Key SAT math formulas to *memorize*
Perfect Square Formulas
(a + b)² = a² + 2ab + b² (a - b)² = a² -2ab + b²
Standard form of a circle (when the center of the circle is located away from the origin)
(x - h)² + (y - k)² = r² r being a constant along with h and k and the value of h and k being the center on a coordinate plane (h,k)
Parallelograms on SAT
* 2 sets of parallel lines * Opposite angles are equal * All interior angles add to 360
Complete the Square (with expression) x² - 4x + 7
1. Create an empty space to add the missing term x² - 4x + 7 Method #1: Take the 2ab term --> Divide the coefficient by 2 ---> Square that number and add it as b² -4 / 2 = -2 (-2)² = 4 x² - 4x + 4 + 7 Method #2: 4x can be represented as 2ab, where a is represented by x if 4x = 2xb b = 2 ---> b² = 4 2. You must use the opposite sign and subtract/add b² to the coefficient x² - 4x + 4 + 7 -4 3. Rewrite by factoring (x - 2)² + 3
Two things to know about reliability of sampling for research
1. Larger samples is often better 2. Sample should include many different parts so that it can represent the population as a whole
Complete the Square of -x² + 3x
1. When there is a negative, you must factor in out first (-(x² - 3x ) ) 2. set -3x = 2xb and solve for b b = 3/2 --> b² = 9/4 (-(x² - 3x + 9/4)) 3. Since a negative has been pulled out, rather than subtracting b², add b² -(x² - 2x + 9/4 )+ 9//4 4. Simplify -(x - 3/2)² + 9/4
PYTHAGOREAN TRIPLES (BIG 4)
3 , 4 , 5 1 1 √2 1 √3 2 5 12 13
45 45 90 Rule
A RIGHT triangle with sides a b c as 1 1 √2 will have 3 angles, 90, and two 45 angles.
30 60 90 Rule
A RIGHT triangle with sides a b c as 1 √3 and 2 will have a 30, 60, and 90 degree angle.
Transversal
A line intersection parallel lines
Inscribed Angle
Angle formed where two chords meet each other at ONE point.
Central Angle
Angle measurement in a circle whose center point (vertex) is at the circle center.
complementary angles
Angles that add to 90
supplementary angles
Angles who add to 180
Minor vs Major Arcs
Arcs are measured with a degree angle. In the picture, two arcs are shown. The major arc is the long distance from point a to be, while the the minor is the short distance from a to b.
What is Completing the Square? How can it help on the SAT?
Completing the Square can be used as a tool for solving quadratic equation easily. It ONLY applies to quadratic equations. Completing the square transforms a quadratic expression that can't be factored into one that can easily. It is used to transform the equation into a perfect square trinomial whose factored form can be written as a single factor squared (such as (x+3)² = x² + 6x + 9 When coefficient in front of x is 1, Step 1: Constant Figure out what the constant which is missing is x² + 6x = -2 ---> x² + 6x + ( ? ) = -2 It's know what we want will be in the form a² + 2ab + b². We are looking for b. However, 2ab, the middle term is known, so an equation can be set up to solve for b. 6x = 2xa 6x/2x = 3 a = 3 what is a²? 9 This number MUST MUST MUST be added to both sides. We did all this to figure out what to add, now the same must be done to the other side. x² + 6x + 9 = 7 The purpose of this process has been fulfilled. Simply factoring will give an expression that can be easily square rooted to solve for x. (x+3)² = 7 sqrt(x+3)² = sqrt(7) x + 3 = +/- sqrt (7) x = +/- sqrt (7) - 3 Additionally, when leading coefficient isn't 1, divide both side (EVERY TERM) by that coefficient. Then repeat the process.
Growth and Decay
Formula: Amount of some quantity = Initial Amount of said quantity * (rate of growth/decay)^amount of time passed Relevant Contexts: 1. Money Growth/Decay could model the total amount of money in one's bank account after some time period given an interest rate
How do you know when a linear equation has no solutions?
If a linear equation has the same variable + coefficient on both sides, but has a different constant, then there will never be any solutions. Ex: 6x + 12 = 6x -10 6x could be cancelled, leaving 12 = -10, which isn't correct.
How to solve a ratio of a:c only given the two ratios of a:b and b:c?
