Part 2: Algebra Flashcards
What is a variable?
A letter that represents an unknown value
What is an algebraic expression?
an arithmetic expression with one or more variables
What is a term?
It’s a single part of an algebraic expression that is separated by an addition or subtraction sign from other parts of the expression.
What are like terms?
Terms are alike when they contain the same variables, with the same exponents.
What is a constant?
A term that does not contain any variables.
What is a coefficient?
A number that is multiplied by variables.
What is a polynomial?
An algebraic expression with a finite number of terms.
What is the degree of a term?
It’s the exponent that’s on the variable.
What is the degree of a polynomial?
The greatest exponent in the polynomial.
Identity 1: ca + cb
c(a + b)
Identity 2: ca - cb
c(a - b)
Identity 3: (a + b) ^2
a^2 + 2ab + b^2
Identity 4: (a - b)^2
a^2 - 2ab + b^2
Identity 5: a^2 - b^2
(a + b)(a - b)
Identity 6: (a + b) ^3
a^3 + 3a^2b + 3ab^2 + b^3
Identity 7: (a - b)^3
a^3 - 3a^2b + 3ab^2 - b^3
Rule 1: x^-a
1/x^a
Rule 2: (x^a)(x^b)
x^(a+b)
Rule 3a: x^a/x^b
x^(a-b)
Rule 3b: x^(a-b)
1/x^(b-a)
Rule 4: x^0
1
Rule 5: (x^a)(y^a)
(xy)^a
Rule 6: (x/y)^a
(x^a)/(y^a)
Rule 7: (x^a)^b
x^(ab)
What is an equation?
It is a statement of equality between two expressions.
What does it mean to solve an equation?
It means to find the values of the variables that satisfy the equation.
What are equivalent equations?
They are equations that have the same solution.
Equivalent equation rule #1 regarding constants:
The same constant can be added or substracted from each side of an equation, equality is preserved and the new equation is equivalent to the old one.
Equivalent equation rule #2 regarding constants:
The same constant can be multiplied or divided on each side of an equation and the new equation will be equivalent to the original one.
Equivalent equation rule #3 regarding equivalent expressions:
An equivalent expression can replace an expression inside an equation and the new equation will be equivalent to the original one.
What is a linear equation?
It’s a 1st degree polynomial where each term is either a constant or a single variable multiplied by a coefficient. None of the variables are multiplied by each other.
What is a linear equation in two variables?
ax + by = c
What is an ordered pair?
An ordered pair is a solution for a linear equation in two variables.
What is a quadratic equation?
ax^2 + bx + c = 0
What is the quadratic formula?
The quadratic formula is used to solve a quadratic equation: x = (-b +- √(b^2 - 4ac))/(2a)
What is the formula for simple interest?
V = P(1 + rt/100)
What is the formula for compound interest, compounded annually?
V = P(1 + r/100)^t
What is the formula for compound interest, compounded n times per year?
V = P(1+r/100n)^nt
How do you determine the distance between two points on the coordinate system?
Use the pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse.
In the equation y = mx + b, what are m and b called?
m is the slope, b is the y-intercept.
What shape is a parabola where a is positive in a quadratic equation?
The parabola opens up and its vertex is its lowest point.
What shape is a parabola where a is negative in a quadratic equation?
The parabola opens downward and its vertex is its highest point.
What is the equation that graphs a circle?
(x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center point, and r is the radius.
What does the graph of the absolute value function h(x) = |x| + C look like?
A “V” shape with its vertex at 0. Or if there is a constant C, then the vertex is at y = C.
What does the positive square root function graph look like?
The upper half of a parabola lying on it side.
How do you achieve a reflection about the y=x line?
By interchanging x and y.
Describe the graph made by the function g(x) = (x + 1)^2
It’s a parabola with the vertex at the origin, but then shifted to the left 1 unit.
Describe the graph made by the function ch(x), for any constant c, in relation to the function h(x).
It’s the graph h(x), stretched vertically if c > 1. And then shrunk vertically, if 0 < c < 1
Find an algebraic expression for:
The square of y is subtracted from 5 and the result is multiplied by 37.
(5 - y^2) * 37
Find an algebraic expression for:
Three times x is squared, and the result is divided by 7
((3x)^2) / 7
Find an algebraic expression for:
The product of (x + 4) and y is added to 18.
y(x + 4) + 18
Simplify each of the following algebraic expressions:
3x^2 - 6 + x + 11 - x^2 + 5x
2x^2 +6x + 5
Simplify each of the following algebraic expressions:
3(5x - 1) - x + 4
14x + 1
Simplify each of the following algebraic expressions:
x^2 - 16 / x - 4, where x != 4
x + 4
Simplify each of the following algebraic expressions:
2x + 5)(3x - 1
6x^2 + 13x -5
What is the value of f(x) = 3x^2 - 7x + 23, when x = -2?
49
What is the value of h(x) = x^3 - 2x^2 + x - 2, when x = 2?
0
What is the value of k(x) = (5/3)x - 7, when x = 0?
