Math Final Review

Solve the following quadratic inequality: x^2 + 2x − 24 ≥ 0 Write your answer in interval notation. Note: Use oo for ∞ and U for union and U for union.

(-oo,-6]U[4,+oo) *-Solve the inequality x^2 + 2x − 24 ≥ 0; factor out the x value to get 6x-4x. -Factor out again to get x(x+6)-4(x+6) ≥ 0. Simplify to (x-4)(x-6)≥0. -Equal both sides to zero and solve.

Simplify: logx(1/x^7)

-7 *Plug in Calculator

Select all of the following graphs which represent y as a function of x.

-Linear Graph> Going down -Linear Horizontal> Straight line going through -2. -Cubic Graph> Went up, down, then up. -Four Curves> Starting from left to right.

The function graphed above is: Increasing on the interval(s): Decreasing on the interval(s):

Increasing on the interval(s): (-2,0.5) Decreasing on the interval(s): (-oo,-2)U(0.5,oo) *- When it curves up it is increasing. - When it goes down it is decreasing. - ALWAYS ( ) - ALWAYS (?,oo) on the last y value for decreasing.

Write the equation in logarithmic form.Assume that all constants are positive and not equal to 1. 4^w = m

log4(m) = w *- Base = 4 ; Exponent = w ; Number = m. - Arrange it in the logarithmic equation. log[base](number) = (exponent).

Evaluate using a calculator and round to four decimal places. log0.47(0.88) ~

~0.1693 *- Use calculator to solve.

Evaluate the partial sum of the series: 16^∑[↓(k=1)] k2=

1496 *-Use equation n(n+1)x(2n+1/6 - Plug was in terms; The top number = n [ex. n=16] -Factor out the num & dem if possible. Solve 2nd ( ). Also factor. -Solve the new equation.

For the sequence an=(−1)^n x 10 / n^2

1st: -10 2nd: 5/2 3rd: -10/9 4th: 5/8 100th: 1/1000 *- Substitute the # term you want in place of "n" in equation. [ex. (-1)^1 x 10 / (1)^2].

For the sequence an=2(an−1−2) and a1=5, find 1, 2, 3, 4, 5 terms

1st: 5 2nd: 6 3rd: 8 4th: 12 5th: 20 *-The 1st term is given=5 -So plug in the fist value into the equation an=2(a(n-1)-2) = 2(5-2) -Solve to get the second term = 6 -Now plug the 2nd term into the equation, and continue that until the 5th term.

Find the first term and the common difference of the arithmetic sequence whose 10th term is −28 and 17th term is −56.

1st: 8 Common Difference: -4 *- Assume the 10th term is the 1st term = a10 = -28 ---> a1 = -28. - Now take the difference between the subscripts ; 10 - 1 = 9. Subtract that difference from a17 ; 17 - 9 = 8. This means that a8 = -56. - Plug numbers into equation an = a1 + (n-1)d to find the difference ; -56 = -28 + (8-1)d. Solve. d = -4 - Now use the same equation to find a1. an is now -28 and n = 10. -28 = a1 + (10-1)-4. Solve ; a1 =8.

A chemist has three different acid solutions. The first acid solution contains 20% acid, the second contains 35% and the third contains 55%. He wants to use all three solutions to obtain a mixture of 60 liters containing 40% acid, using 33 times as much of the 55% solution as the 35% solution. How many liters of each solution should be used?

= 20 liters of 20% solution = 10 liters of 35% solution = 30 liters of 55% solution *- Identify the x, y, and z values. The 20, 35, and 55 percentages respectively. - Identify the equations needed. 1) x + y + z = 60. 2) 0.20x + 0.35y + 0.55z = 0.4(60). 3) z = 3y. - Start with the equation x + y + z = 60. Replace the z value with 3y as indicated by the third equation. New equation is x + y + 3y = 60. Solve to get 4y + x = 60 -Use the equation 0.20x + 0.35y + 0.55z = 0.4(60). Replace the z value with 3y. New equation is 0.20x + 0.35y + 0.55(3y) = 0.4(60). Solve to get 0.20x + 2y = 24. - Use the equation 4y + x = 60 to solve for x. Move the y value over. New equation is x = -4y + 60. - Plug the x value into the second equation 0.20x + 2y = 24. New equation 0.20(-4y + 60) + 2y = 24. Solve for y ; y = 10. - Go back to the equation x = -4y + 60. Plug in the y value 10. New equation is x = -4(10) + 60. Solve for x ; x = 20. - Use the equation x + y + z = 60. Plug in x an y values to solve for z. New equation is 20 + 10 + z = 60. Solve for z ; z = 30.

Given that f(x)=1/x−1, which of the following is the graph of f^-1(x)?

B = Cubic Graph/Centered around (-1,0) *- Use calculator to graph the original and choose the other one.

