Math I UNIT QUESTION: What is a quadratic function? Standard: MM2A3, MM2A4 Today’s Question: How do you graph quadratic functions in vertex form? Standard: MM2A3.b.

6.6 Graphing Quadratic Functions in Vertex or Intercept Form Definitions Definitions 3 Forms 3 Forms Steps for graphing each form Steps for graphing each form Examples Examples

Daily Check 1.Factor: 3x x Factor and Solve: 2x 2 - 7x + 3 = 0

Parent Name:Quadratic Parent Equation: xy Domain: Range : General Equation (in Vertex Form) Domain: Range : Vertex: Axis of Symmetry: Vertex: Axis of Symmetry: Review: Quadratic Parent

AAT-A Date: 2/17/14 SWBAT write quadratic equations in vertex form.  Do Now: Review Questions pg 336 #1-50, evens  HW Requests: pg 303 #42-49; Pg 310 #15-37 odds Worksheets for homework Skills Practice pg 340 Worksheet Quadratics Graphing TBD  HW: Skills Practice Vertex form 6.6  Announcements: Life Is Just A Minute Life is just a minute—only sixty seconds in it. Forced upon you—can't refuse it. Didn't seek it—didn't choose it. But it's up to you to use it. You must suffer if you lose it. Give an account if you abuse it. Just a tiny, little minute, But eternity is in it! By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Writing a Quadratic Function in Vertex Form 5. Solve for y  the equation will be in vertex form. Steps: 1. Write the function in standard form. 2. Set it up to complete the square. 3. Add the square to both sides of the = sign. 4. Write the trinomial as a binomial squared.

Writing a Quadratic Function in Vertex Form Example 1: Write the function in vertex form and identify its vertex. 1. Write the function in standard form. 2. Set it up to complete the square.

Writing a Quadratic Function in Vertex Form 3. Add the square to both sides of the = sign. 4. Write the trinomial as a binomial squared. 5. Solve for y  the equation will be in vertex form. Vertex:

Practice: Write a quadratic function in vertex form and identify its vertex. P1:

P2:

Writing a Quadratic Function in Vertex Form Example 2: Factor, write the function in vertex form, and identify its vertex. 1. Write the function in standard form. 3. Set it up to complete the square. 2. Factor the first two terms.

Writing a Quadratic Function in Vertex Form 4. Add the square to both sides of the = sign. 5. Write the trinomial as a binomial squared. 6. Solve for y  the equation will be in vertex form. Vertex:

Practice: Write a quadratic function in vertex form and identify its vertex. P3:

P4: Vertex :

Writing a Quadratic Function in Vertex Form 1. Write the function in standard form. 2. Set it up to complete the square. Example 3: Write the function, using fractions, in vertex form, and identify its vertex.

Writing a Quadratic Function in Vertex Form 3. Add the square to both sides of the = sign. Look! Be careful with the added term when a<1 4. Write the trinomial as a binomial squared. 5. Solve for y  the equation will be in vertex form. Vertex:

Practice: Write a quadratic function in vertex form and identify its vertex. P5: Vertex:

Independent Practice Write each function in vertex form and identify its vertex.

Quadratic Function A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:

Vertex- The lowest or highest point of a parabola. Vertex Axis of symmetry- The vertical line through the vertex of the parabola. Axis of Symmetry

Vertex Form Equation y=a(x-h)2+k If a is positive, parabola opens up If a is negative, parabola opens down. The vertex is the point (h,k). The axis of symmetry is the vertical line x=h. Don’t forget about 2 points on either side of the vertex! (5 points total!)

Vertex Form  Each function we just looked at can be written in the form (x – h) 2 + k, where (h, k) is the vertex of the parabola, and x = h is its axis of symmetry.  (x – h) 2 + k – vertex form EquationVertex Axis of Symmetry y = x 2 or y = (x – 0) (0, 0) x = 0 y = x or y = (x – 0) (0, 2) x = 0 y = (x – 3) 2 or y = (x – 3) (3, 0) x = 3

Example 1: Graph y = (x + 2) Analyze y = (x + 2) Analyze y = (x + 2) Step 1 Plot the vertex (-2, 1) Step 1 Plot the vertex (-2, 1) Step 2 Draw the axis of symmetry, x = -2. Step 2 Draw the axis of symmetry, x = -2. Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 4 Use symmetry to complete the graph, or find two points on the Step 4 Use symmetry to complete the graph, or find two points on the left side of the vertex. left side of the vertex.

Your Turn! Analyze and Graph: Analyze and Graph: y = (x + 4) y = (x + 4) (-4,-3)

Example 2: Graph y= -.5(x+3) 2 +4 a is negative (a = -.5), so parabola opens down. a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Axis of symmetry is the vertical line x = -3 Table of values x y Table of values x y Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

Now you try one! y=2(x-1) 2 +3 Open up or down? Open up or down? Vertex? Vertex? Axis of symmetry? Axis of symmetry? Table of values with 4 points (other than the vertex? Table of values with 4 points (other than the vertex?

(-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

Intercept Form Equation y=a(x-p)(x-q) The x-intercepts are the points (p,0) and (q,0). The x-intercepts are the points (p,0) and (q,0). The axis of symmetry is the vertical line x= The axis of symmetry is the vertical line x= The x-coordinate of the vertex is The x-coordinate of the vertex is To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. If a is positive, parabola opens up If a is positive, parabola opens up If a is negative, parabola opens down.

Example 3: Graph y=-(x+2)(x-4) Since a is negative, parabola opens down. The x-intercepts are (-2,0) and (4,0) To find the x-coord. of the vertex, use To find the y-coord., plug 1 in for x. Vertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) x=1 (-2,0)(4,0) (1,9)

Now you try one! y=2(x-3)(x+1) Open up or down? Open up or down? X-intercepts? X-intercepts? Vertex? Vertex? Axis of symmetry? Axis of symmetry?

(-1,0)(3,0) (1,-8) x=1

Changing from vertex or intercepts form to standard form The key is to FOIL! (first, outside, inside, last) The key is to FOIL! (first, outside, inside, last) Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 y=-x 2 +5x+36 =3(x 2 -2x+1)+8 =3x 2 -6x+3+8 =3x 2 -6x+3+8 y=3x 2 -6x+11 y=3x 2 -6x+11

Challenge Problem Write the equation of the graph in vertex form. Write the equation of the graph in vertex form.

Assignment Day 1 -p. 65 #4,6,7,9,13,16 and Review for Quiz Day 2 – p. 67 #4,5,7,9,11-14 We will not do intercept form.