1. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions The quadratic function is a second- order polynomial function It is always written in this format, with the coefficient parameters: a, b, c f(x) = ax 2 + bx + c OR ax 2 + bx + c = f(x)

2. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions Example 1: Identify the coefficient parameters (a, b, c) in the following quadratic functions: a.f(x) = 5x 2 + 10x + 15 b.f(n) = n 2 – 10n + 22 c.h(r) = 7r 2 + 14r – 7 d.g(x) = x 2 – 25 e.g(x) = 7x 2 f. h(n) = 5x – 24 a = 5, b = 10, c = 15 a = 1, b = -10, c = 22 a = 7, b = 14, c = -7 a = 1, b = 0, c = -25 a = 7, b = 0, c = 0 a = 0, b = 5, c = -24 (linear)

3. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions This is a graph of a quadratic function. We call it a parabola What are some observations we can make about the graph? x y

4. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions The axis of symmetry is the x- coordinate that forms the middle and splits the parabola in two halves Axis of Symmetry: x = - b Formula 2a The vertex is the (x,y) ordered pair on the axis of symmetry Plug in the axis of symmetry for x to find the y coordinate

5. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions 1.Find axis of symmetry X = - b 2a 2.Find the vertex plug in the axis of symmetry for x to find the y- coordinate of the vertex Ex 2: How to Graph a Quadratic Function? 1.f(x) = x 2 – 10x + 24 a=1, b=-10, c=24 x = -b = -(-10) = 10 = 5 2a 2(1) 2 2. f(x) = x 2 – 10x + 24 f(5) = (5) 2 – 10(5) + 24 f(5) = 25 – 50 + 24 f(5) = -1 vertex: (5, -1) 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function

6. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions 3.Make a data table (x/y or in/out table) Put the vertex in the middle value Plug-in two x- values lower and two x-values higher than the axis of symmetry You should notice some symmetry in your output values XY 3 4 5 6 7 f(x) = x 2 – 10x + 24 a=1, b=-10, c=24 vertex: (5, -1) Ex 2: How to Graph a Quadratic Function? 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function

7. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions 4. First plot the vertex. Then plot all the other points on your graph. Draw your parabola. Done! XY 33 40 5 60 73 f(x) = x 2 – 10x + 24 a=1, b=-10, c=24 vertex: (5, -1) Ex 2: How to Graph a Quadratic Function? 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function

8. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions XY f(x) = x 2 – 6x + 8 a=, b=, c= vertex: ( __, __ ) Ex 3: How to Graph a Quadratic Function? 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function x = -b = = ___ 2a f(x) = x 2 – 6x + 8 f( ) = ( ) 2 – 6( ) + 8 f( ) = ___ Vertex goes here

9. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions f(x) = (x – 4) 2 + 1 f(x) = (x – 4)(x – 4) + 1 f(x) = x 2 – 8x + 16 + 1 f(x) = x 2 – 8x + 17 Ex 4: How to Graph a Quadratic Function? 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function Use the box method to multiply the binomials x2x2 -4x 16 x-4 x f(x) = x 2 – 8x + 17 a=, b=, c= vertex: ( __, __ ) x = -b = = ___ 2a f(x) = x 2 – 8x + 17 f( ) = ( ) 2 – 8( ) + 17 f( ) = ___

10. Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions XY Ex 4: How to Graph a Quadratic Function? 1.Identify a, b, c 2.Find the axis of symmetry 3.Find the vertex (x,y) 4.Fill in data table values 5.Graph the function Vertex goes here f(x) = x 2 – 8x + 17 a=, b=, c= vertex: ( 4, 1 )

11. Evaluating and Graphing Quadratic Functions 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions What is true about where the curve intercepts the x- axis? How many times does it intercept the x-axis? x y Y = f(x) = 0 at the x-intercepts (curve crosses x-axis) The x-coordinates where y=0 are called solutions, or the roots, or the zeroes of the quadratic function

12. Evaluating and Graphing Quadratic Functions I DO: Find the solution of the following function: f(x) = (x – 3)(x + 2) 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions (x – 3)(x + 2) = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 3)(x + 2) = 0 x – 3 = 0 +3 +3 x = 3 x + 2 = 0 -2 -2 x = -2 3. Write the solution Solutions: (3,0) and (-2,0) The parabola crosses the x-axis at x = -2 and x = 3 1. Factor the polynomial and set the function = 0

