1 1.9.1: proving the interior angle sum theory. 2

Starter MON, SEP 29, 2014

Given the equation

1. Find y when x = 2

2. Find y when x = -3

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1.9.1: Proving the Interior Angle Sum Theory

Starter MON, SEP 29, 2014

Given the equation

1. Find f(2)

2. Find f(-3)

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WORDS TO KNOW

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IntroductionYou may recall that a line is the graph of a linear function and that all linear functions can be written in the formf(x) = mx + b, in which m is the slope and b is the y-intercept. The solutions to a linear function are the infinite set of points on the line. In this lesson, you will learn about a second type of function known as a quadratic function.

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2.1.1: Graphing Quadratic Functions

Key Concepts

• A quadratic function is a function that can be written in the form f(x) = ax2 + bx + c, where x is the variable, a, b, and c are constants, and a ≠ 0.

• This form is also known as the standard form of a quadratic function.

• Quadratic functions can be graphed on a coordinate plane and will have a U-shape called a parabola.

• Characteristics of a parabola include: the y-intercept, x-intercepts, the maximum or minimum, the axis of symmetry, and vertex.

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Key Concepts, continuedf(x) = x2 – 2x – 3

• y-intercept (0, -3)• x-intercepts (-1, 0) & (3, 0)• Minimum at the Vertex (1, -4)• Axis of Symmetry is the line

x = 1

Creating a table of values allows

you to plot more points on the

graph

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x y–2 5–1 00 –31 –42 –3

3 0

Key Concepts, continued• A quadratic function either has a

maximum or a minimum.

• The vertex of a parabola is the point on a parabola that is the maximum or minimum of the function.

• The extrema of a graph are the minima or maxima of a function. In other words, an extremum is the function value that achieves either a minimum or maximum. 7


Finding the Vertex Algebraically

1. ()

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f(x) = x2 – 2x – 3a= 1, b= -2, c= -3

𝑥=−𝑏2𝑎

=22(1)

=22=1

The vertex is a point.

We just found the x value.

How do we find the y value?

𝑦=𝑥2−2𝑥−3=(1 )2−2 (1 )−3=1−2−3=−4

Vertex (1, -4)

Finding the Vertex Algebraically2. If you know the x-intercepts of the graph, or any two points on the

graph with the same y-value, the x-coordinate of the vertex is the point halfway between the values of the x-coordinates. For the x-intercepts (p, 0) and (q, 0) the x-coordinate of the vertex is

If you know the equation, you can

plug this x value in to find the y!

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(-1, 0) (3, 0)𝑥=

𝑝+𝑞2

=−1+32

=22=1

Determining whether a Quadratic Function () has a Maximum or Minimum

If a > 0 If a < 0

Graph opens up Graph opens down

Vertex is a minimum Vertex is a maximum

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Using a Graphing Utility• To graph a function using a graphing calculator, follow

these general steps for your calculator model.

On a TI-83/84:

Step 1: Press the [Y=] button.

Step 2: Type the function into Y1, or any available equation. Use the [X, T, θ, n] button for the variable x. Use the [x2] button for a square.

Step 3: Press [WINDOW]. Enter values for Xmin, Xmax, Ymin, and Ymax. The Xscl and Yscl are arbitrary. Leave Xres = 1.

Step 4: Press [GRAPH]. 11


Using a Graphing Utility, continuedOn a TI-Nspire:

Step 1: Press the [home] key.

Step 2: Arrow over to the graphing icon and press [enter].

Step 3: Type the function next to f1(x), or any available equation, and press [enter]. Use the [X] button for the variable x. Use the [x2] button for a square.

Step 4: To change the viewing window, press [menu]. Select 4: Window/Zoom and select A: Zoom – Fit.

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Guided Practice

Example 3Given the function f(x) = x2 – 4x + 3, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph.

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How can we find the x-intercepts?

Guided Practice

Example 4Given the function f(x) = –2x2 + 4x + 16, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph.

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1 1.9.1: proving the interior angle sum theory. 2

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