PSAT Study

Area of a sphere

4πr^2

Fractional Exponents

Within the exponent fraction, the numerator tells us what power to raise the base to, and the denominator tells us which root to take

f(x) is another way of saying

Y

The negative sign inside the parentheses indicates a __________ reflection (across the y-axis), so the parabola should still _____ __.

horizontal, open up

Decreasing functions have y-values that decrease as the corresponding x-values _________.

increase

Tangent a =

opposite/adjacent

Sin a =

opposite/hypotenuse

A line slanted to the right would be

positive

Linear equations and linear graphs can be used to model relationships and changes such as those concerning

time, temperature, or population

af(x) (0 < a < 1) means that f(x)

undergoes vertical compression

Negative Exponent Rule

x ⁻ⁿ = 1/xⁿ

slope-intercept form

y=mx+b, where m is the slope and b is the y-intercept of the line.

Anything to the power of 0 will equal

1

Area of a square

A=s²

Area of a circle

A=πr²

Area Of A Cone

A=πr×s

"My own belief is, although I have never before said so in so many words, that the time will come when African Americans in the South will be accorded the political rights which his ability, character and material possessions entitle him to. I think, though, that the opportunity to freely exercise such political rights will not come in any large degree through outside or artificial forcing, but will be accorded to African Americans by white people themselves, and that they will protect him in the exercise of those rights. Just as soon as the South gets over the old feeling that it is being forced by "foreigners," or "aliens" to do something which it does not want to do, I believe that the change in the direction that I have indicated is going to begin. In fact, there are indications that it is already beginning to a slight degree." This passage can best be described as (A) a description of a state of of affairs intolerable to the author. (B) a statement of belief about society and how it will change. (C) a declaration of basic rights and a roadmap to achieve them. (D) a call to action to correct an injustice.

(B) a statement of belief about society and how it will change. The author's topic is his belief, the scope is gaining rights, and the purpose is to express hope for change.

The number k can be determined in the following way: Multiply m by 2, add 3n to the result, and subtract (4m − 5n) from this sum. What is the value of k in terms of m and n? (A) −2m − 3n (B) −2m + 2n (C) −2m + 8n (D) 6m − 2n

(C) −2m + 8n

The following passage is an adaptation of an excerpt from Up From Slavery: An Autobiography by Booker T. Washington. "My own belief is, although I have never before said so in so many words, that the time will come when African Americans in the South will be accorded the political rights which his ability, character and material possessions entitle him to. I think, though, that the opportunity to freely exercise such political rights will not come in any large degree through outside or artificial forcing, but will be accorded to African Americans by white people themselves, and that they will protect him in the exercise of those rights. Just as soon as the South gets over the old feeling that it is being forced by "foreigners," or "aliens" to do something which it does not want to do, I believe that the change in the direction that I have indicated is going to begin. In fact, there are indications that it is already beginning to a slight degree." According to the passage, "the political rights" mentioned in line 4 will come about through (A) increased political pressure on those denying the rights. (B) additional laws mandating those rights. (C) peer pressure designed to embarrass anyone denying those rights. (D) a natural evolution of society.

(D) a natural evolution of society.

Power of a Power Property

(a^m)^n = a^mn

Equation of an Ellipse

(x-h)^2/a^2 + (y-k)^2/b^2 = 1; Center (h,k) Axis a and b

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

1/2 (3x + 17) = 1/6 (8x - 10) Which value of x satisfies the equation above? A) -61 B) -55 C) -41 D) -35

A) -61

Area of a triangle

A=1/2bh

Area of a regular polygon

A=P/2×a

The General Equation for a Conic Section is

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Loan Balance Formula

B = A (1 + r/n)NT - P((1 + r/n)NT - 1/(1 + r/n) - 1)

Which ordered pair (, x y) satisfies the system of equations shown below? 2x-y = 6 x + 2y = −2 A) (−6, 2) B) (−2, 2) C) (2, −2) D) (4, 2)

C) (2, −2)

Tyra subscribes to an online gaming service that charges a monthly fee of $5.00 and $0.25 per hour for time spent playing premium games. Which of the following functions gives Tyra's cost, in dollars, for a month in which she spends x hours playing premium games? A) C(x) = 5.25x B) C(x) = 5x + 0.25 C) C(x) = 5 + 0.25x D) C(x) = 5 + 25x

C) C(x) = 5 + 0.25x

110x + y = 1,210 A coffee shop is running a promotion where a number of free coffee samples are given away each day. The equation above can be used to model the number of free coffee samples, y, that remain to be given away x days after the promotion began. What does it mean that (11, 0) is a solution to this equation? A) During the promotion, 11 samples are given away each day. B) It takes 11 days during the promotion to see 1,210 customers. C) It takes 11 days during the promotion until none of the samples are remaining. D) There are 11 samples available at the start of the promotion.

C) It takes 11 days during the promotion until none of the samples are remaining.