If it was given that a:b is 2:3 and b:c is 6:5, the key to solving this problem is recognizing that you want both ratios to have a common list between them. This link is the variable that appears in both ratios: the b variable. To get both b to match, multiply the entire a:b ratio by 2 to get 4:6. Now the ratios can be simply compared. a:c is 4:5 Note that you could always multiply both ratios by factors that get them equal to each other. 2:3 (x6) = 12:18 6:5 (x3) = 18:15 a:c would be 12:15, which is also equal to 4:5
For a systems of equality problem, what can the intersect point be used for?
If the equations were to be solved as if they were a regular systems of equations problem, you get the intersect of two lines, which will represent the highest value that x will be. (Highest value means furthest left/right on the x line for an intuitive explanation*). X will not go higher than this, so this can be applied to problems requiring a maximum number of x or y. You can also use the graphing calculate and graph the two equations. The minimum y value will be, well, where the lowest y value is. The maximum is vice versa. It may help to convert to slope intercept form*
When should you use the slope (m value) to answer linear equation interpretation problems?
If you see the words increase/decrease by some quantity, you should know to look at (or solve for as needed) the slope value. You can ignore the y-intercept. Identify what variables represent (change x) / (change y). Create an equation that sets the slope equal to the desired change. Solve for this variable.
What trick should you know for answering percent based questions?
If you want to increase a quantity by a percentage multiply (1 + percentage as decimal) x (quantity) If you want to decrease a quantity by a percentage multiply (1 - percentage as decimal) x (quantity) For compound interest problems, the same concept can be applied. If a percent % of compound interest is earned, it's just a way of saying that you are increasing a quantity by the same percentage for x amount of years. (principle amount) * (1 +/- rate as percentage)^x The x is just a way of condensing all the factors.
Solving for an inequality
Is like solving for y in a regular equation. Keep in mind when you multiply or divide by a negative value when needed, flip the direction of the sign.
Ratio Scale Factor rule
Let's say you are given a ratio of two items: 2:3 Recognize that adding the two numbers gets you the "total" factor that is scaled with the ratio. 2:3:5 Ex: If it is given that in a classroom, the ratio of boys to girls is 2:3, and you are asked to find how many boys/girls in a 25 student classroom, you would recognize that "5" corresponds to the total number of students. The key to solve is to figure out the scale factor between the proportion and the total number of students. 25/5 = 5. So there should be 10:15:25, or 10 boys and 15 girls.
Chord
Line segment inside a circle. The ends lie on opposite sides on the circumference.
When do two Inscribed Angles have the same degree measure?
Look at the diagram. Two of the 3 points for BOTH CHORDS are at the same location. If only point is moved, the angles will be equal.
IN ANY TRIANGLE, THE LONGEST SIDE IS
Opposite the biggest angle
Vertical Angles
Pair of angles that open opposite to each other and are congruent
K ≤ x ≤ 3k + 12 How would you prove that -6 ≤ K?
Recognize that K ≤ 3k + 12 Then solve for K. In other words, you can set the lower bound equal to the upper bound to solve inequality problems in this format. -2k ≤ 12 -6 ≤ K
DO NOT MISREAD GRAPH ....
SCALES
How do you solve two-inequality in one problems? Ex: -7 ≤ -2x + 3 ≤ 15
Split the inequality into to and solve for separately First: -7 ≤ -2x + 3 Then: -2x + 3 ≤ 15 Last: Put both the results together 5 ≥ x x ≥ -6 -6 ≤ x ≤ 5
What are two methods to know for solving a systems of equality word problem?
The first method involves an system containing an equation and inequality r + b = 48 r < 2b You would solve for one variable to create an equation you can substitute into the equality r = 48 - b so 48 - b < 2b 48 < 3b 16 < b Essentially, we know r had to be some value that is 48 - b then, whatever 48 - b is, it is less than 2b. When there are 2 systems of equalities, if the equalities face the same direction you can add the two equations together. This can be done by multiplying the one equation by a negative. r + p ≤ 12 2r + 3p ≥ 30 -2 (r + p ≤ 12) = -2r - p ≥ -24 -2r - 2p ≥ -24 2r + 3p ≥ 30 p ≥ 6
Sector of a circle and it's area
The sector of a circle is the "pizza slice" area resulting from a CENTRAL angle. 1. To find the area, first find the TOTAL area of the circle. 2. Get the angle measurement of the central angle and divide it by 360 (as if to find a proportion) 3. Multiply this by the total area to get the sector area.
