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Module 3 Lesson 5:
a
hk
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 EquationsIn 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 ItGame
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= x2
Graph y = -x2
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) = Ax2+ 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 = -1 In this equation, A = 1
When "a" is positive, the graph of y = ax2 + 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 larger, |a| > 1, the parabola narrows.
As the value of the coefficient “a” gets smaller, | a | is between 0 and 1, the parabola widens.
As the value of the coefficient “a” gets smaller, | a | is between 0 and 1, the parabola widens.
For the graph of y = Ax2 + Bx + C
If C is positive, the graph shifts up C units
If C is negative, the graph shifts down C units
56.9 2 xy
122 2 xxy
823
2
xx
y
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=Ax2 + Bx + C (Standard Form)y=Ax2 + Bx + C (Standard Form)
Can be factored
y = a(x - h)2 + k (Vertex Form)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.
4284
1)(.3
135)(.2
422)(.1
xxf
xxxf
xxxf 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.
4284
1)(.3
135)(.2
422)(.1
xxf
xxxf
xxxf a. Quadratic Form
b. Vertex Form
c. Neither
Transformation affects of a, h, and k.
khxay 2)(
When a is positiveGraph opens up. As a increases, graph gets narrowerAs a decreases, graph gets wider
When a is positiveGraph opens up. As a increases, graph gets narrowerAs a decreases, graph gets wider
When a is negativeGraph opens down. As a increases (approaches 0), graph gets wider
As a decreases (approaches negative infinity), graph gets narrower
When a is negativeGraph opens down. As a increases (approaches 0), graph gets wider
As a decreases (approaches negative infinity), graph gets narrower
When k is positiveGraph moves up regardless of what a and h are.
When k is positiveGraph moves up regardless of what a and h are.
When k is negativeGraph moves down regardless of what a and h are.
When k is negativeGraph 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 positiveGraph opens up.
When a is positiveGraph opens up.
khxy 21
khxy 25
khxy 2
2
1
As a increases, graph gets narrowerAs a increases, graph gets narrower
As a decreases, graph gets widerAs a decreases, graph gets wider
khxay 2)(
When a is negativeGraph opens down. When a is negativeGraph opens down.
khxy 21
khxy 25
khxy 2
2
1
As a increases (approaches 0), graph gets widerAs a increases (approaches 0), graph gets wider
As a decreases (approaches negative infinity), graph gets narrowerAs a decreases (approaches negative infinity), graph gets narrower
khxay 2)(
When k is positiveGraph moves up regardless of what a and h are.
When k is positiveGraph moves up regardless of what a and h are.
When k is negativeGraph moves down regardless of what a and h are.
When k is negativeGraph moves down regardless of what a and h are.
khxay 2)(
khxay 2)(
+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.
khxay 2))((
khxay 2))((
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.
khxay 2))((
khxay 2))((
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
Find the equation of the following graph.
Start by writing down the generic vertex form equation. khxay 2)(
Identify the vertex:Identify the vertex:(1, 2)
Note: a will be a negative since the graph is upside down.Note: a will be a negative since the graph is upside down.
Identify the y-intercept:Identify the y-intercept: (0, -1)
Fill in all the information you just found and solve for a.
Fill in all the information you just found and solve for a.khxay 2)(
2)1( 2 xay From the vertex: h = 1 and k = 2From the vertex: h = 1 and k = 2
2)10(1 2 a From the y-intercept: x = 0 and y = -1From the y-intercept: x = 0 and y = -1
2)1(1 2 a Simplify and Solve for aSimplify and Solve for a
2)1(1 a
21 a
a 3
2)1(3 2 xyTherefore, the equation in vertex form is:
Identify the vertex and y-intercept of the following equation.
6)3(2
1 2 xy
Identify the vertex:Identify the vertex:
(h, k)
(-3, -6)
Identify the y-intercept:Identify the y-intercept: (0, -10.5)
6)3(2
1 2 xy
6962
1 2 xxy
65.432^2
1 xxy
5.1032^2
1 xxy
5.10032^02
1y
5.10y
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