1. Direction: _____________ Width: ______________ AOS: _________________ Set of corresponding points: _______________ Vertex: _______________ Max or Min? __________ y – int: _____________ x – int: _____________ Function? __________ Domain: ___________ Range: _____________ Rising: _____________ Falling: ____________ Opens down Narrow (-0.5, -7) & (2.5, -7) x = 1 (1, 0) Maximum (0, -3) (1, 0) Yes (- ,  ) (- , 0] (- , 1) (1,  )

2. xx 2 – 4x – 5 0-5 1-8 2-9 3-8 4-5 Direction: Opens Up Width: Standard AOS: x = 2 Corresponding Point: (6, 7) Vertex: (2, -9); Minimum y – intercept: (0, -5) x – intercept: (-1, 0) & (5, 0) Function? Yes Domain: (- ,  ) Range: [-9,  ) Rising: (2,  ) Falling: (- , 2)

3. x-2(x+2) 2 + 8 -40 -36 -28 6 00 Direction: Opens Down Width: Narrow AOS: x = -2 Corresponding Point: (1, -10) Vertex: (-2, 8); Maximum y – intercept: (0, 0) x – intercept: (-4, 0) & (0, 0) Function? Yes Domain: (- ,  ) Range: (- , 8] Rising: (- , -2) Falling: (-2,  )

4. x½(x+4) 2 -62 -5½ -40 -3½ -22 Direction: Opens Up Width: Wide AOS: x = -4 Corresponding Point: (-6, 2) Vertex: (-4, 0); Minimum y – intercept: (0, 8) x – intercept: (-4, 0) Function? Yes Domain: (- ,  ) Range: [0,  ) Rising: (-4,  ) Falling: (- , -4)

5. x-3x 2 + 6x - 4 -13 0-4 1 2-4 3-13 Direction: Opens Down Width: Narrow AOS: x = 1 Corresponding Point: (4, -28) Vertex: (1, -1); Maximum y – intercept: (0, -4) x – intercept:  Function? Yes Domain: (- ,  ) Range: (- , -1] Rising: (- , 1) Falling: (1, -  )

6. Need Help? Look in textbook in Section 5.1: Modeling Data w/ Quadratic Functions Section 5.2: Properties of Parabolas Section 5.5: Quadratic Equations Section 5.8: The Quadratic Formula Worksheet: Properties of Parabolas

7. Day 14: Finding Properties of Parabolas Using Algebra

8. Objectives: To identify properties of parabolas using algebra

9. Begin with the equation in standard form. Standard form of a quadratic equation:

10. Direction: Parabolas open up or open down Direction is determined by the sign of “a” Open “up” a is positive Open “down” a is negative y = ax 2 + bx +c

11. Width: Parabolas can be narrow, standard or wide Width is determined by the value of a (not including the sign) Narrow |a| > 1 Standard |a| = 1 Wide |a| < 1 y = ax 2 + bx +c

12. Axis of Symmetry: The line that divides the parabola into two parts that are mirror images AOS is found using the formula: Equation: a = 1, b = 4, c = 1 AOS: x = -2 y = ax 2 + bx +c

13. Vertex: The point where the parabola passes through the AOS Vertex is found by plugging the AOS into the equation. Equation: AOS: x = -2 Vertex: (-2, -3) Vertex is a minimum because a is positive and parabola opens up. y = ax 2 + bx +c

14. y – intercept: The point on the graph where the parabola intersects the y-axis. y – intercept is found by, making x = 0 and solving for y Y – intercept will be “c” value Equation: y -intercept: (0, 1) y = ax 2 + bx +c

15. Number of Real Solutions: The number of times the parabola intersects the x-axis on the real coordinate plane. Use the disriminant to determine the number of solutions The discriminant is b 2 – 4ac 2 Real Roots b 2 – 4ac > 0 1 Real Root b 2 – 4ac = 0 0 Real Roots 2 Imaginary Roots b 2 – 4ac < 0

16. x – intercept(s): The point(s) on the graph where the parabola intersect the x - axis. Other names include: roots, zeroes and solutions. To find x – intercepts, make y = 0 and solve. Solve quadratics by taking square roots, factoring or using the quadratic equation. y = ax 2 + bx +c Equation: x -intercept: (-2+ √3, 0 ) & (-2- √3, 0 )

17. Function: Yes, passes the VLT Domain: The domain of a parabola is (- ,  ) Range: Depends on how parabola opens, includes max or min and infinity. Always use bracket w/ #. Use y – value of vertex and direction to determine range. (- ,  ) Equation: Opens Up Vertex: (-2, -3) Function? Domain: Range: y = ax 2 + bx +c Yes [-3,  )

18. Intervals of Rising/Falling: The interval of the domain where the graph is rising or falling as x increases Use x – value of vertex and direction to find intervals Rising: ______________ Falling: _____________ Equation: Opens Up Vertex: (-2, -3)

19. Direction: __________ Width: _____________ AOS: ______________ Vertex: _____________ Max or Min? __________ y – int: _____________ # of Real Solutions: ___ x – int: _____________ Function? __________ Domain: ___________ Range: _____________ Rising: _____________ Falling: ____________ Opens up Standard x = -2 (-2, -9) Minimum (0, -5) (-5, 0) & (1, 0) Yes (- ,  ) [-9,  ) (-2,  ) (- , -2) a is positive a =1 2 Or

21. Direction: _____________ Width: ______________ AOS: _________________ Vertex: _______________ Max or Min? __________ y – int: _____________ # of Real Solutions: ____ x – int: _____________ Function? __________ Domain: ___________ Range: _____________ Rising: _____________ Falling: ____________ Opens Down Narrow x = -1 (-1, -2) Maximum (0, -5) Yes (- ,  ) (- , -2] (- , -1) (-1,  ) a is negative a = 3 0

23. Name three of the properties you learned about today and how to find them.