Math Project
1. Medium Problem
Multiply the matrices:
[3 -9] [4 6 1] [(3)(4) +(9)(5) (3)(6)+(-9)(7) (3)(1)+(-9)(3)]
[2 1] * [5 7 -3] = [(2)(4)+(1)(5) (2)(6)+(1)(7) (2)(1)+(1)(-3)]
2*2 2*3
Step 1: Multiply row by column
Step 2: Add to get the sum and answer for the each number
Step 3: Simplify
= [(12+-45) (18+-63) (3+-27)] = [-33 -45 -25]
[(8+5) (12+7) (2+-3)] [13 19 -1]
2*3
2. Hard Problem
Find the Determinant of a 2 by 2 Matrix
A= [a b] det(A) = [a b]= a*b-c*b
[c d] [c d]
A= [4 9]
[3 2]
Step 1: Multiply together A and B
Step 2: Take the product of A and B and subtract the product of C and D from the the product of A and B.
Step 3: Simplify to get the determinant
3. Ridiculous Problem
Find the Determinant of a 3*3 Matrix
Det(A) = [ a b c] = a(ei-hf)-d(bi-hc)+g(bf-ec)
[d e f]
[g h I ]
3x3
Det(A) = [ -2 4 7]
[5 6 8]
[-1 3 0]
Step 1: Plug in the numbers for each letter
Step 2: Simplify to get the determinant
4. Medium Problem
Solve for x.
Problem: x2- 3= 2x
Step 1: Stet the problem up equal to zero: x2– 2x – 3= 0
Step 2: Factor: (x – 3) (x +1)= 0
Step 3: Find x: x – 3= 0 or x +1 =0, x=3 or x= -1
5. Easy Problem
Graph 4x- 8y= 16
Step 1: Subtract 4x from both sides of the equation
Step 2: Divide 8 from both sides of the equation, which will give you an equation in y-intercept: y= -2 – 2/4x
Step 3: Graph the equation
6. Hard Problem
Solve x 2+bx+c by factoring
Problem: x 2- 10x – 24 = 0
Step 1: Write original equation: x 2-10x- 24=0
Step 2: Factor
Step 3: Zero product property
Step 4: Solve for x.
7. Medium Problem
Graph a quadratic Function in Vertex Form: y= 1/2 (x+2) squared +4
Step 1: Identify the constants: a= ½, h= -2, k=4
Step 2: Plot vertex: (h, k)= (-2,4)
Step 3: Find axis of symmetry: x= -2
Step 4: Plug a number in for x to find x and y
Step 5: Plot the vertex, and the point you found
Step 6: Graph
8. Evaluate an algebraic expression:
Evaluate: -8x-3x+2, when x=-10
Step 1: Substitute -10 into the equation
Step 2: Evaluate and multiply
Step 3: Add together to get sum
9. Tell whether a relation is a function:
Directions: To tell whether a relation is a function, an input can go to one output. If an input is mapped onto two outputs, than it is not a function, however an output can have to two inputs.
10: Simplify the square root of 36 over 2
Step 1: Separate the fraction with two square root signs, one over 36 and the other over 2.
Step 2: Simplify the square root of 36
Step 3: Multiply the square root of 2 to the top and bottom of the equation to get rid of the square root in the denominator
11. Linear Programing Problem:
Sophie makes wedding rings. She doesn’t want to run out. She can make one regular size ring with one diamond in one hour. She can make a ring with 3 diamonds in two hours. She only has 50 diamonds and only 6 hours of time to work. She wants to make at least 4 rings with 3 diamonds. How many regular rings can she make and how many large rings can she make with 3 diamonds to maximize her profit?