-7
If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(2)
1
If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(-2)
-1
If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(2) - g(-2)
2
Simplify: (n^5)(n^-3)
n^2
Simplify: (s^7)(t^7)
(st)^7
Simplify: r^12/r^4
r^8
Simplify: (2a/b)^5
32a^5/b^5
Simplify: (w^5)^-3
w^-15
Simplify: (5^0)(d^3)
d^3
Simplify: ((x^10)(y^-1)) / ((x^-5)(y^5))
(x^15)(y^-6)
Simplify: (3x/y)^2 / (1/y)^5
9x^2y^3
Solve for x: 5x -7 = 28
x = 7
Solve for x: 12 - 5x = x + 30
x = -3
Solve for x: 5(x + 2) = 1 - 3x
x = -9/8
Solve for x: (x + 6)(2x - 1) = 0
x = -6 or x = 1/2
Solve for x: x^2 + 5x - 14 = 0
x = 2 or x = -7
Solve for x: x^2 - x - 1 = 0
x = (1 + √5)2 and x = (1 - √5)/2
Solve for x and y:
x + y = 24
x - y = 18
x =21, y = 3
Solve for x and y:
3x - y = -5
x + 2y = 3
x = -1, y = 2
Solve for x and y:
15x - 18 - 2y = -3x + y
10x + 7y + 20 = 4x + 2
x = 1/2, y = -3
Solve for x:
-3x > 7 + x
x < -7/4
Solve for x:
25x + 16 >= 10 - x
x >= -3/13
Solve for x:
16 + x > 8x -12
x < 4
For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer.
83
If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y ?
15: 8
Kathleen’s weekly salary was increased by 8 percent to $712.80. What was her weekly salary before the increase?
660
A theater sells children’s tickets for half the adult ticket price. If 5 adult tickets and 8 children’s tickets cost a total of $81, what is the cost of an adult ticket?
9
Pat invested a total of $3,000. Part of the money was invested in a money market account that paid 10 percent simple annual interest, and the remainder of the money was invested in a fund that paid 8 percent simple annual interest. If the total interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10 percent and how much at 8 percent?
$800 at %10
$2200 at %8
Two cars started from the same point and traveled on a straight course in opposite directions for 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed of each car for the 2-hour trip?
car 1: 48m/h
car 2: 56m/h
A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12.
(a) What is the fixed total cost to charter the aircraft?
(b) What is the cost per person if 40 people charter the aircraft?
Total cost: $4320
CPP for 40: $108
An antiques dealer bought c antique chairs for a total of x dollars. The dealer sold each chair for y dollars.
(a) Write an algebraic expression for the profit, P, earned from buying and selling the chairs.
(b) Write an algebraic expression for the profit per chair.
P = cy - x PPC = y - x/c
Algebra Figure 16 that follows shows right triangle PQR in the xy-plane. Find the following where P is (-2, 6), R (5,0), and Q is unknown but is at y = 0 and x -2.
(a) The coordinates of point Q
(b) The lengths of line segment PQ, line segment QR, and line segment PR
(c) The perimeter of triangle PQR
(d) The area of triangle PQR
(e) The slope, y-intercept, and equation of the line passing through points P and R
Q (-2, 0) PQ =6 QR = 7 PR = √85 perimeter = 13 + √85 area = 21 y = -6/7x + 30/7
In the xy-plane, find the following.
(a) The slope and y-intercept of the line with equation 2 y + x = 6
(b) The equation of the line passing through the point (3, 2) with y-intercept 1
(c) The y-intercept of a line with slope 3 that passes through the point (−2, 1)
(d) The x-intercepts of the graphs in parts (a), (b), and (c)
slope = -1/2 y-intercept = 3
y = 1/3x + 1
y-intercept = 7
x-intercepts: 6, -3, -7/3
For the parabola y = x^2 − 4x −12 in the xy-plane, find the following.
(a) The x-intercepts
(b) The y-intercept
(c) The coordinates of the vertex
x-intercepts = 6, -2 y-intercept = -12 vertex = 2, -16
For the circle ( x − 1)^2 + ( y + 1)^2 = 20 in the xy-plane, find the following.
(a) The coordinates of the center
(b) The radius
(c) The area
center = (1, -1) radius = √20 area = 20pi
For each of the following functions, give the domain and a description of the graph y = f (x) in the xy-plane, including its shape, and the x- and y- intercepts.
(a) f(x)= −4
(b) f(x)=100 - 900x
(c) f(x)=5− (x+ 20)^2
(d) f(x)= √(x +2)
(e) f(x)=x + |x|
a) domain = all real numbers Looks like a horizontal line, passing through y = -4. Does not intercept x.
b) , x-intercept = 1/9. y-intercept = 100, domain all real numbers, slope of -900.
c) inverted parabola. y-intercept = -395, x-intercepts = -20 +- √5, domain all real numbers
d) half parabola, x-intercept at -2, y-intercept at √2, x >= -2
e) steep linear line starting at origin extending to the upper right. x >= 0 then the negative x-axis
What is a domain?
It is the set of numbers in x.
What is a range?
The set of numbers in y.
