1. Chapter 9 Quadratic Equations And Functions By Chris Posey and Chris Bell

2. 9-1 Square Roots You already know how to find the square of a number i.e. 4²=16 To find the square root, you basically do the opposite of squaring a number. i.e. √(16)=4

3. Apply Skills Learned  Now that you know how find the square root, let’s try some problems for practice! Find the square roots of the following problems. 1. √(144)2. √(16)3. √(400)4 √(36) 4. √36=6 3. √400=20 2. √16=4 1. √122=12

4. 9-2 Solving by Quadratic Equations by Finding Square Roots Example 1> x 2 =4 Write the original equation x= + √4 or - √4 Find square roots x= 2 or -2 2 2 =4 and (-2 2 )=4 Example 2> 3x 2 -48=0 Write the original equation 3x 2 =48 Add 48 to each side x 2 =16 Divide each side by 3 x= √16 Find square roots x=4 or -4 4 2 =16 and (-4) 2 =16

5. Apply Skills Learned Now that you know how to solve quadratic equations by finding square roots, find the answers to the following problems. 1.5t 2 -125=02. x 2 =2253. x 2 -15=10 3. x=5 2. x=15 1. t=5

6. 9.3 Simplifying Radicals Product Property of Radicals: √ab= √a ∙ √b Example> √50= √(25∙2)= 5 √2 Example> √48= √(4∙12)= 4 √3 Quotient Property of Radicals: √a/b= √a/√b Example> √32/50= √(2∙16)/√(2∙25) Factor using square roots =√(16/25) Divide common factors =√(16)/√(25) Use Quotient Property =4/5 Simplify

7. Apply Skills Learned Now that you know how to simplify radicals, try it out on your own! 1.√(9/49)  √(18) 3. √(196)  √2 3)14

8. 9.4 Graphing Quadratic Functions A quadratic function is a function that can be written as a formula Y=ax 2 +bx+c, where a≠0 This will give a graph with a u shape called a parabola. If a is grater than 0, it opens up. If it is negative, then it opens down.

9. Graphing (9.4) Find the x coordinate of the vertex, which is x=-b/2a Make a x,y table and use the x values Plot the points and connect them with a smooth curve to form the parabola

10. Example Sketch graph of y=x 2 -2x-3 Find x coordinate of vertex. -b/2a=-2/2(1)=1 Make a table x| -2-101234 y|50-3-4-305 (1,-4) is the vertex, plot the rest of the points and draw a curve connecting them.

11. Practice Find vertex coordinates, and make a x,y value table using x values to the right and left of the vertex. y=-4x 2 -4x+8  x|-3 -2 -1 -.5 0 1 2 y|-16 0 8 9 8 0 -16

12. 9.5 Solving Quadratic Equations by Graphing Solutions for the quadratic graphs are the x-axis intercepts, where y=0. This number can be checked in the original equation, by setting it equal to 0.

13. Example y=x 2 -2x-3 is shown here. Note that the x intercepts are located at -1 and 3. If substituted for x, the equation would result in zero, the solutions for the equation.

14. Practice Solve the equation algebraically, check your answers by graphing. 2x 2 +8=16 Solutions are ±2

15. 9.6 Solving Quadratic Equations by the Quadratic Formula The solutions of the quadratic equation, ax 2 +by+c=0, are (-b+/-√(b 2 -4ac) x= --------------------- (2a) when a≠0 and b 2 -4ac≥0

16. Example Solve x 2 +9x=14=0 Solution 1x+9x+14=0 (-b+/-√(b 2 -4ac) x= --------------------- (2a) (-(9)+/-√((9) 2 -4(1)(14)) x= ------------------------------- (2(1)) -9+/-√(25) x=-------------------- 2

17. -9+/-5 X=---------------- 2 There are 2 solutions x=-2, and x=-7

18. Practice Solve the quadratic equation. 2x 2 -3x=8 X=2.89, and x=-1.39

19. 9.7 Using the Discriminant The discriminant is the radical expression in the quadratic formula ie. (-b+/-√(b 2 -4ac)) x= --------------------- (2a) If the discriminant is positive, then the solution has 2 solutions. If it is zero, it has one solution. If it is negative, there are no real solutions.

20. Example Find value of the discriminat and determine if it has two solutions, one solution, or no solutions. x 2 -3x-4=0 Use the equation, ax 2 +bx+c=0 to identify values, ie. a=1, b=-3, c=-4 Substitute into the discriminant b 2 -4ac=(-3) 2 -4(1)(-4) =9+16 =25 Discriminant is positive, therefore two solutions.

21. Practice Determine whether the graph will intersect the x-axis at one, two, or zero points. y=x 2 -2x+4 It does not intersect the axis

22. 9.8 Graphing Quadratic Inequalities The graph of a quadratic inequality consists of all the points (x,y) that are part of the inequality. Quadratic inequalities can be represented by; y> or < or  or  ax 2 +bx+c

23. When graphing, use a dashed line for the parabola when the equality is > or< Use a solid line when it is  or  The parabola separates the graph into two sections. A test point is a point that is not on the graph. If the test point is a solution, then shade the region, if not shade the other region.

24. Example Solve –x 2 + 4 < 0. Find the x-axis intercepts –x 2 + 4 = 0 x2 – 4 = 0 (x + 2)(x – 2) = 0 x = –2 or x = 2 Use the origional inequality to find the area to shade y<0, therefore shade everything outside the parabola, below the x-axis.

25. Practice Determine whether the orderd pair is a solution of the inequality. y≤x 2 +7, (4,31) Point is a solution outside of the parabola

26. Now remember to study hard, because we’ll be watching you….