Step1: Create a key: r= regular ring, l= large ring w/ 3 diamonds
Step 2: Create an equation: 60r+ 120l is less than or equal to 360, 1x+3y is less than or equal to 50
Step 3: Write out things you know from the equation, so that you can begin to graph: y greater than or equal to 4, x greater than or equal to 2, x greater than or equal to 0, y greater than or equal to 0
Step 4: Graph lines and shade in on graph
Step 5: Create an equation for her profit: P= 400r +700l
Step 6: Pug in points from graph into equation to get the profit: P= 400(2)+700(2)= $2,220
Multiply the matrices:
[3 -9] [4 6 1] [(3)(4) +(9)(5) (3)(6)+(-9)(7) (3)(1)+(-9)(3)]
[2 1] * [5 7 -3] = [(2)(4)+(1)(5) (2)(6)+(1)(7) (2)(1)+(1)(-3)]
2*2 2*3
Step 1: Multiply row by column
Step 2: Add to get the sum and answer for the each number
Step 3: Simplify
= [(12+-45) (18+-63) (3+-27)] = [-33 -45 -25]
[(8+5) (12+7) (2+-3)] [13 19 -1]
2*3
2. Hard Problem
Find the Determinant of a 2 by 2 Matrix
A= [a b] det(A) = [a b]= a*b-c*b
[c d] [c d]
A= [4 9]
[3 2]
Step 1: Multiply together A and B
Step 2: Take the product of A and B and subtract the product of C and D from the the product of A and B.
Step 3: Simplify to get the determinant
3. Ridiculous Problem
Find the Determinant of a 3*3 Matrix
Det(A) = [ a b c] = a(ei-hf)-d(bi-hc)+g(bf-ec)
[d e f]
[g h I ]
3x3
Det(A) = [ -2 4 7]
[5 6 8]
[-1 3 0]
Step 1: Plug in the numbers for each letter
Step 2: Simplify to get the determinant
4. Medium Problem
Solve for x.
Problem: x2- 3= 2x
Step 1: Stet the problem up equal to zero: x2– 2x – 3= 0
Step 2: Factor: (x – 3) (x +1)= 0
Step 3: Find x: x – 3= 0 or x +1 =0, x=3 or x= -1
5. Easy Problem
Graph 4x- 8y= 16
Step 1: Subtract 4x from both sides of the equation
Step 2: Divide 8 from both sides of the equation, which will give you an equation in y-intercept: y= -2 – 2/4x
Step 3: Graph the equation
6. Hard Problem
Solve x 2+bx+c by factoring
Problem: x 2- 10x – 24 = 0
Step 1: Write original equation: x 2-10x- 24=0
Step 2: Factor
Step 3: Zero product property
Step 4: Solve for x.
7. Medium Problem
Graph a quadratic Function in Vertex Form: y= 1/2 (x+2) squared +4
Step 1: Identify the constants: a= ½, h= -2, k=4
Step 2: Plot vertex: (h, k)= (-2,4)
Step 3: Find axis of symmetry: x= -2
Step 4: Plug a number in for x to find x and y
Step 5: Plot the vertex, and the point you found
Step 6: Graph
8. Evaluate an algebraic expression:
Evaluate: -8x-3x+2, when x=-10
Step 1: Substitute -10 into the equation
Step 2: Evaluate and multiply
Step 3: Add together to get sum
9. Tell whether a relation is a function:
Directions: To tell whether a relation is a function, an input can go to one output. If an input is mapped onto two outputs, than it is not a function, however an output can have to two inputs.
10: Simplify the square root of 36 over 2
Step 1: Separate the fraction with two square root signs, one over 36 and the other over 2.
Step 2: Simplify the square root of 36
Step 3: Multiply the square root of 2 to the top and bottom of the equation to get rid of the square root in the denominator
11. Linear Programing Problem:
Sophie makes wedding rings. She doesn’t want to run out. She can make one regular size ring with one diamond in one hour. She can make a ring with 3 diamonds in two hours. She only has 50 diamonds and only 6 hours of time to work. She wants to make at least 4 rings with 3 diamonds. How many regular rings can she make and how many large rings can she make with 3 diamonds to maximize her profit?
Step1: Create a key: r= regular ring, l= large ring w/ 3 diamonds
Step 2: Create an equation: 60r+ 120l is less than or equal to 360, 1x+3y is less than or equal to 50
Step 3: Write out things you know from the equation, so that you can begin to graph: y greater than or equal to 4, x greater than or equal to 2, x greater than or equal to 0, y greater than or equal to 0
Step 4: Graph lines and shade in on graph
Step 5: Create an equation for her profit: P= 400r +700l
Step 6: Pug in points from graph into equation to get the profit: P= 400(2)+700(2)= $2,220