Module 3 Lesson 5: Basic Quadratic Equation Standard Form and Vertex Form Translations Affect of changing a Translations Affect of changing h Translations Affect of changing k a h k

Table of Contents  Slides 3-9: Review Standard Form  Slides 10-14: Review Vertex Form  Slides 15-21: Review Transformations  Slide 22: Find the Equation From a Graph  Slide 23: Find the Vertex and y-Intercept from an Equation Audio/Video and Interactive Sites  Slide 4: Gizmos  Slide 6: Gizmos  Slide 12: Gizmos

Explore Quadratic Equations In Standard Form

To explore the affect the variables A, B, and C have on the graph, play the Zap It game found at the following link. Zap It Game For a more straightforward explanation on the affect of A, B, and C. Look at the following Gizmo. Gizmo: How A, B, and C affect the quadratic function.

On your graphing calculator… Graph y= x 2 Graph y = - x 2 What do you see? What is A in the above two equations and how does it affect the graph?

In this equation, A = 1In this equation, A = -1 Gizmo: Standard Form, Vertex, and Intercepts Gizmo: Standard Form, Vertex, and Intercepts (B)

If A = 0, the graph is not quadratic, it is linear. If A is not 0, B and/or C can be 0 and the function is still a quadratic function. If A = 0, the graph is not quadratic, it is linear. If A is not 0, B and/or C can be 0 and the function is still a quadratic function. f(x) = Ax 2 + Bx + C If A < 0, The parabola opens down, like an umbrella. If A > 0, The parabola opens up, like an bowl. If A < 0, The parabola opens down, like an umbrella. If A > 0, The parabola opens up, like an bowl. In this equation, A = -1In this equation, A = 1

When "a" is positive, the graph of y = ax 2 + bx + c opens upward and the vertex is the lowest point on the curve. As the value of the coefficient "a" gets larger, |a| > 1, the parabola narrows. As the value of the coefficient “a” gets smaller, | a | is between 0 and 1, the parabola widens.

For the graph of y = Ax 2 + Bx + C If C is positive, the graph shifts up C units If C is negative, the graph shifts down C units Since C is negative, -5, the graph shifts down 5 units Since C is positive, 12, the graph shifts up 12 units. Note: This graph would open upside down since A is -2 Since C is negative, -8, the graph shifts down 8 units

Vertex Form of the Quadratic Equation

The Vertex Form of quadratic functions is often used, especially when we want to see how a graph behaves as values change. y=Ax 2 + Bx + C (Standard Form) Can be factored y = a(x - h) 2 + k (Vertex Form) Notice: If you are given the equation in Vertex Form, you can FOIL the (x-h) part, distribute a, and then combine like terms to get back to Standard Form. Notice: If you are given the equation in Vertex Form, you can FOIL the (x-h) part, distribute a, and then combine like terms to get back to Standard Form.

Gizmo: Vertex Form, Vertex, Intercepts Gizmo: Vertex Form, Vertex, Intercepts Gizmo: Vertex Form, Vertex, Intercepts (B) Gizmo: Vertex Form, Vertex, Intercepts (B)

Practice Assignment 1—State whether each Quadratic Function is in Vertex Form, Quadratic Form, or Neither. a.Quadratic Form b.Vertex Form c.Neither

Practice Assignment 1 Answers—State whether each Quadratic Function is in Vertex Form, Quadratic Form, or Neither. a.Quadratic Form b.Vertex Form c.Neither

Transformation affects of a, h, and k.

When a is positive Graph opens up. As a increases, graph gets narrower As a decreases, graph gets wider When a is positive Graph opens up. As a increases, graph gets narrower As a decreases, graph gets wider When a is negative Graph opens down. As a increases (approaches 0), graph gets wider As a decreases (approaches negative infinity), graph gets narrower When a is negative Graph opens down. As a increases (approaches 0), graph gets wider As a decreases (approaches negative infinity), graph gets narrower When k is positive Graph moves up regardless of what a and h are. When k is positive Graph moves up regardless of what a and h are. When k is negative Graph moves down regardless of what a and h are. When k is negative Graph moves down regardless of what a and h are. When a and h are positive: a(x – (+h)) 2 = a(x – h) 2 Graph moves right h units. When a and h are positive: a(x – (+h)) 2 = a(x – h) 2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h)) 2 = -a(x – h) 2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h)) 2 = -a(x – h) 2 Graph moves right h units. When a and h are negative: -a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a and h are negative: -a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a is positive and h is negative: a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a is positive and h is negative: a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units.

When a is positive Graph opens up. When a is positive Graph opens up. As a increases, graph gets narrower As a decreases, graph gets wider

When a is negative Graph opens down. When a is negative Graph opens down. As a increases (approaches 0), graph gets wider As a decreases (approaches negative infinity), graph gets narrower

When k is positive Graph moves up regardless of what a and h are. When k is positive Graph moves up regardless of what a and h are. When k is negative Graph moves down regardless of what a and h are. When k is negative Graph moves down regardless of what a and h are. +k

When a and h are positive: a(x – (+h)) 2 = a(x – h) 2 Graph moves right h units. When a and h are positive: a(x – (+h)) 2 = a(x – h) 2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h)) 2 = -a(x – h) 2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h)) 2 = -a(x – h) 2 Graph moves right h units.

When a and h are negative: -a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a and h are negative: -a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a is positive and h is negative: a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units. When a is positive and h is negative: a(x – (-h)) 2 = a(x + h) 2 Graph moves left h units.

Find the equation of the following graph. Start by writing down the generic vertex form equation. Identify the vertex: (1, 2) Note: a will be a negative since the graph is upside down. Identify the y-intercept: (0, - 1) Fill in all the information you just found and solve for a. From the vertex: h = 1 and k = 2 From the y-intercept: x = 0 and y = -1 From the y-intercept: x = 0 and y = -1 Simplify and Solve for a Therefore, the equation in vertex form is:

Identify the vertex and y-intercept of the following equation. Identify the vertex: (h, k) (-3, -6) Identify the y-intercept: (0, -10.5)