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Quadratic and polynomial functions
Lesson 4 Source : Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice Hall, ISBN (Chapter 3 pp )

Part 1: Quadratic Functions
Properties of Quadratic function Graphing Quadratic Function Vertex form

Example: f(x) = 3x2 + 3x + 5 g(x) = 5  x2

Quadratic Function Properties
Leading Coefficient The graph of a quadratic function is a parabola—a U shaped graph that opens either upward or downward. A parabola opens upward if its leading coefficient a is positive and opens downward if a is negative. The highest point on a parabola that opens downward and the lowest point on a parabola that opens upward is called the vertex. (The graph of a parabola changes shape at the vertex.) The vertical line passing through the vertex is called the axis of symmetry. The leading coefficient a controls the width of the parabola. Larger values of |a| result in a narrower parabola, and smaller values of |a| result in a wider parabola.

Examples of different parabolas

Quadratic Functions Vertex Vertex
y x The graph of a quadratic function is a parabola. Vertex A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. Vertex NOTE: if the parabola opened left or right it would not be a function!

Standard Form a > 0 a < 0
y x The standard form of a quadratic function is a < 0 a > 0 y = ax2 + bx + c The parabola will open up when the a value is positive. The parabola will open down when the a value is negative.

The line of symmetry ALWAYS passes through the vertex.
Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side. The line of symmetry ALWAYS passes through the vertex.

Finding the Line of Symmetry
When a quadratic function is in standard form For example… Find the line of symmetry of y = 3x2 – 18x + 7 y = ax2 + bx + c, The equation of the line of symmetry is Using the formula… This is best read as … the opposite of b divided by the quantity of 2 times a. Thus, the line of symmetry is x = 3.

Finding the Vertex Therefore, the vertex is (2 , 5) y = –2x2 + 8x –3
But we know the line of symmetry always goes through the vertex. y = –2x2 + 8x –3 STEP 1: Find the line of symmetry Thus, the line of symmetry gives us the x – coordinate of the vertex. STEP 2: Plug the x – value into the original equation to find the y value. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. y = –2(2)2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5)

Example Use the graph of the quadratic function shown to determine the sign of the leading coefficient, its vertex, and the equation of the axis of symmetry. Leading coefficient: The graph opens downward, so the leading coefficient a is negative. Vertex: The vertex is the highest point on the graph and is located at (1, 3). Axis of symmetry: Vertical line through the vertex with equation x = 1.

Graphing Quadratic Function: Method 1

Graphing Quadratic Function: Method 1

Graphing Quadratic Function: Method 2

Graphing Quadratic Function: Method 2

Graphing Quadratic Function: Method 3
The standard form of a quadratic function is given by y = ax2 + bx + c STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

Thus the line of symmetry is x = 1
Graphing Quadratic Function: Method 3 Let's Graph ONE! Try … y = 2x2 – 4x – 1 y x STEP 1: Find the line of symmetry Thus the line of symmetry is x = 1

Graphing Quadratic Function: Method 3
Let's Graph ONE! Try … y = 2x2 – 4x – 1 y x STEP 2: Find the vertex Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex. Thus the vertex is (1 ,–3).

Graphing Quadratic Function: Method 3
Let's Graph ONE! Try … y = 2x2 – 4x – 1 y x STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. 3 2 y x –1 5

The quadratic function f(x) =ax2 + bx + c can be written in an alternate form that relies on the vertex (h, k). Example: f(x) = 3(x - 4)2 + 6 is in vertex form with vertex (h, k) = (4, 6). What is the vertex of the parabola given by f(x) = 7(x + 2)2 – 9 ? Vertex = (-2,-9)

Example Write the formula f(x) = x2 + 10x + 23 in vertex form by completing the square. Given formula Subtract 23 from each side. Add (10/2)2 = 25 to both sides. Factor perfect square trinomial. Subtract 2 form both sides.. What is the vertex? Vertex is h= -5 k = -2.

Example: Convert the quadratic f(x) = 3(x + 2)2 – 8 which
is in vertex form to the form f(x) = ax2 + b + c Given formula. f(x) = 3(x + 2)2 – 8 Expand the quantity squared. f(x) = 3(x2 + 4x + 4) - 8 Multiply by the 3. f(x) = 3x2 + 12x Simplfy. f(x) = 3x2 + 12x + 4

Part 2: Polynomial Functions
Properties of Basic Polynomial Function

Basic Polynomial Function
NOTE: Linear and Quadratic functions we considered earlier belong to the family of polynomial functions!

Basic Polynomial Function

Basic Polynomial Function

Basic Polynomial Function
Example: Draw the graph of the following functions:

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