SAT Math Concepts & Desmos

Parallel lines

- Never intersect

- Have the same slope

Perpendicular lines

- intersect at a 90 degree angle

- slopes are negative inverses of each other (m & -1/m)

how to quickly recognise properties of linear functions

in the form y = mx +b:

- b = y intercept (0, b)

- if m/slope is negative, the line/y value will decrease as x increases

relationship between lines & number of solutions

- 2 different slopes: intersect once, 1 solution

- same slope, diff y-incpt: 0 solutions (parallel)

- same slope, same y-incpt: same line, infinite solutions

mean

the average of a set of numbers

- in desmos: mean(1,2,3)

- removing a value > mean: new mean decreaeses

- removing a value < mean: new mean increases

- if removed value = old mean: new mean remains the same

median

the middle # when all numbers are organised in numerical order

- in desmos: median(1,2,3)

- only changes if the removed number forces the remaining middle numbers to shift (depends on data set)

mode

the number that appears the most

range

the space between the smallest to largest number

- calculated by: (max-min value)

standard deviation

average distance between mean and value in data set

- in desmos: stdev(1,2,3)

- in a bar graph:

=> clustered near centre (ie. a pyramid-shape) -> low standard deviation

=> spread out toward edges (ie. flat shape) -> high standard deviatoin

- on a line graph, the larger range (y), the higher standard deviation is

- removing an outlier (the highest/lowest number) makes the data set less spread out, so standard deviation decreases

- removing a number that is very close or equal to the mean, the average distance grows, so standard deviation increases

samples —> entire populations

sample proportion/percentage x total population

range

estimate ± margin of error

axⁿ ± bxⁿ

(a+b) xⁿ

axᵐ × bxⁿ

ab + xᵐ⁺ⁿ

(axᵐ) / (bxⁿ)

(a / b)xᵐ⁻ⁿ

(aⁿ) / (bⁿ)

(a/b)ⁿ

(axᵐ)ⁿ

aⁿ × xᵐⁿ

(aᵐ)ⁿ

aᵐⁿ

a-ᵐ

1 / (aᵐ)

a⁰

1

√x = 16

Can never equal a negative number; √ (square roots) are always asking to find a POSITIVE number

x² = 256

has a positive and negative number

standard form of quadratic functions

f(x) = ax^2 + bx + c

- directly tell you the y intercept (0, ?)

vertex form of quadratic functions

y = a(x-h)^2 + c

- h: horizontal shift

=> if (x-h): shift right

=> if (x+h): shift left

- k: vertical shift

=> -k: shift down

=> +k: shift up

(h, k) is the vertex, whether it is max or min depends on whether the slope is positive or negative

- a: vertical stretch

=> -a: parabola opens down, vertex is the max

=> +a: parabola opens up, vertex is the min

=> a > 1: vertical stretch

=> 0 < a < 1: vertical compression

factored form of quadratic functions

y = a(x-b)(x-c)

- tells you all the solutions/x intercepts/zeroes of the graph

- the exponent on each bracket tells you the behaviour at each 0

finding vertex

- x = -b/2a

- plug in x to find y

What is the discriminant?

(b²- 4ac)

a quadratic equation in standard form (ax^2 + bx + c) as ___ solutions if...

- 0 solutions if its discriminant (b²- 4ac) < 0

- 1 real solution if (b²- 4ac) 0

- 2 real solutions if (b²- 4ac) > 0

Exponential functions

y = a(b) ᶜ⁽ˣ⁻ʰ⁾ + d

- a: vertical stretch/compression

=> |a| > 1: stretches

=> 0 < |a| < 1: vertical compression

=> positive a: increasing

=> negative a: decreasing

- b: base

=> b > 1: exponential growth

=> 0 < b < 1: exponential decay

- c: horizontal stretch/compression/reflection

=> |c| > 1: compression by 1/|c|

=> 0 < |c| < 1: stretch horizontally by 1/(|c|)

=> positive c: increasing right

=> negative c: increasing left

- h: horizontal translation

=> (x-h): right

=> (x+h): left

- d: vertical translation

=> +d: up

=> -d: down

=> horizontal asymptote at y = d

Exponential equations can never show...

