Graphing Function Calculator

Quadratic Function Plotting Data Calculator

Understanding Quadratic Functions and Their Graphs

A quadratic function is a polynomial function of degree two, meaning the highest exponent of the variable (usually 'x') is 2. Its general form is y = ax² + bx + c, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero.

The graph of a quadratic function is a U-shaped curve called a parabola. The direction in which the parabola opens depends on the sign of the coefficient 'a':

• If a > 0, the parabola opens upwards (like a smile), and the vertex is a minimum point.
• If a < 0, the parabola opens downwards (like a frown), and the vertex is a maximum point.

Key Features of a Parabola:

1. Vertex: This is the turning point of the parabola. It's either the lowest point (minimum) or the highest point (maximum) on the graph. The x-coordinate of the vertex can be found using the formula x = -b / (2a). Once you have the x-coordinate, you can substitute it back into the quadratic equation to find the y-coordinate.
2. Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / (2a).
3. Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when x = 0. Substituting x = 0 into the equation y = ax² + bx + c gives y = c. So, the y-intercept is always (0, c).
4. X-intercepts (Roots/Zeros): These are the points where the parabola crosses the x-axis. They occur when y = 0. Finding these points involves solving the quadratic equation 0 = ax² + bx + c, often using the quadratic formula: x = [-b ± sqrt(b² – 4ac)] / (2a). A parabola can have two, one, or no x-intercepts.

How to Use This Calculator:

This calculator helps you understand the behavior of a quadratic function by generating a table of (x, y) coordinates and identifying key features. Simply input the coefficients 'a', 'b', and 'c' for your desired quadratic function (e.g., for y = 2x² + 3x – 1, input a=2, b=3, c=-1). Then, define the range of x-values you want to explore (Starting X Value, Ending X Value) and the increment for x (X Step Size). The calculator will then provide a list of points that can be used to plot the function, along with its vertex, axis of symmetry, and y-intercept.

Example:

Let's consider the function y = x² – 4x + 3.

• Coefficient 'a' = 1
• Coefficient 'b' = -4
• Coefficient 'c' = 3
• Starting X Value = 0
• Ending X Value = 4
• X Step Size = 0.5

Upon calculation, you would find:

• Function: y = 1x² – 4x + 3
• Vertex: (2, -1)
• Axis of Symmetry: x = 2
• Y-intercept: (0, 3)
• Plotting Points:
  • (0, 3)
  • (0.5, 1.25)
  • (1, 0)
  • (1.5, -0.75)
  • (2, -1)
  • (2.5, -0.75)
  • (3, 0)
  • (3.5, 1.25)
  • (4, 3)

This data clearly shows the parabola opening upwards (since a=1 > 0), with its minimum point at (2, -1).

Function Details:

Function:

Vertex:

Axis of Symmetry:

Y-intercept:

Plotting Points:

X	Y
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