Chapter=3.3- 5.4

1. Chapter=3.3- 5.4 Tya Hyde

2. This Unit……..3.3-5.4 • You Know You have mastered this unit when you can do this without any problems. • One example of how you can apply chapter 3 to real life is by using linear programming to keep organized. When I say that I mean that you can keep organized by being able to figure out your profit/ income ect. • One example of how you can apply chapter 4 to real life is by using matrices. You use matrices to make software and with computers and for codes. • One example of how you can apply chapter 5 to the real world is by using factoring to figure out the area of thing like the area of a room.

3. Objectives • To Be able to • Solve quadratic equations by factoring • Write quadratic equations in intercept form • Find the intercept and max and min of quadratic equations • Write matrix equations • Find the inverse of matrices • Classify matrices • Use Cramer’s Rule • Multiply, add, and subtract matrices • Organize data in matrices • Graph • Quadratic functions • System inequalities

4. Definitions of the unit • Feasible region- The intersection of a graph of a system constrains. • Constraints- Inequalities. • Bounded- Closed in mean the inside of a shape. • Unbounded- open meaning the there is a space to go out. • Vertex- The point where to lines intersect. • Linear programming- the process of finding the max/ min values of a function for a region defined by inequalities. • Matrix- a rectangular array organized in rows and columns with variables and/or variables in enclosed brackets. • Element- The value of each matrix. • Zero matrix- every element is zero. • Square matrix- Has the same number and rows. • Second- order determinant- 2 by 2 • Third- order determinant- 3 by 3 • Parabola- The graph of any quadratic function. • Roots- The solutions of a quadratic equation.

5. Example 1: • Your school is planning to make toques and mitts to sell at the winter festival as a fundraiser.  The school’s sewing classes divide into two groups – one group can make toques, the other group knows how to make mitts.  The sewing teachers are also willing to help out.  Considering the number of people available and time constraints due to classes, only 150 toques and 120 pairs of mitts can be made each week.  Enough material is delivered to the school every Monday morning to make a total of 200 items per week.  Because the material is being donated by community members, each toque sold makes a profit of $2 and each pair of mitts sold makes a profit of $5.  • Step1: Variables x = the number of toques made weeklyy = the number of pairs of mitts made weekly • Step2: Constrains x ≤ 150 x ≥ 0 y ≤ 120 y ≥ 0 x + y ≤ 200 Objective function: f(x)= p= $2x+$5y 2(80) + 5(120) • Step 3: Graph • Step 4: This a Maximum 80 toques and 120 pairs of mitts each week. Vertices: (0, 120), (150, 0), (150, 50), and (80, 120) The profit is $760 

6. Example 1: Graph

7. Example 2: • Find 7 2 6 9 4 3 5 3 1 2 , if possible = 48 65 32 33 32 22 These are a 2by2 and 2by3 matrix Plug into the calculator 2nd matrix Over to edit, enter Then put in the matrix Then go back in press 2nd matrix Then press 1 press times then 2nd matrix then press 2 Then enter

8. Example 3: • Org. equation 2a-1-3c=-20 4a+2b+1=6 2a+1-1=-6 A B C 2-1-3 -20 a = .5 42 1 6 b 1 211 -6 c 6 Formula A^-1* B

9. Example 4: • Find the determinant. 4 8 (4)(3)-(8)(6) – Multiply 6 3 12-48 -Subtract 36 - Answer

10. Example 5: • Use The graph to determine the solutions This is Two solutions because it hits the x-axis in to different spots.

11. Example 6: • Solve x^2=4x by factoring. x^2- 4x=0 –the org. equation x^2-4x=0 –subtract 4x from each side x(x-4)=0 –factor the binomial x=0, y=4 or x-4=0 The solution set is {0,4}