1. The quadratic function is a second-order polynomial function 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) = ax2 + bx + c OR ax2 + bx + c = f(x)

1. Graphing Quadratic Functions Example 1: Identify the coefficient parameters (a, b, c) in the following quadratic functions: a. f(x) = 5x2 + 10x + 15 b. f(n) = n2 – 10n + 22 c. h(r) = 7r2 + 14r – 7 d. g(x) = x2 – 25 e. g(x) = 7x2 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)

1. This is a graph of a quadratic function. We call it a parabola 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? y x

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

1. Ex 2: How to Graph a Quadratic Function? f(x) = x2 – 10x + 24 Graphing Quadratic Functions Ex 2: How to Graph a Quadratic Function? f(x) = x2 – 10x + 24 a=1, b=-10, c=24 x = -b = -(-10) = 10 = 5 2a 2(1) 2 2. f(x) = x2 – 10x + 24 f(5) = (5)2 – 10(5) + 24 f(5) = 25 – 50 + 24 f(5) = -1 vertex: (5, -1) Find axis of symmetry X = - b 2a Find the vertex plug in the axis of symmetry for x to find the y-coordinate of the vertex Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function

1. Ex 2: How to Graph a Quadratic Function? X Y 3 4 5 -1 6 7 Graphing Quadratic Functions Ex 2: How to Graph a Quadratic Function? 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 f(x) = x2 – 10x + 24 a=1, b=-10, c=24 vertex: (5, -1) X Y 3 4 5 -1 6 7 Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function

1. Ex 2: How to Graph a Quadratic Function? X Y 3 4 5 -1 6 7 Graphing Quadratic Functions Ex 2: How to Graph a Quadratic Function? 4. First plot the vertex. Then plot all the other points on your graph. Draw your parabola. Done! f(x) = x2 – 10x + 24 a=1, b=-10, c=24 vertex: (5, -1) Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function X Y 3 4 5 -1 6 7

1. Ex 3: How to Graph a Quadratic Function? X Y f(x) = x2 – 6x + 8 Graphing Quadratic Functions Ex 3: How to Graph a Quadratic Function? f(x) = x2 – 6x + 8 a= , b= , c= vertex: ( __ , __ ) f(x) = x2 – 6x + 8 f( ) = ( )2 – 6( ) + 8 f( ) = ___ x = -b = = ___ 2a Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function X Y Vertex goes here

1. Ex 4: How to Graph a Quadratic Function? x2 -4x 16 Graphing Quadratic Functions Ex 4: How to Graph a Quadratic Function? f(x) = (x – 4)2 + 1 f(x) = (x – 4)(x – 4) + 1 f(x) = x2 – 8x + 16 + 1 f(x) = x2 – 8x + 17 Use the box method to multiply the binomials x -4 x x2 -4x 16 Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function -4 f(x) = x2 – 8x + 17 a= , b= , c= vertex: ( __ , __ ) f(x) = x2 – 8x + 17 f( ) = ( )2 – 8( ) + 17 f( ) = ___ x = -b = = ___ 2a

1. Ex 4: How to Graph a Quadratic Function? X Y f(x) = x2 – 8x + 17 Graphing Quadratic Functions Ex 4: How to Graph a Quadratic Function? f(x) = x2 – 8x + 17 a= , b= , c= vertex: ( 4 , 1 ) Identify a, b, c Find the axis of symmetry Find the vertex (x,y) Fill in data table values Graph the function X Y Vertex goes here

2. Solving/ Finding X-Intercepts 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? 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 y x

The parabola crosses the x-axis at x = -2 and x = 3 2. Solving/ Finding X-Intercepts Finding Roots/ Finding Zeroes Of Quadratic Functions I DO: Find the solution of the following function: f(x) = (x – 3)(x + 2) 1. Factor the polynomial and set the function = 0 (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 Solutions: (3,0) and (-2,0) The parabola crosses the x-axis at x = -2 and x = 3 3. Write the solution

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

The parabola crosses the x-axis at x = 1 and x = 3 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions I DO: Find the solution of the following function: f(x) = x2 – 4x + 3 x2 – 4x + 3 = 0 (x – 1)(x – 3) = 0 1. Factor the polynomial and set the function = 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 Solutions: (1,0) and (3,0) The parabola crosses the x-axis at x = 1 and x = 3 3. Write the solution

The parabola crosses the x-axis at x = -2 and x = 6 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions I DO: Find the solution of the following function: x2 – 4x + 3 = 15 x2 – 4x + 3 = 15 -15 -15 x2 – 4x – 12 = 0 (x – 6)(x + 2) = 0 1. Factor the polynomial and set the function = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 6)(x + 2) = 0 x – 6 = 0 +6 +6 x = 6 x + 2 = 0 -2 -2 x = -2 Solutions: (-2,0) and (6,0) The parabola crosses the x-axis at x = -2 and x = 6 3. Write the solution

The parabola crosses the x-axis at x = 3 and x = 8 2. Solving/ Finding Roots/ Finding Zeroes Of Quadratic Functions WE DO: Find the solution of the following function: x2 – 11x + 15 = -9 x2 – 11x + 15 = -9 +9 +9 x2 – 11x + 24 = 0 (x – 8)(x – 3) = 0 1. Factor the polynomial and set the function = 0 2. Solve for variable. You will have two solutions/roots/zeroes (in most cases) (x – 8)(x – 3) = 0 x – 8 = 0 +8 +8 x = 8 x – 3 = 0 +3 +3 x = 3 Solutions: (3,0) and (8,0) The parabola crosses the x-axis at x = 3 and x = 8 3. Write the solution

3. Find the Max/Min of a Quadratic Function What is Concavity? f(x) = x2 – 4x – 5 a= 1, b=-4, c=-5 f(x) = x2 a= 1, b=0, c=0 f(x) = -x2 – 2x + 3 a= -1, b=-2, c=3 a= -1, b=0, c=0

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

3. Find the Max/Min of a Quadratic Function What is Concavity? Ex 1 (I DO): What is the vertex of the quadratic function? Is it a relative max or min? f(x) = x2 – 6x + 8 a= , b= , c= f(x) = x2 – 6x + 8 f( ) = ( )2 – 6( ) + 8 f( ) = ___ x = -b = = ___ 2a vertex: ( __ , __ ) 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

3. Find the Max/Min of a Quadratic Function What is Concavity? Ex 2 (WE DO): What is the vertex of the quadratic function? Is it a relative max or min? f(x) = -x2 + 8x – 20 a= , b= , c= f(x) = -x2 + 8x – 20 f( ) = -( )2 + 8( ) – 20 f( ) = ___ x = -b = = ___ 2a vertex: ( __ , __ ) 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