13. Evaluating and Graphing Quadratic Functions WE DO: Find the solution of the following function: f(n) = (2n + 5)(n – 4) 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions (2n + 5)(n – 4) = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (2n + 5)(n – 4) = 0 2n + 5 = 0 -5 -5 2n = -5 2 2 n = -2.5 n – 4 = 0 +4 +4 x = 4 3. Write the solution Solutions: (-2.5, 0) and (4, 0) The parabola crosses the x-axis at x = -2.5 and x = 4 1. Factor the polynomial and set the function = 0

14. Evaluating and Graphing Quadratic Functions I DO: Find the solution of the following function: f(x) = x 2 – 4x + 3 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions 1. Factor the polynomial and set the function = 0 x 2 – 4x + 3 = 0 (x – 1)(x – 3) = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 1)(x – 3) = 0 x – 1 = 0 +1 +1 x = 1 x – 3 = 0 +3 +3 x = 3 3. Write the solution Solutions: (1,0) and (3,0) The parabola crosses the x-axis at x = 1 and x = 3

15. Evaluating and Graphing Quadratic Functions I DO: Find the solution of the following function: x 2 – 4x + 3 = 15 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions 1. Factor the polynomial and set the function = 0 x 2 – 4x + 3 = 15 -15 -15 x 2 – 4x – 12 = 0 (x – 6)(x + 2) = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 6)(x + 2) = 0 3. Write the solution Solutions: (-2,0) and (6,0) The parabola crosses the x-axis at x = -2 and x = 6 x – 6 = 0 +6 +6 x = 6 x + 2 = 0 -2 -2 x = -2

16. Evaluating and Graphing Quadratic Functions WE DO: Find the solution of the following function: x 2 – 11x + 15 = -9 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions 1. Factor the polynomial and set the function = 0 x 2 – 11x + 15 = -9 +9 +9 x 2 – 11x + 24 = 0 (x – 8)(x – 3) = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 8)(x – 3) = 0 3. Write the solution Solutions: (3,0) and (8,0) The parabola crosses the x-axis at x = 3 and x = 8 x – 8 = 0 +8 +8 x = 8 x – 3 = 0 +3 +3 x = 3

17. Evaluating and Graphing Quadratic Functions 3. Find the Max/Min of a Quadratic Function What is Concavity? f(x) = x 2 – 4x – 5 a= 1, b=-4, c=-5 f(x) = x 2 a= 1, b=0, c=0 a= -1, b=0, c=0 f(x) = -x 2 – 2x + 3 a= -1, b=-2, c=3

18. Evaluating and Graphing Quadratic Functions 3. Find the Max/Min of a Quadratic Function What is Concavity? Coefficient Parameter “a” ConcavityShape of Parabola Vertex is Max or Min? a > 0Concave Up Vertex is Minimum a < 0 Concave Down Vertex is Maximum A = 0 Linear No Concavity No max/min No vertex

19. Evaluating and Graphing Quadratic Functions f(x) = x 2 – 6x + 8 a=, b=, c= Ex 1 (I DO): What is the vertex of the quadratic function? Is it a relative max or min? x = -b = = ___ 2a f(x) = x 2 – 6x + 8 f( ) = ( ) 2 – 6( ) + 8 f( ) = ___ 3. Find the Max/Min of a Quadratic Function What is Concavity? Is it a max or min?Is a>0 or a<0? a=1, greater than 0, concave up, vertex is minimum The minimum of -1 is where x = 3 vertex: ( __, __ )

20. Evaluating and Graphing Quadratic Functions f(x) = -x 2 + 8x – 20 a=, b=, c= Ex 2 (WE DO): What is the vertex of the quadratic function? Is it a relative max or min? x = -b = = ___ 2a f(x) = -x 2 + 8x – 20 f( ) = -( ) 2 + 8( ) – 20 f( ) = ___ 3. Find the Max/Min of a Quadratic Function What is Concavity? Is it a max or min?Is a>0 or a<0? a=-1, less than 0, concave down, vertex is maximum The maximum of -4 is where x = 4 vertex: ( __, __ )