What does the shaded region of the graph of inequalities represent?
The shaded region (overlapping region) represents ALL THE POINTS that are (x,y) solutions to the graph.
What is the significance of the discriminate?
The value has no meaning. The sign does. If D is zero, there is 1 real root ( two solutions where y = 0) If D is a negative, there is no solution that equals y = 0 If D is positive, there are two solutions.
Root shortcut for finding vertex of parabola
The vertex x coordinate is always the average between the x values of the two roots (keep in mind to still use negatives if needed). To find the y coord of this vertex, simply plug in this avg. value
y = ab^t/k
This equation models exponential growth/decay. However, it is in sets of time periods. a is the original amount b is the growth/decay rate (1 +/- percentage as decimal) k is the time required to increase/decrease by an amount t represents how many sets of the time required to increase/decrease by an amount. t and k must be the same units. if t is in years, so is k Sometimes, k is written as a reciprocal; If the exponent is 3t for instance, math it to t/k; t / 1/3 = 3t, so k is 1/3. In other words, it would be every 1/3 day/month/year/etc.
What are the common rules for transformations of graphs?
To reflect a graph across the x axis, multiply the equation by -1. To move a function up, add a constant. To move a function down, subtract a constant. To move a function left add some number to x and substitute for x (so sub. (x + a) for x) To move a function right subtract a number to x and substitute for x (so sub. (x - a) for x)
When does a linear equation no solutions/infinitely many solutions?
When both sides are the same, the equation has infinitely many solutions: 3x + 1 = 3x + 1 If the question says, "True for all values of x" be mindful of this concept. When both sides have the same coefficient + x term, but different constants, the solution has no solutions. 3x + 1 = 3x + 5 If the question says, "True for no values of x" be mindful of this concept
When do systems of equations have no solutions? What about infinite solutions?
When both terms for x and y between the two equations are the same, however the constants are different. 3x + 2y = 5 3x + 2y = -4 Also, If one or both of the equations can be multiplied by some factor, there will still be no solutions. Ex: 2(3x + 2y = -4) ==> 6x + 4y = -8 still has no solution. For infinite solutions, both equations have to match, or be a factor of one anough 3x + 2y = 5 3x + 2y = 5 or 3x + 2y = 5 9x + 6y = 15
How should inequalities and systems of inequalities be interpreted according to < > ≤ and ≥?
When you get inequalities into slope form such as y ≥ 1/2x + 24, For when y is > or ≥, shade the region above the line. For ≥, it's solid; for > it's dashed For when y is < or ≤, shade the region below the line. For ≤, it's solid; for < it's dashed.
If a = b/3, what is a:b?
a:b = 1:3 solve for a:b (which is a/b) a/b = 1/3
Multiplying Pythagorean triples each by a constant,
gets you another triple.
The degree measure of AN INSCRIBED ANGLE is
half whatever the central angle is. Remember, central angle has it's vertex in the center. If a 90 degree angle is the central angle, the inscribed will be 45.
Two or more points are collinear
if they lie on the same line
The number of solutions can be found on the graph of a system of equations by
looking at the number of intersections. The number of intersections is equal to the number of solutions.
For a triangle to be made,
one of the 3 sides MUST be longer than the others.
x intercepts on ti 84
press zoom standard to reset graph 2nd trace 2: zero move the left right and guess near the intercept
For inequality functions such as y < 2x + 1, the shaded region that gets graphed the shaded region where two inequality functions overlap
represents all the possible points (x,y) that can be plugged into the inequality and produce a "true statement". For the less than/greater than, the curve will be dashed indicated the points lying on the curve aren't counted. points that, when plugged in two both equations, yield valid inequality statements for both equations.
For max/min word problems, if a whole humber answer is implied for a minimum problem for a maximum problem...
round the answer up. round down
Special Polygon formula for finding the total interior angle measurement
s represents the number of sides sum of interior angle = (s - 2) * 180 Ex: For a triangle, (3 - 2) * 180 = 180*
Two triangles are ___ if they
similar if they have the same angle measurements. On top of this, the relationship (the factor the sides differ between triangle) will be the same for each corresponding sides.
How to calculate Chance and Compound probability
total number of outcomes that is desired /total number of possible outcomes For two or more events happening in a sequence, multiply the probabilities.
Standard form of a circle (when at the center at the origin)
x² + y² = r² With r being a constant
