starterwed, oct 1 given the function f(x) = 3x 2 – 12x – 36, identify these key features of the...
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STARTER WED, OCT 1
Given the function f(x) = 3x2 – 12x – 36, identify these key features of the graph:
1. the extrema
2. vertex
3. y-intercept
4. x-intercepts
5. sketch the graph
When you finish: Graph y = (2x – 1)(x + 3) on the calculator. Use the CALC feature to find the x-intercepts. What are they?
Can you see a way to look at the factors and get the same x-intercepts?
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1.9.1: Proving the Interior Angle Sum Theory
IntroductionQuadratic equations can be written in several forms, including standard form, vertex form, and factored form.
While each form is equivalent, certain forms easily reveal different features of the graph of the quadratic function.
In this lesson, you will learn to use the various forms of quadratic functions to show the key features of the graph and determine how these key features relate to the characteristics of a real-world situation.
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2.1.2: Interpreting Various Forms of Quadratic Functions
Standard Form
KEY FEATURES SHOWN:• Y-Intercept is the value of c, so (0, c) is the y-intercept.
• Vertex can be found by using , then plugging that value in to find y.
• Maximum or Minimum can be determined by If , the graph opens up and has a minimum. If the graph opens down and has a maximum.
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2.1.2: Interpreting Various Forms of Quadratic Functions
Example 1:
Find the y-intercept.
Find the vertex.
Does this graph open up/minimum or open down/maximum?
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1.9.1: Proving the Interior Angle Sum Theory
c = -7, so y-intercept is (0, -7)
𝒙=−𝒃𝟐𝒂
=−𝟖
𝟐(−𝟐)=−𝟖−𝟒
=𝟐
Vertex is at the point (2, 1)
a = -2, therefore graph opens down and has a maximum
Vertex Form
KEY FEATURES SHOWN:• Vertex is the point (h, k)
• Axis of Symmetry is the vertical line
• Maximum or Minimum can be determined by If , the graph opens up and has a minimum. If the graph opens down and has a maximum
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2.1.2: Interpreting Various Forms of Quadratic Functions
Example 2:
Find the vertex.
Find the axis of symmetry.
Does this graph open up/minimum or open down/maximum?
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1.9.1: Proving the Interior Angle Sum Theory
Vertex is (2, 5)
Vertical line x = 2
a = 3, therefore the graph opens up and has a minimum.
Factored Form (aka Intercept Form)
KEY FEATURES SHOWN:• X-Intercepts are the values p & q, so (p, 0) and (q, 0)
• Axis of Symmetry occurs at the midpoint between the x-intercepts, therefore
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1.9.1: Proving the Interior Angle Sum Theory
Factored Form continued on next page
Factored Form (aka X-Intercept Form)
KEY FEATURES SHOWN:
• Vertex can be found now by finding the y value. Take the x value you just found from the axis of symmetry and plug it into the equation.
• Y-Intercept can be found by plugging in 0 for x and finding y. Remember the y-intercept occurs on the y-axis, therefore x = 0.
• Maximum or Minimum can be determined by If , the graph opens up and has a minimum. If the graph opens down and has a maximum
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1.9.1: Proving the Interior Angle Sum Theory
Example 3:
Find the x-intercepts.
Find the axis of symmetry.
Find the vertex.
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1.9.1: Proving the Interior Angle Sum Theory
p = -2, q = 8 so x-intercepts are at (-2, 0) and (8, 0)
𝒙=𝒑+𝒒𝟐
=−𝟐+𝟖𝟐
=𝟔𝟐
=𝟑
So axis of symmetry is the vertical line x = 3
Plug in x =3 into the equation and find y.
y = -(3 + 2)(3 – 8) = -(5)(-5) = 25, therefore the vertex is (3, 25)
Continue on next page
Example 3:
Find the y-intercept.
Does this graph open up/minimum or open down/maximum?
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1.9.1: Proving the Interior Angle Sum Theory
Find y when x = 0. So y = -(0 + 2)(0 – 8) = -(2)(-8) = 16So the y-intercept is (0, 16)
a = -1, therefore graph opens down and has a maximum
Guided Practice
Example 4Suppose that the flight of a launched bottle rocket can be modeled by the function f(x) = –(x – 1)(x – 6), where f(x) measures the height above the ground in meters and x represents the horizontal distance in meters from the launching spot at x = 1.
(A) How far does the bottle rocket travel in the horizontal direction from launch to landing?
(B) What is the maximum height the bottle rocket reaches?
(C) How far has the bottle rocket traveled horizontally when it reaches its maximum height?
(D) Graph the function.
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2.1.2: Interpreting Various Forms of Quadratic Functions
Guided Practice
Example 5A football is kicked and follows a path given by f(x) = –0.03x2 + 1.8x, where f(x) represents the height of the ball in feet and x represents the horizontal distance in feet. (A) What is the maximum height the ball reaches? (B) What horizontal distance maximizes the height? (C) Graph the function.
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2.1.2: Interpreting Various Forms of Quadratic Functions