Marking+Period+2+Review+Page

= **Marking Period 2 Review** =



You are expected to sign up for AT LEAST 2 sections. Besides giving some explanation about the section you must answer the BIG IDEA Question for your section. You may need to do some work and use photobooth to take a picture of your work and post it on the page.
 * Section || Name || Name ||  ||
 * 3-2 - || Sariah || Lauren ||  ||
 * 3-3 || Lauren || Trisha || Sierra ||
 * 3-4 || Sierra || Josh ||  ||
 * 5-1 || Keosha || Steph ||  ||
 * 5-2 || Sariah || Tia || Steph ||
 * 5-3 || Tia || Josh ||  ||
 * 5-4 || Amy || Trish ||  ||
 * 5-5 || Keosha || Josh ||  ||

3-2 You would like to minimize the amount of work required to solve a system of equations. Tell whether you would solve each system using substitution or elimination and why. What is Substitution? Where you solve for one variable and then put it in where that variable was. What is elimination? Where you get id of one variable by adding or subtracting equations.

Solve: Substitution: Elimination:

2x+3y=34 2x+4y=-10 __4x-3y=-4__ 3x+3y=-3 6x=30 3(2x+4y)= 3(-10) x=5 -2(3x+3y)= -2(-3) 6x+12y=-30 -6x-6y= 6 6y=-24 y=-4 y = 2x || 2) 3x + y = -7 x-y = -5 || 3) 3x -2y =0 9x + 8y = 7 || 1) You would use substitution because the "y" is already solved for. Steps to Solve: 4x + (2x) = 6 Substitute y into the equation and solve. 6x = 6 6x/6 = 6/6 x = 1
 * 1) 4x+y =6

4(1) + y = 6 Now that x is solved, plug it into the equation you used. In our case 4x + y = 6, and solve. 4 + y = 6 4 - 4 + y = 6 - 4 y = 2

(1, 2) is the solution.

2) You would use elimination because there is both a negative and positive "y" meaning that the "y's" in both equations will cancel. Steps to Solve:

Step 1: Find the value of one variable. 3x + y = -7 The y terms in both equations have oppisite coefficents. __+ x - y = -5__ 4x = -12 Add the equation to eliminate y. x = -3 First part of the equation.

Step 2: Substitute x into the original equation. (-3) - y = -5 -y = -2 y = 2 Second part of the solution.

(-3, 2) is the solution.

3) you would use elimination again, and to do that you would multiply the top equation by a -3 and that would eliminate the "x's" leaving the "y's" for you to solve for.

Step 1: -3(3x -2y = 0) Multiply for one of the variables in this equation so that they would cancel out. (In our case, x will cancel.) + 9x + 8y = 7

-9x + 6y = 0 Cancel the x variable. __+ 9x + 8y = 7__ 14y = 7 y = 1/2

Step 2: Find the other variable.

Classify and determine the number of solutions: 2x + y = 8 6x + 3y = -15

Step 1: Isolate y. y = -2x + 8 6x + 3y = -15 6x + 3(-2x + 8) = -15 6x -6x + 24 = -15 24 = -15 This is a contradiction. So its consistant with no solution.

Extra Examples (see Sariah for the answers): Use substitution for this problem: 1. 2x - y = 2 3x - 2y = 11

Use elimination to solve this: 1. 2x + 6y = -8 5x - 3y = 88

Classify and determine the number of solutions: 1. x - 2y = -8 4x = 8y - 56

3-3 Explain how to determine which region to shade to indicate the solution set of a system of linear inequalities. you plug in points into the equation and if the equation proves true then you shade the side that has that point on it. To see which side to shade, do the zero- zero method. 
 * When you need to shade your inequality all you do is plug a point on the graph in for "x and y" and if the statement is true (ex. 2>0 or -10<0) then you shade towards the point on your graph. But if the statement is not true then you shade away from the point. ** 

EXAMPLES and PROBLEMS: - What is a system of inequalities? - Try this problem: y__<__ -2x+ 4 y> x- 3 y__>__ 3/2x +2 x< 3
 * to find out the answer see page 199!
 * to find out the answer see page 200!

3-4 Explain the meaning of the constraints, feasible region, vertices, and objective function in a Linear Programming problem. CONSTRAINTS: The given set of conditions that the line or lines must meet in linear programing. FEASIBLE REGION: The solution thats meets the constraints in graph form (with a shaded region) VERITICES: The points that two lines meet to form an angle. it is at specific spots in a linear programing problem. OBJECTIVE FUNCTION: The answer to the linear programing problem. It is the series of lines with a shaded region. It also shows the maximums and minimums.

5-1 Explain the vertical and horizontal translations, reflection, vertical stretch and compression in parabolas. For a horizontal translation if f(x-h)=(x-h)^2 h <0 the parabola would move to the left and if h>0 the parabola would move to the right. If you are working with a vertical translation f(x)+k=x^2+k the parabola would move down for k<0and it would move up for k>0. If there were to be a reflection across the y-axis the input value would change an if it were to e reflected across the x-axis the output values would be the one to change.When a vertical compression occurs the output value would be the one to change a if f(x)=x^2 a x f(x)=ax^2. Only if the compression or stretch is by a factor of |a|. Then would a<|a|<1 compresses toward the y-axis and |a|>1 stretches away from the x-axis.

