: Proving the Interior Angle Sum Theory

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WORDS TO KNOW : Proving the Interior Angle Sum Theory

Introduction You may recall that a line is the graph of a linear function and that all linear functions can be written in the form f(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 : Graphing Quadratic Functions

Key Concepts A quadratic function is a function that can be written in the form f(x) = ax 2 + 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 : Graphing Quadratic Functions

Key Concepts, continued f(x) = x 2 – 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 : Graphing Quadratic Functions xy –25 –10 0–3 1–4 2–3 30

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 : Graphing Quadratic Functions

8 f(x) = x 2 – 2x – 3 a= 1, b= -2, c= -3 The vertex is a point. We just found the x value. How do we find the y value? Vertex (1, -4)

: Graphing Quadratic Functions (-1, 0)(3, 0)

: 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 [x 2 ] 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] : Graphing Quadratic Functions

Using a Graphing Utility, continued On 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 [x 2 ] button for a square. Step 4: To change the viewing window, press [menu]. Select 4: Window/Zoom and select A: Zoom – Fit : Graphing Quadratic Functions

Guided Practice Example 3 Given the function f(x) = x 2 – 4x + 3, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph : Graphing Quadratic Functions How can we find the x-intercepts?

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