Given A = 4 x 4 matrix ; B = 2 x 4 matrix ; C = 4 X 2 matrix Which of the following products can be computed?

CB, AC, BC, BA *- Equal the dimensions of the matrices. [ex. CA = 4 x 2 = 4 x 4]. - Matrix dimensions we set up as rows x columns. - Columns = rows

Given the series: 1 − (1/7) + (1/49) − (1/343) + ⋯ Does this series converge or diverge?

Coverages Partial Sum = 7/8 *-Use equation S=a1/1-r (ratio=r; ratio=difference) -Plug 1st term into a1 and difference into r -Solve S=1/1-(1/7)

Find the domain and range of the function graphed below.

Domain = [-2,3) Range = (-5,4] *- Domain: x axis values ; L ---> R. 1st: [ ; 2nd: ). - Range: y axis values ; D ---> U. 1st: ( ; 2nd: ].

Find the domain of the function f(x)=1/3x+5. What is the only value of x not in the domain?

Only Value= -5/3 *Set equal to zero and solve.

Find a formula for the general term "an" of the sequence assuming the pattern of the first few terms continues. {7,6,5,4,3,...}

an = -n + 8 *-Use equation an = a1 +d(n-1) -Plug first term in a1; find difference(-1) and plug it in -Solve

Given the geometric sequence: 6, 421/7, 294/289,... Find an explicit formula for an: Find a9:

an = 6(7^n-1)/17(n-1) a9 = 0.0049584301 *- Find the ratio (or difference). Divide the first term by the second term. Then flip them to get the ratio. - Use the equation a1r^n-1. Can put in as fraction ; 6(7^n-1)/17^n-1. Needs n-1 on top and bottom. - Plug the term wanted into the equation. 6(7^9-1)/17^9-1. Use calculator and solve.

Let f(x)=7x−4 and g(x)=x2−6x+5

(f ∘ g)(x) = 7x^2-42x+31 (g ∘ f)(x)= 49x^2-98x+45 *- (f(g)): Plug g(x) into f(x). - (g(f)): Plug f(x) into g(x). - (a-b)^2 = a^2-2ab+b^2

Sketch a graph of ((x+1)^2/9)-(y+3)^2=1

(h,k) ---> (-1,3) *-Use equation: ((x-h)^2/a^2)-((y-k)^2/b^2) -Plug in values. -Plug in Calc. ->< = + (positive) -˅^ = - (negative)

Solve the system. Give your answer as (x,y,z): 2x−y−6z=2 6x+6y+4z=−4 2x+5y+2z=−18

(x,y,z) = (4,-6,2) *-First, take second line and divide by 2. -Take the first line and add both second and third line in 2 separate equations. -First equation: multiply 1st by 3; 2nd of -2; add both terms together. -Second equation: multiply third line by -1; add both together. -Take answers and add them- first multiply 1st term by -4 and 2nd by 11. Add them to get y. -Take the second new term and plug in y to get z. -Take first term and plug in values for y and z to get x.

Write the equation of the circle centered at (3,−7) with radius 9.

(x-3)^2 + (y+7)^2 = 81 *Use equation (x-h)^2 + (y-k)^2 = (r)^2

Find all zeros of f(x)=x3−3x2−11x+5. Enter the zeros separated by commas. Enter exact value, not decimal approximations.

-1-sqrt/2; -1+sqrt/2; 5 *-Equal equation to 0. Solve to get (x-5)(x^2+2x-1)=0 - FACTOR. x-5=0 = 5; Take the quadratic equation of (x^2+2x-1)=0.

Given the matrices A=⎡−1 −2 −3 0 −3 −3⎤and B=[0 5] A. It is______to multiply these matrices in this order AB. B. If multiplication is possible what is AB.

-Possible -[-10 0 -15] *-Multiply the matrices together. -Multiply the first column by the first value of the 2nd matrix. -Multiply the second column by the 2nd value of the 2nd matrix. -In each row, add solutions together to get answer.

Suppose you have $9200 deposited in an account with interest compounded monthly. After 10 years, the account has grown to $10400. What is the interest rate on this account?

1.23% *- Use equation A = P(1 + r/n)^tn. Plug in given values. 10400 = 9200(1 + r/12)^10(12). -Using equation divide the P number on both sides. And combine the 10(12) = 120. - Remove the exponent by turning it into a fraction ; 120 = 1/120 and put it on both sides. - Solve the (1.30)^1/120 side ; = 1.001. - Subtract the 1 from both sides. - Multiply the denominator 12 on both sides. - Solve and get 1.23 = r.