Questions: What is the vertax form of a quadratic function? What form can a quadratic function be written in? 17. F(x)= -x^2+4

5-2 What are the properties of a parabola. (see the Get Organized section of this chapter). If the equation has a negative a then it will open downward, if positive then it will open upward. To find the axis of symmetry use the formula -b/2a. Then to find the vertex plug your axis of symmetry into the original problem the answer you get is the y-intercept. That and the x-intercept is the axis of symmetry.

1.What is standard? f(x)=ax^2+bx+c

2.How do you know if a parabola is going to face up or down? Facing up the equation will have a positive a. Facing down a will be negative.

3.What is the axis of symmetry? The line that goes through the vertex of the parabola.

To figure out if it opens upward: f(x) = **a** x^2 + bx + c. If the variable a which is highlighted above is positive it opens up. If the variable a is negative it opens down. The variable will never equal zero.

To find the axis of symmetry: f(x) = **a** x^2 + **b** x + c. Use the formula - (x = -**b **/2**a** ) __//Just remember its always x = (vertex)//__
 * In the function: f(x) = a(x - h)^2 + k, the axis of symmetry is x = h.

To find the vertex: The x coordinate is already solved for you when you find the axis of symmetry. So to find the y coordinate you plug the x into the original equation and solve. Therefore the vertex is the point (-b/2a f(-b/2a))

To find the y-intercept: Well, its simple. Its just c: f(x) = ax^2 + bx +  **c

All you have to do is look at the variable a. If its positive, it has a minimum. If its negative, it has a maximum.
 * To determine if the graph has a minimum or a maximum:

Example: x^2 - 8x + 6 a.Determine if the graph opens upward or downward. It opens upward, because a is positive.  **(1)**x^2 - 8x + 6 b.Find the axis of symmetry. x = (-b/2a)** (1) **x^2  **- 8**x + 6 x =**<span style="color: rgb(4,15,231);"> 8 **/2**<span style="color: rgb(255,26,26);">(1) **x =** <span style="color: rgb(187,62,249);">4 c. Find the vertex. <span style="color: rgb(187,62,249);"> **4**<span style="color: rgb(3,3,3);">^2 - 8(<span style="color: rgb(21,19,22);"> **<span style="color: rgb(187,62,249);">4 **) + 6 16 - 32 + 6 = <span style="color: rgb(249,31,229);"> **-10 d.Find the y-intercept. x^2 - 8x <span style="color: rgb(2,3,2);"> **<span style="color: rgb(68,251,35);">+ 6
 * The axis of symmetry is 4.
 * (<span style="color: rgb(187,62,249);"> **4**<span style="color: rgb(5,5,5);"><span style="color: rgb(6,5,5);">,<span style="color: rgb(249,31,229);">  **-10**)
 * y-intercept: <span style="color: rgb(2,3,2);"><span style="color: rgb(68,251,35);"> **+ 6**

Examples (see Sariah for the answers): Find if it opens up or down, the axis of symmetry, the vertex, the y-intercept, and determine if it has a maximum or a minimum. 1. g(x) = x^2 - 3x + 2 2. h(x) = -x^2 - 2x - 8

5-3 Explain how to factor using the x-box method. When do you find a zero of a function and how? When do you find a root of an equation and how? In the top of the X is where a•c goes. The bottom is where b goes. Then you must find a factor of the top number that ads up to answer the bottom sum. To find the zeros of a function you set the equation to zero then do the same as you what to find the roots. To find the roots of an equation the equation must be in standard form to start. Then if the equation has a GCF factor that out. Then rewrite it as (a)(a)-2ab+(b)(b). Then factor the sides. Then answer the equation. Find the zeros: 1.f(x)=x^2+11x+24 -8

2.f(x)=x^2-9 0,9

Find the roots 3.4x^2=81 -9/2

4.36x^2-9=36 x=-1/2 x=1/2

5. 49x^2=28x-4 x=2/7

<span style="color: rgb(0,150,255); font-family: 'Arial Black',Gadget,sans-serif;">5-4 Explain how to convert a quadratic equation from standard to vertex form? -by completing the square 1. collect variables to one sides 2. divide to make the coefficient of x² 3. complete by adding 4. factor the variable expression as a perfect □ 5. take the square root of both sides 6. solve!

EXAMPLES and PROBLEMS: - x²+6x=15=9 - What is completing the □ ? - What do you solve by completing the square?

5-5 Complete the Get Organized problem on pg 352. Explain the parts of a complex number. A complex number is a number that can written in the form a+bi. Every complex number has a real part and an imaginary part b.

Questions: What is an imaginary number? 29. g(x)=4x^2 -3x+1 What is the complex conjugate of any complex number a+bi?