1. Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form:

f(x) = ax^2 + bx + c

Where:

• ( a ), ( b ), and ( c ) are constants.
• ( a not eqauls to 0 ).

2. Purpose of Creating a Table

Creating a table helps to:

• Visualize the behavior of the quadratic function.
• Identify key points such as the vertex, axis of symmetry, roots (if any), and the y-intercept.
• Understand how the function changes as ( x ) varies.

3. Steps to Create a Table for a Quadratic Equation

Step 1: Choose a Range of ( x ) Values

Select a range of ( x ) values that are evenly spaced and cover both negative and positive values. This ensures that the table captures the behavior of the quadratic function around its vertex and across the x-axis.

For example, if the quadratic equation is ( f(x) = 2x^2 – 3x + 1 ), you might choose ( x ) values such as (-2, -1, 0, 1, 2).

Step 2: Substitute ( x ) Values into the Equation

For each chosen ( x ) value, substitute it into the quadratic equation to find the corresponding ( f(x) ) value. This involves simple arithmetic to evaluate the expression.

Step 3: Calculate ( f(x) ) for Each ( x ) Value

Perform the calculation for each ( x ) value and record the result.

4. Example of Completing a Table

Let’s consider the quadratic equation:

f(x) = x^2 – 4x + 3

Step-by-Step Table Creation

1. Select ( x ) values: Choose values around the expected vertex, such as ( -1, 0, 1, 2, 3, 4, 5 ).
2. Create the table:

( x )	Calculation	( f(x) )
-1	((-1)^2 – 4(-1) + 3)	( 1 + 4 + 3 = 8 )
0	(0^2 – 4(0) + 3)	( 0 + 0 + 3 = 3 )
1	(1^2 – 4(1) + 3)	( 1 – 4 + 3 = 0 )
2	(2^2 – 4(2) + 3)	( 4 – 8 + 3 = -1 )
3	(3^2 – 4(3) + 3)	( 9 – 12 + 3 = 0 )
4	(4^2 – 4(4) + 3)	( 16 – 16 + 3 = 3 )
5	(5^2 – 4(5) + 3)	( 25 – 20 + 3 = 8 )

Final Table:

( x )	( f(x) )
-1	8
0	3
1	0
2	-1
3	0
4	3
5	8

5. Interpreting the Table

• Vertex: The lowest (or highest for negative (a)) point on the table is the vertex. In this case, at ( x = 2 ), ( f(x) = -1 ) suggests the vertex is at (2, -1).
• Axis of Symmetry: The x-value of the vertex (x = 2) acts as the line of symmetry for the parabola.
• Roots: The x-values where ( f(x) = 0 ) are the roots of the equation. In this case, ( x = 1 ) and ( x = 3 ).
• Y-Intercept: The value of ( f(x) ) when ( x = 0 ) is the y-intercept. Here, it’s 3.

6. Graphing Using the Table

Plotting these ( (x, f(x)) ) points on a coordinate plane helps visualize the parabolic shape of the quadratic equation. You can use the table data to sketch the curve, marking key points like the vertex, intercepts, and roots.