6.2 Graphing Parabolas Function Language and Graphing with a Table

“F of X equals three “x” squared plus two “x” plus 5” 6.2 Graphing Parabolas We will start using a new math term called “Function Language”. We use the term “Function” when we work with a table using INPUT (called Domain) and OUTPUT (called Range). A Function is a math term that means…”An equation that has exactly one output for every input”. A function looks like this… f(x) is read as “F of X” And looks like this… f(x) = 3x2 + 2x + 5 “F of X equals three “x” squared plus two “x” plus 5”

x y 1 2 3 4 6.2 Graphing Parabolas How do we use it? Please graph the equation y = 3x2 - 12x + 7 using at least 5 points. Start by drawing a table that leaves space for 5 points. I will give you the 5 “x” points (Domain) that I want you to use… (0, 1, 2, 3, 4) Now replace the y = 3x2 - 12x + 7 with f(x) = 3(x)2 – 12(x) + 7 x y 1 2 3 4 Re-write your function 5 times, replacing your “x” with a number. It will now look like this…

f( x ) = 3( x )2 – 12( x ) + 7 x y 1 2 3 4 6.2 Graphing Parabolas 7 1 2 3 4 7 f(0) = 3( )2 – 12( ) + 7 = 0 0 f(1) = 3( )2 – 12( ) + 7 = f(2) = 3( )2 – 12( ) + 7 = f(3) = 3( )2 – 12( ) + 7 = f(4) = 3( )2 – 12( ) + 7 = Now guess what goes in the ( )??! f(0) “F of Zero” means replace “0” where the “X” was. 7 is the Range for the 0 Domain! Here is f(0) Now do the math! f(0) = 3( )2 – 12( ) + 7 = 0 0 f(0) = 0 – 0 + 7 = 7

x y 1 2 3 4 6.2 Graphing Parabolas Domain Range f( x ) = 3( x )2 – 12( x ) + 7 x y 1 2 3 4 7 f(0) = 3( )2 – 12( ) + 7 = 0 0 0 - 0 + 7 = 7 -2 f(1) = 3( )2 – 12( ) + 7 = 1 1 3 - 12 + 7 = -2 -5 f(2) = 3( )2 – 12( ) + 7 = 2 2 12 - 24 + 7 = -5 -2 f(3) = 3( )2 – 12( ) + 7 = 3 3 27 - 36 + 7 = -2 7 f(4) = 3( )2 – 12( ) + 7 = 4 4 48 - 48 + 7 = 7 Now you finish the rest of the table… (use a calculator if needed) Great! You can now… 1. Change “y = “ into “f(x) =“ 2. Create a table and input your Domain (x values) 3. Input your x values (domain) into the function to get your y values (range)

x y 1 2 3 4 6.2 Graphing Parabolas Now how do you graph the points? 7 1 2 3 4 Now how do you graph the points? 7 One of the most important things to know about graphing any type of line (straight, curved, or absolute value) is to have an idea of what it will look like Before You Start. -2 -5 Our present equation is of the form y = ax2 + bx + c. This is a Quadratic and the shape of the graph looks like a horseshoe called a Parabola. -2 7 Positive quadratics make a happy parabola (+x2) Negative quadratics make a sad parabola (-x2)

6.2 Graphing Parabolas x y 1 2 3 4 Our Parabola is happy (y = 3x2 is positive). Let’s graph the points from the table. 7 We are going to “Sketch” a graph. This means our points will not be exact. When you use graph paper, your points should be perfect. -2 -5 -2 Y 7 (0, 7) (4, 7) First, make your graph… Now plot your points from the table… X (0, 7) (1, -2) (3, -2) (1, -2) (2, -5) (2, -5) Now draw your parabola through the points… (3, -2) (4, 7)

6.2 Graphing Parabolas x y 1 2 3 4 Notice the Parabola looks exactly the same on both sides of itself. This attribute is call Symmetry. 7 -2 -5 -2 Y 7 (0, 7) (4, 7) In a later lesson, we will learn to graph without using a table. We will look for special points called y intercept, x intercepts (zeros), Axis of Symmetry (AOS), and Vertex. (1, -2) (3, -2) (2, -5)