Starter MON, SEP 29, 2014 Given the equation 1.Find y when x = 2 2.Find y when x = -3 1 1.9.1: Proving the Interior Angle Sum Theory
Jan 16, 2016
Starter MON, SEP 29, 2014
Given the equation
1. Find y when x = 2
2. Find y when x = -3
1
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)
2
1.9.1: Proving the Interior Angle Sum Theory
WORDS TO KNOW
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1.9.1: Proving the Interior Angle Sum Theory
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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2.1.1: Graphing Quadratic Functions
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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2.1.1: Graphing Quadratic Functions
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
2.1.1: Graphing Quadratic Functions
Finding the Vertex Algebraically
1. ()
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2.1.1: Graphing Quadratic Functions
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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2.1.1: Graphing Quadratic Functions
(-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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2.1.1: Graphing Quadratic Functions
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
2.1.1: Graphing Quadratic Functions
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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2.1.1: Graphing Quadratic Functions
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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2.1.1: Graphing Quadratic Functions
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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2.1.1: Graphing Quadratic Functions