What is the minimum value of the function graphed on the xy-plane for −4 ≤ x ≤ 6? A. Infinity B. -4 C. -2 D. 1

C. -2

Ms. Walser's class had 18 students. She used three equally weighted tests to determine their final grades. The class average for the first test was 92, and the class average for the second test was 77. If the overall class average was 84, what was the average score for the third test? A. 74.3 B. 77 C. 83 D. 84.3

C. 83; (92 + 77 + c)/3 = 84(3) = 252 - 169 = 83

Jacques Charles was a French scientist who discovered the relationship between the temperature and volume of a gas. Specifically, Charles found that gases expand when heated. This relationship was formalized in Charles's Law, which illustrates a linear relationship between temperature and volume in gases. The graph here shows the volume of a sample of gas as it is cooled. If T is the temperature of the gas in °C and V is the volume in liters, which of the following equations, when plotted, could produce a graph? A. V = 0.004T + 100 B. V = 0.004T C. V = 0.004T + 1 D. V = 0.004T - 0.25

C. V = 0.004T + 1; The skeleton of the equation you're looking for is V = mT + b. Because each answer choice has a different y-intercept, finding this is enough to get you to the right answer; there is no need to determine the slope. Although the graph is not centered like you're used to, you can still find the y-intercept. In this case, it's 1.

________ functions have y-values that stay the same as the x-values _________.

Constant; increase

If f (x) = -x + 5 and g(x) = x 2, which of the following is not in the range of f (g(x)) ? A. -11 B. 0 C. 1 D. 9

D. 9. When dealing with a composition, the range of the inner function becomes the domain of the outer function, which in turn produces the range of the composition. In the composition f(g(x)), the function g(x) = x^2is the inner function. Every value of x, when substituted into this function, will result in a nonnegative value (because of the square on x). This means the smallest possible range value of g(x) is 0. Now look at f(x). Substituting large positive values of x in the function will result in large negative numbers. Consequently, substituting the smallest value from the range of g, which is 0, results in the largest range value for the composition, which is -0 + 5 = 5. Because 9 > 5, it is not in the range of f(g(x)), making D correct.

Given that f(x)=2x+1 and g(x)=x+2/3 what is the value for of (fg)(-5)? A. -6 B. -2 C. 3 D. 9

D. 9; f(-5) = 2(-5)+1 = -9 x -1 = 9

A municipality charges two types of local taxes: a per capita tax, which is a flat fee that every person pays, and a local income tax, which is a percentage of the person's annual income, i. If a taxpayer's total tax bill is given by the function T = 0.02i + 25, then the value 0.02 best represents which of the following? A. The per capita tax B. The taxpayer's annual income C. The total tax bill minus the per capita tax D. The amount of the income tax as a percentage

D. The amount of the income tax as a percentage; 0.02i represents the percentage income tax times the annual income (which the question tells you is i). Therefore, 0.02 must represent the amount of the income tax as a percentage

What is the formula for density?

D=m/V (Density=mass/volume)

What is E=mc^2 used for?

Einstein's equation E = mc2 shows that energy and mass are interchangeable. The theory of special relativity explains how space and time are linked for objects that are moving at a consistent speed in a straight line. One of its most famous aspects concerns objects moving at the speed of light.

Euler's Polyhedral Formula

F+V=E+2 : Face = F Vertex = V Edge = E

_________ functions have y-values that __________ as the corresponding x-values increase.

Increasing; increase

The Compound Interest Equation is written as

P = C (1 + r/n)^nt P = future value C = initial deposit r = interest rate (expressed as a fraction: eg. 0.06) n = number of times per year interest is compounded t = number of years invested

Continuous Compound Interest Equation

P = C x e^rt P = future value C = initial deposit r = interest rate (expressed as a fraction: eg. 0.06) t = number of years invested

Simplified Compound Interest Equation (when interest is only compounded once per year)

P = C(1 + r)^t P = future value C = initial deposit r = interest rate (expressed as a fraction: eg. 0.06) t = number of years invested

Volume of a cone/pyramid

V=1/3b(h)

Volume of a cylinder

V=πr²h

f(x) is an example of an _________ ________.

algebraic change

exponental growth

as a population gets larger, it also grows at a faster rate

If B2 - 4AC is...then the curve is

a... < 0 ellipse, circle, point or no curve. = 0 parabola, 2 parallel lines, 1 line or no curve. > 0 hyperbola or 2 intersecting lines

What is the Pythagorean Theorem formula?

a^2+b^2=c^2

cosine a

adjacent/hypotenuse

A tangent is perpendicular to the radius at the point of ________.

contact

tangent circles

coplanar circles that intersect in one point

a^2 - b^2 = (a+b)(a-b) is formula for

difference of squares

af(x)(a > 1) is when

f (x) undergoes vertical expansion

f(x) − a means

f(x) moves down a units

f(x − a) would require

f(x) moves right a units

f(x) + a means

f(x) moves up directly a units

f(ax)(a > 1) means that

f(x) undergoes horizontal compression

f(ax) (0 < a < 1) is

f(x) undergoing horizontal expansion

That straight line means you're dealing with a

linear equation

Solve for r, d = r/t

r = d/t

-f(x) is f(x)

reflected over the x-axis (top to bottom)

f(−x) is f(x)

reflected over the y-axis (left to right)

Fundamental Identitiy

sin^2 a + cos^2 a = 1

b in y = mx + b represents

the point at which the line intersects the y-axis

If ax^2 + bx + c = 0 then x = Quadratic Formula, which is

x = -b ± √(b² - 4ac)/2a

Quadratic Formula is

x = -b ± √(b² - 4ac)/2a

sin(−α)=

−sin α

adding radicals

√(a+b)≠√a+√b

multiplying radicals

√ab = √a x √b