Basic Functions
The location (x-coordinate) of the absolute maximum or absolute minimum (specify which) of the quadratic function f(x)=3x2+4x-1
Has an absolute min because LC=3, which is positive, so the parabola opens up
Min is at x-coordinate of vertex, x=-4/2(3) = -2/3
Find the vertex and the x-intercept(s) of the function f(x) = 25 - x^2
Vertex: (0, 25) x-intercepts: (+/-5, 0)
The solution to the system
10x+4y=8
15x+6y=-6
No solution
The solution to 4x=7
x=log47 because log and exponential functions of same base are inverses
The sum of the matrices
2 1 3 + -3 -1 7
4 5 6 0 -2 -5
-1 0 10
4 3 1
The equation of a line in point-slope form that passes through the points (1,2) and (-3,4)
m=2/-4 = -1/2
y-2=-1/2 (x-1)
The domain of (2x+1) / (x-5)(2x) in interval notation
x can't be zero or 5 but can be everything else
(-inf, 0) U (0,5) U (5, inf)
A coffee shop sells drinks for $4 each. It costs them $.50 per cup to make each drink. Their monthly fixed costs total $2,000. Write their revenue, cost, and profit functions.
R(x)=4x
C(x)=.5x+2,000
P(x)=4x-(.5x+2,000)=3.5x-2,000
The solution to 53x-1=25x
53x-1=52x
3x-1=2x
x=1
(or can take log5 on both sides)
The product of
2 3 1 0
0 -1 -2 -1
4
1
-7
The domain of (2x+1)1/2 in interval notation
[-1/2,inf)
The leading term, leading coefficient, and degree of the polynomial f(x)=3x2-5x4+2x-9
LT: -5x4
LC: -5
Degree: 4
The amount of time it will take an investment to double if it is compounded continuously at 1.5%
2P=Pe.015t
2=e.015t
ln(2)=.015t
t=ln(2)/.015 = 46.21 years
The value of log3(815)
log3((92)5)
log3(((32)2)5)
log3(320)
=20
The system of equations expressed as a matrix equation
5y-2x=-1
3x-6=y
-2 5 x = -1
3 -1 y 6
The quadratic function f(x) = x2-2x+5 in standard/vertex form
x-coordinate of vertex = 2/2(1) = 1
y-coordinate is f(1)=4
f(x)=(x-1)2+4
The hole(s) of the rational function
f(x)=(x+1)(x-1) / (3x+2)(x-1)
(must give both the x and y coordinate)
x=1 and y=(1+1)/(3(1)+2)=2/5
(1,2/5)
The amount of money that should be invested in an account earning 3.25% compounded monthly to produce a final balance of $25,000 in 15 years
A=P(1+r/m)mt
25,000=P(1+.0325/12)12(15)
P=$15,364.12
The solution to log7(x-2)+log7(x+3)=log714
x=4
The inverse of the matrix
2 4
-4 10
5/2 1
-1 -1/2
The graph of the piecewise function
f(x) = -x-7 if x is less than or equal to -2
x2 if x is greater than -2 and less than or equal to 2
6 if x>2
Closed circle on (-2, -5), with slope of -1, arrow going to the left
Parabola centered at the origin with open circle on (-2, 4) and closed circle on (2,4)
Horizontal line with open circle on (2,6) and arrow going to the right
The equation(s) of all asymptote(s) and the x-coordinate of all holes of the rational function
f(x)=(x+1)(2x-1) / (3x+2)(2x-1)(-x+3)
VA: x=-2/3 and x=-3
HA: y=0
Hole: occurs at x=1/2
The number of items needed to be sold to maximize profit, if revenue is R(x)=-x2+24x and cost is C(x)=12x+28.
P(x)=R(x)-C(x)
=-x2+24x-(12x+28)
=-x2+12x-28
Parabola opens down, so max at vertex x=-12/2(-1)=6
The solution of log2(x+1)-log2(x-4)=3
x=33/7
The solution (given as a matrix) of the system of equations below, found by using elementary row operations
3x+4y=1
x-2y=7
1 0 3
0 1 -2