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

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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

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)

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