under 5 as x, assume it is 7. then find the f(x) value under 7 as x.
in the xy-plane, the points (a,2) and (b,-4) are the endpoints of a diameter of a circle. if the center of that circle is (h.k) and h+k = 4, what is the value of a+b?
to find the center given two endpoints, use the midpoint formula (2-4)0.5 = -1 this means k = -1 h - 1 = 4 h = 5 (a+b)0.5 = 5 a + b = 10
if f(x+3) = 4x-5 for all values of x, what is the value of f(5)
to get f(5), you have to plug in 2 into the equation and NOT 5
one serving of popcorn provides 150 calories, 90 of which are from fat. one serving of prezels have 120 calories, 12 of which are from fat. if both were served in a 100 calorie serving, how many more calories would be in a popcorn serving than a pretzel serving?
to get from 150 popcorn serving to 100 popcorn serving 150 * x = 100 x= 100/150 multiply calories by 10/15 90 * 10/15 = 60 calories of fat to get from 120 pretzel serving to 100 popcorn serving 120 * x = 100 x= 10/12 multiply calories by 10/12 12 * 10/12 = 10 calories of fat
converting bar graph to box and whisker
total number of people * 0.25 = x the xth person should be where the 25th percentile mark is total number of people * 0.75 = y the yth person should be where the 75th percentile mark is the median should be where the middle line is
anita created a batch of green paint by mixing 2 ounces of blue paint with 3 ounces of yellow paint. she must mix a second batch using the same ratio of blue and yellow paint as the first batch. if she uses 5 ounces of blue paint for the second batch, how much yellow paint should see use?
understand that this is not a ratio of blue:yellow paint that is incorrect instead this ratio is of blue:total paint the correct ratio is 2:5 = 5: x x is total paint x= 12.5 12.5 is total paint total paint- blue = yellow paint 12.5 - 5 = 7.5
a manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. which of the following expresses the volume, V, in cubic inches of such containers where r is the radius?
volume of cylinder= πr²h optimal cylinder= πr²(r+4) πr³+4πr²
what is the probability that a visitor will stay more than 4 hours if the visitor is 14 or older
whatever comes after "if" is the denominator
A sphere has a diameter of 12. What is the volume of the sphere?
why did i miss this? I used the formula how can i correct this? see that the formula is 4/3πr³ the radius is 6 4/3π(216) = 288π
increasing a number by a percent
x + (0.change in percent)x = new number
y intercept means
x has no value
if one of the values at x is chosen at random means
x is denominator
how to set up proportions with circles and arcs
x/360 = minor arc/ circumference
g(x) = x²(x+3)²(x²+4) how many distinct real zeroes does the function have?
x² = 0 x+3, x=-3 x²+4, x² = -4, no real solution
f(x) = x²(x²-2)(x²+1) how many distinct real zeroes are in the equation above
x² = 0 x²-2 = 0, x²=2, x = √2 x²+1 = 0, x² = -1 no solution
find the zeroes of 2x³-x²-8x+4 = 0 no calculator
x²(2x-1) - 4(2x-1) = 0 (x²-4)(2x-1) = 0 x= 2,-2,1/2
x intercept means
y has no value
f(x) = -500x² + 25000x the revenue f(x), in dollars, that a company receives from sales of a product is given by the function above, where x is the unit price, in dollars, of the product. the graph of y=f(x) intersects the x axis at 0 and a, what does a represent?
y= revenue x= unit price a is a unit price where the revenue is 0
how to find zero vs how to find vertex
zeroes: quadratic vertex: -b/2a