Write the following as a single logarithm. Assume all variables are positive. 2(log4(4)+log4(a))−log4(2)=

3/2 +log2(a) *- In the first part of the equation, cancel the log4(4) and replace it with 1. 2(1+log4(a))-log4(2) - Use the equation loga^4(b) = 1/y(loga(b)). Turn log4(2) into log2^2(2). - Distribute 2(1+log4(a)). -Simplify log2^2(2) into 1/2. - With the new equation of 2 + 2log4(a) - 1/2, combine 2 and 1/2 to get 3/2. - From new equation 3/2 + 2log2^2(a), cancel the power of 2 and turn it into 1/2. Combine 2 and 1/2 - both 2's cancel out. This leaves the answer be 3/2 + log2(a).

NASA launches a rocket at t=0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t)=−4.9t^2 + 43t + 122. Assuming that the rocket will splash down into the ocean, at what time does splashdown occur? How high above sea-level does the rocket get at its peak?

= 11.032 seconds. = 216.337 meters *- t = 0 ---> h(t) = 0 - A: Take equation and = to 0= -4.9t^2 + 43t + 122 ; use the quadratic equation. (Calculator - NEEDS to be "-"(subtract)). - B: Use equation x = -b/2a; -43/2(-4.9) ; plug value for t into original equation then solve.

The equation of the ellipse that has a center at (10,4), a focus at (14,4), and a vertex at (5,4), is (x−C)^2/A^2 + (y−D)^2/B^2 = 1

A = 5 B = 3 C = 10 D = 4 *- The center (10,4)---> h = 10 ; k = 4 so C = 10 & D = 4. - Graph center then graph focus (14,4) & vertex (5,4). Do the same on the other side of the center. - Count how far the focus and vertex is away from center. That value is A(vertex) & B(focus). - Maybe the + in the equation ='s an one off foci?

The height y (in feet) of a ball thrown by a child is y = (−1/14)x^2 + 2x + 5 where xx is the horizontal distance in feet from the point at which the ball is thrown.

A. 5 feet B. 19 feet C. 30.31 feet *- A) Use the equation y = (−1/14)x^2 + 2x + 5. The y value is the last value in the equation. y = 5 - Horizontal distance is 0, therefore x = 0. Substitute 0 for x, then y = 5' high. - B) Use the equation y = (−1/14)x^2 + 2x + 5. (Max height = axis of sym). Use the equation -b/2(a). Plug values from problem into new equation. a = -1/14 b = 2. - Solve for x. -2/2(-1/14). x = 14 - Use the original equation y = (−1/14)x^2 + 2x + 5. Plug the x value into equation. Solve for y ; 19. - C) Set the equation to 0. Use the quadratic equation. x = 30.31. - NEEDS "-" subtract

Given the function f(x)={9x+6 x < 0 {9x+12 x ≥ 0 Calculate the following values: f(-1) = f(0) = f(2) =

f(-1) = -3 f(0) = 12 f(2) = 30 *- Plug the x values into both equations and soIve. - The answer is the that is true. - < = less than - > greater than

Find the time required for an investment of 5000 dollars to grow to 6900 dollars at an interest rate of 7.5 percent per year, compounded quarterly.

t = 4.33 years *- Use the equation A = P(1 + r/n)^nt. -Plug in values into the equation. - Equation is 6900 = 5000(1 + 0.075/4)^4t. - Divide 5000 by both sides. New equation is 1.38 = 1.01875^4t. - Take the natural log (ln) of both sides. ln(1.38) = ln(1.01875)^4t. - Move the 4t to in front of the ln(1.01875). - Divide 4ln(1.01875) by both sides. - t = ln(1.38)/4ln(1.01875). Use calculator. t = 4.33.

Solve for x: log(x)+log(x+3)=4

x = (-3+sqrt/40009)/2 *-Combine logs: log(x(x+3)) = 4 ----> log(x^2+3x) = 4 -Cancel log by changing 4 into 10^4: x^2 + 3x = 10^4. -Solve to get value x^2+3x-10000 = 0. -Use quadratic formula to get answer (if first value negative then +, if not then use -) .

Let f(x)=2x^2 + 11x + 15 / 2x^2 − 11x + 5 This function has: 1) Vertical asymptotes at x= 2) Horizontal asymptote at y=

x = 1/2 ; 5 y = 1 *- Graph on calculator.

Solve the given equation for x. 3^3x-6 = 47

x = 3.1682 *- Take the equation and take the log3 out of it. log3(3^3x-6 = 47) = log3(47). - Solve the equation 3x + 6 = log3(47). Move the 6 over to the other side. 3x = log3(47) - 6. - Divide 3 by both sides. This creates fractions ; 1/3 & 6/3. -New equation is x = 1/3log3(47) + 6/3. - Solve to get x = 3.1682.

The following is data for the first and second Quiz scores for 8 students in a class. First Quiz | Second Quiz 13 9 20 16 21 19 38 35 39 36 40 38 42 40 47 42 Plot the points in the grid below, then sketch a line that best fits the data.

{Graph with each plot point and linear line that best fits the data points.} *-Each row of points = (x,y) [ex. (13,9)] -Plot each point -Use the 1st point to start the line then end it as accurately as can be.