Graphing Calculator

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Graphing Calculator (Quadratic Functions)
Quadratic: f(x) = ax² + bx + c
Results:
Enter coefficients and click Graph.
function calculateGraph(){var a=parseFloat(document.getElementById('inputA').value);var b=parseFloat(document.getElementById('inputB').value);var c=parseFloat(document.getElementById('inputC').value);var canvas=document.getElementById('graphCanvas');var ctx=canvas.getContext('2d');if(isNaN(a)||isNaN(b)||isNaN(c)){alert('Please enter valid numeric coefficients.');return;}if(a===0){alert('Coefficient "a" cannot be zero for a quadratic function.');return;}ctx.clearRect(0,0,canvas.width,canvas.height);var width=canvas.width;var height=canvas.height;var centerX=width/2;var centerY=height/2;var scale=20;ctx.beginPath();ctx.strokeStyle='#ccc';ctx.moveTo(0,centerY);ctx.lineTo(width,centerY);ctx.moveTo(centerX,0);ctx.lineTo(centerX,height);ctx.stroke();ctx.beginPath();ctx.strokeStyle='#428bca';ctx.lineWidth=2;var first=true;for(var x=-10;x=0&&canvasY0){var x1=(-b+Math.sqrt(disc))/(2*a);var x2=(-b-Math.sqrt(disc))/(2*a);rootsText='Real Roots: x = '+x1.toFixed(2)+', x = '+x2.toFixed(2);}else if(disc===0){var x1=-b/(2*a);rootsText='One Real Root: x = '+x1.toFixed(2);}else{rootsText='No Real Roots (Complex).';}document.getElementById('answerDetail').innerHTML='Function: f(x) = '+a+'x² + '+b+'x + '+c+'
Vertex: ('+vertexX.toFixed(2)+', '+vertexY.toFixed(2)+')
Y-Intercept: (0, '+c+')
Discriminant: '+disc.toFixed(2)+'
'+rootsText+'';}

Using the Graphing Calculator

A graphing calculator is an essential tool for students and professionals to visualize mathematical functions. This specific tool focuses on quadratic functions, which take the form f(x) = ax² + bx + c. By entering the coefficients, you can immediately see the parabola's shape, its direction, and critical points like the vertex and intercepts.

Coefficient (a)
Determines the width and direction of the parabola. If a is positive, it opens upward; if negative, it opens downward.
Coefficient (b)
Affects the horizontal position of the vertex and the slope of the curve as it passes the y-axis.
Constant (c)
The y-intercept of the function. This is where the curve crosses the vertical axis.

How the Graphing Math Works

To plot a quadratic function manually, the graphing calculator uses several algebraic formulas. The most critical is the Quadratic Formula to find where the graph crosses the x-axis:

x = [-b ± sqrt(b² – 4ac)] / 2a

  • Discriminant (b² – 4ac): Determines if there are two, one, or zero real roots.
  • Vertex (h, k): The peak or valley of the curve, where h = -b/2a and k = f(h).
  • Axis of Symmetry: The vertical line x = h that splits the parabola into two mirror images.

Graphing Calculator Example

Scenario: Graph the function f(x) = 1x² – 4x + 3.

Step-by-Step Analysis:

  1. Identify coefficients: a = 1, b = -4, c = 3.
  2. Find the Vertex: h = -(-4) / (2*1) = 2. Plug 2 into the function: (2)² – 4(2) + 3 = 4 – 8 + 3 = -1. Vertex is (2, -1).
  3. Find the Roots: Using the quadratic formula, roots are x=1 and x=3.
  4. Find the Y-Intercept: Set x=0, so y=3. Intercept is (0, 3).
  5. Result: A parabola opening upward with its lowest point at (2, -1).

Common Questions

Why is the graphing calculator showing no roots?

If the discriminant (b² – 4ac) is negative, the parabola does not touch the x-axis. This happens when the vertex is above the axis and the curve opens upward, or when the vertex is below the axis and it opens downward.

What happens if "a" is zero?

If "a" is zero, the squared term disappears and the function becomes linear (f(x) = bx + c). A standard graphing calculator for quadratics requires "a" to be non-zero to form a parabola.

How do I find the maximum or minimum value?

The maximum or minimum value is always the y-coordinate of the vertex. If a > 0, the vertex is a minimum. If a < 0, the vertex is a maximum.

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