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Quadratics By: Aditi Singh, Maegan Lloyd, Bella 4th period Un-Factoring Methods of Un-factoring: 1.FOIL 2.Box Method Un-Factoring:FOIL First (x+a)(x+b) Outer x2+ax+bx+ab Inner x2+(a+b)x+(ab) Last Un-Factoring:Box Method x x x2 bx a ax ab x2+ax+bx+ab x2+(a+b)x+ab b Un-Factoring:Example 1.(x-9)(x+2) (2x+4)(x-5) 2. Factoring x2+bx+c b Factor Factor Factor + Factor Factor Factor c Factor * Factor ➔What 2 numbers do you add to get b and multiply to get c? Factoring:Example 1.x2-7x-18 Factoring:With Co-Efficient ax2+bx+c=0 b Factor Factor Factor + Factor Factor (ax2+(factor)x) (+(factor)x+c) Factor ac Factor * Factor **Grouping method **will make more sense in example ➔What 2 numbers do you add to get b and multiply to get a*c? Factoring:Example 4x2-19x+12=0 **want to take out same factor from both Factoring: Worksheet Example 7x2 + 9x Zeros Zero:x-intercepts of graph; when y=0 (x+a)=0 (x+b)=0 **solve for x Zeros:Example 1.(7x+4)(x-7)=0 2. 5x2-x-16=2 Zero: Worksheet Example 5r2 − 44r + 120 = −30 + 11r Quadratic Formula **x2+bx+c ➔Use Quadratic Formula to find x-intercepts when equation is unfactorable ➔Simply plug in values into equation and solve Quadratic Formula:Example 9x2+7x-56 Quadratic Formula: Worksheet Example 5x2 + 9x = −4 Discriminant b2-4ac If discriminant is: Negative-no real solutions to equation 0-1 real solution to equation Positive-2 real solutions to equation **Solution:x-intercept Discriminant: Example 1.Discriminant of: 9x2+7x-56 2.Number of solutions:7x2-32x-6 Discriminant: Worksheet Example −9b2 = −8b + 8 AOS Discriminant Axis of Symmetry (AOS) x=(-b)/(2a) Maximum Maximum- highest point on the graph Minimum Minimum- lowest point on the graph Vertex Vertex- the maximum or minimum of a graph How to find vertex: 1.Find axis of symmetry (x value) 2.Plug in that value into quadratic equation (y value) 3.Coordinates (x, y) Vertex:Example Find vertex & state if max/min: x2+5x+7 Vertex: Worksheet Example y = -2x2 + 4x + 1 Graphing Quadratics 1.Calculate and graph Axis of Symmetry (points can be reflected over AOS) 2.Calculate and graph vertex 3.Calculate and graph zeros 4.Calculate and graph y-intercepts 5.Plug in other x-values into equation to complete the shape of parabola Graphing Quadratics:Example x2-9x+8 Quadratic Inequalities < or > use dotted line < or > use solid line < or < shade below line > or > shade above line **Choose a test point to double-check shading; usually (0,0) Quadratic Inequalities:Example y<x2+5x+10 Quadratic Inequalities:Example y>x2-7x-18 Quadratic Inequalities: Worksheet Example y<-2x2-8x-12 Vertex Form y=a(x-h)2+k vertex:(h,k) AOS:h Vertex Form:Transform X+h: translate left h units (x-h) X-h: translate right h units (x+h) +k: translate up k units (y+k) -k: translate down k units (y-k) |a|>1: stretch vertically by a units (y*a) |a|<1: compress vertically by a units (y*a) -a: reflect across x-axis (y*-1) Vertex Form:Transform Example y=-0.5(x-2)2+3 ➔Reflected over x-axis ➔Compressed by 0.5 ➔Right 2 ➔Up 3 Vertex Form:Transform Example Original Reflected (x) Compress (.5) Right (2) Up (3) (-2,4) (-2,-4) (-2,-2) (0,-2) (0,1) (-1,1) (-1,-1) (-1,-0.5) (1,-0.5) (1,2.5) (0,0)**vertex (0,0) (0,0) (2,0) (2,3) (1,1) (1,-1) (1,-0.5) (3,-0.5) (3,2.5) (2,4) (2,-4) (2,-2) (4,-2) (4,1) **Can also plug in x-values into equation to get y-values and coordinates for graphing Vertex Form:Transformation Worksheet Example y = −0.5(x + 1)2 − 7 Vertex Form: To Standard Form y=a(x-h)2+k y=a(x-h)(x-h)+k y=a(x2-hx-hx+hh)+k y=a(x2-(h+h)x+hh)+k y=ax2-a(h+h)x+ahh+k Vertex Form: To Standard Form Example y=3(x-1)2+5 Word Problem Example 1. You throw a bowling ball off of a cliff into the ocean. Its height as a function of time could be modeled by the function h(t)=-16x2+16x+480, where t is the time in seconds and h is the height in feet. a. How long did it take for the bowling ball to reach his maximum height? b. What was the highest point that the bowling ball reached? c.The bowling ball hit the water after how many seconds? Word Problem Worksheet Example The number of bacteria in a refrigerated food is given by n(t)= 20t2 - 20t + 120, where T is the temperature of the food in Celsius and n is the number of bacteria in the food. At what temperature will the number of bacteria be minimal?