SLOPES; the slopes of exponential functions are always changing

Even functions

f(x) = f(-x)

Odd functions

-f(x) = f(-x)

# of Zeroes

- max = degree (largest exponent)

- min:

=> even degrees: 0

=> odd degrees: 1

# of Turning Pints

- max: degree - 1

- min:

=> even degrees: 1

=> odd degrees: 0

End behaviours

- Even functions:

=> positive: Q2 - Q1

=> negative: Q3 - Q4

- Odd functions:

=> positive: Q3 - Q1

=> negative: Q2 - Q4

Remainder theorem

- When a polynomial function is divided by (x-a), the remainder = p(a)

- if p(a) = 0, then (a,0) is a root, and (x-a) is a factor of p(x)

Formulas for prisms

Volume = B x H

=> Surface area of base (B) depends on the shape of the base

Surface area = 2B + (pxh)

Congruent angles

Have equal measure

Similar triangles

Have the same shape, but are not necessarily the same size

-> They have corresponding angles; equal, corresponding angles are proportional to one another

Angle rules

- All angles in a triangle add up to 180 degrees

- All angles in quadrilateral add up to 360 degrees

- Angles in straight lines add up to 180 degrees

- Opposite angles are ALWAYS EQUAL

Pythagorean theorem

a² + b² = c²

2 complementary angles add up to:

90 degrees

cos of 1 angle =

sin of its complement angle (180 - first angle)

addiding them up together always = 90 degrees

Circle theorems

- arc length / circumference = centre angle / 360 degrees

- sector angle / total area = centre angle / 360 degrees

converting radians to degrees

- pi = 180 degrees

- (central angle / 2pi) = (arc length / circumference) = (sector area / circle area)

Special triangles

Special angles

Circle equation

(x-h)² + (y-k)² = r²

- the circle's centre is at (h, k), and its radius is r long

- to shift the circle, find the centre and do transformations there; then rewrite the equation

- if there are 2 points, one at the centre an one at r, plug x and y in respectively to find r

Evaluating statistical claims

- positive association: a directly proportional relationship, as 1 thing goes up, so does the other

- negative association: an inversely proportional relationship, as 1 thing goes up, the other goes down

-

Rules to solving rational equations

- Always check for extraneous solutions when the rational equation has 2+ answers

- Remember to check for holes --> if x is an hole, than it is an extraneous solution

Circle distance formula

Altitude of a triangle

= the height of a triangle

Perimeter formula of an isosceles triangle

p = 2x + x(sqrt2)

2x = two equal legs

x(sqrt2) = hypotenuse

Perimeter formula of an equilateral triangle

p = 3s

Limitations of Regression on desmos

- regression only finds one solution, not ALL; therefore, if there are multiple valid answers and u have to find the smallest/largest, then you have to use another method

- when a question says 'positive interger', try to restrict the variable to be >1; >0 will onlyget an inf

Tangent line in the circle

- touches a circle at exactly one point

- line is always perpendicular to h=the radius and forms a 90 degree angle (remember perpendicular lines have slopes that are negative reciprocals)

A circle's diameter must...

- pass through the centre of the circle

- touch 2 opposite points on the circle's edge

2 tangent theorem

- if 2 tangent segments are drawn to a circle from the same external point, those two segments are always equal in lengh

• p can be any value

• always remember that radius is both distances added together / 2

• add a slider for a —> manipulte it into the two graphs are touching / have 1 intersection

• remember that when they are asking for ‘what is the smallest value of b’, you OMIT THE NEGATIVE SIGN; because the factor is actually -(b); they are not asking the smallest overall factor, just the smallest absolute value of b itself

• always remember to check if desmos is on DEGREE MODE

Sum of 4 Equations

Create euations out of all the info you know first:

• s = j

• p (seniors) = 20 + 2(s+j)

• s + j + p = 0.8(s + j + p +f)

Then, put all of these values into brackets and run regression in desmos

Sum of Solutions

• Move equation to one side to find all solutions (which are x--intercepts) → this is your f(x) equation

  • In this case, f(x) = (m-4b)(x-m) - (x-m)²

• In this euation of f(x), there are 2 different values of x here that con cause f(x) = 0 → you can then use a list system to do regression on desmos and find values

• If the sum of the solutions to the euation is 3m + 27, that means the sum of all zeros = 3m + 27

• Rememebr to set {x1 > x2}; this forces desmos to treat the different solutions as 2 different numbers —> they cannot equal each other

• Set m = 1 (can be whatever number as this wil not change the graph)

• b

Simple Quadratic Regression / Quadratic word problems

• Use a table & insert all known values

• y1 ~ax²+bx + c → regression will find all the vlalues you need, unless you need a product of some numbers/etc

Quadratic constants and clauses

• Since f(-9) = f(3, then these are x intercepts, and you can find the vertex thorugh
  (-9+3)/2 = -3

• Since the formula for the x value of the vertex = -b/2a, then
  -4/2a ~ -3

• You can use regression to find the value of a

• When you find the value of a using regresssion, see if II is right; it is NOT

• To test if c < 0, set c to an arbitrary number that is bigger than 0; then, set f(x) to the vertex, and see the y value is < 0

  • f(-3) = -2.2; so even though when c > 0, k is still < 0, so I is not necessarily true

Function of Geometric Series 4

• All of the answers are equivalent forms of each other, just structured differently

• Choose the simpest form to plug into desmos