Graphing Quadratics: putting it all together

43210 In addition to level 3, students make connections to other content areas and/or contextual situations outside of math. Students will sketch graphs of quadratics using key features and solve quadratics using the quadratic formula. - Students will be able to write, interpret and graph quadratics in vertex form. Students will be able to use the quadratic formula to solve quadratics and are able to identify some key features of a graph of a quadratic. Students will have partial success at a 2 or 3, with help. Even with help, the student is not successful at the learning goal. Focus 10 Learning Goal – (HS.A-REI.B.4, HS.F-IF.B.4, HS.F-IF.C.7, HS.F-IF.C.8) = Students will sketch graphs of quadratics using key features and solve quadratics using the quadratic formula.

Strategies & steps to graphing a quadratic. y = 3x 2 + 2x Determine the vertex. Use x = - b / 2a 1. x = - 2 / 2(3) 2. x = - 1 / 3 3. y = 3(- 1 / 3 ) 2 + 2(- 1 / 3 ) y = 11 / 3 or 3 2 / 3 5. The vertex is (- 1 / 3, 3 2 / 3 ) 2.Use the discriminant to determine the number of solutions. 1.(2) 2 – 4(3)(4) 2.4 – No real solutions. This means the graph will NOT cross the x-axis.  Factoring or using the quadratic formula will not help you graph this quadratic.  You need to use an “x” point left and right of - 1 / 3 in order to graph this.

Strategies & steps to graphing a quadratic. y = 3x 2 + 2x Point left of and right of - 1 / 3 : (-2, _____) and (1, _____) 1.y = 3(-2) 2 + 2(-2) y = 12 – y = 12 4.(-2, 12) 5.y = 3(1) 2 + 2(1) y = y = 9 8.(1, 9) 4.Graph the vertex and the two points.

Strategies & steps to graphing a quadratic. y = x 2 + 8x Determine the vertex. Use x = - b / 2a 1. x = - 8 / 2(1) 2. x = y = (-4) 2 + 8(-4) y = 0 5. The vertex is (-4, 0) 2.Use the discriminant to determine the number of solutions. 1.(8) 2 – 4(1)(16) 2.64 – One solution. This means the graph will intercept the x-axis at the VERTEX.  Factoring or using the quadratic formula will not help you graph this quadratic. (Your answers will be -4 & -4).  You need to use an “x” point left and right of -4 in order to graph this.

Strategies & steps to graphing a quadratic. y = x 2 + 8x Point left of and right of -4: (-5, _____) and (-3, _____) 1.y = (-5) 2 + 8(-5) y = 25 – y = 1 4.(-5, 1) 5.y = (-3) 2 + 8(-3) y = y = 1 8.(-3, 1) 4.Graph the vertex and the two points.

Strategies & steps to graphing a quadratic. y = x 2 + 6x Determine the vertex. Use x = - b / 2a 1. x = - 6 / 2(1) 2. x = y = (-3) 2 + 6(-3) y = The vertex is (-3, -4) 2.Use the discriminant to determine the number of solutions. 1.(6) 2 – 4(1)(5) 2.36 – Two solutions. This means the graph will intercept the x-axis two times.  Factoring or using the quadratic formula will HELP you.  Which one do you want to use?

Strategies & steps to graphing a quadratic. y = x 2 + 6x Factoring will be the quickest way to find the x-intercepts. 1. (x +5)(x + 1) = 0 2. x + 5 = 0 1. x = x + 1 = 0 1. x = -1 4.The x-intercepts are (-5, 0) & (-1, 0). 4.Graph the vertex and the x-intercepts.

Strategies & steps to graphing a quadratic. y = -2x 2 + 6x Determine the vertex. Use x = - b / 2a 1. x = - 6 / 2(-2) 2. x = 3 / 2 3. y = -2( 3 / 2 ) 2 + 6( 3 / 2 ) y = -2( 9 / 4 ) The vertex is (1½, 5 ½ ) 2.Use the discriminant to determine the number of solutions. 1.(6) 2 – 4(-2)(1) Two solutions. This means the graph will intercept the x-axis two times.  Because 44 is not a perfect square, the x-intercepts will be irrational.  Use the quadratic formula to find the x- intercepts.

Strategies & steps to graphing a quadratic. y = -2x 2 + 6x Substitute into the quadratic formula and solve to find your x- intercepts. 4.Divide 6, 2 and 4 by 2 (the GCF).

Strategies & steps to graphing a quadratic. y = -2x 2 + 6x Estimate the value of the irrational x-intercepts. 6.Graph the vertex and the x-intercepts.