Desmos Graphing Calculator

by
Desmos Function Property Calculator
Quadratic Analysis (ax² + bx + c)Linear Properties (y = mx + b)
Analysis Results:
Enter values above to see critical points and graphing data.
© Calculator Soup Style Grapher
function calculateDesmosProperties(){var a=parseFloat(document.getElementById('input_a').value);var b=parseFloat(document.getElementById('input_b').value);var c=parseFloat(document.getElementById('input_c').value);var type=document.getElementById('calc_type').value;var showSteps=document.getElementById('show_steps').checked;var ans=document.getElementById('answer');if(isNaN(a)||isNaN(b)||isNaN(c)){ans.innerHTML='Error: Please enter valid numeric coefficients.';return;}if(type==='quadratic'){if(a===0){ans.innerHTML='Error: "a" cannot be 0 in a quadratic. Use Linear mode.';return;}var disc=(b*b)-(4*a*c);var vX=-b/(2*a);var vY=a*(vX*vX)+(b*vX)+c;var result='Vertex: ('+vX.toFixed(2)+', '+vY.toFixed(2)+')
';result+='Y-intercept: (0, '+c.toFixed(2)+')
';result+='Discriminant (Δ): '+disc.toFixed(2)+'
';if(disc>0){var x1=(-b+Math.sqrt(disc))/(2*a);var x2=(-b-Math.sqrt(disc))/(2*a);result+='Roots (X-intercepts): x₁='+x1.toFixed(2)+', x₂='+x2.toFixed(2);}else if(disc===0){var x=-b/(2*a);result+='Root (X-intercept): x='+x.toFixed(2);}else{result+='Roots: No real roots (complex roots only).';}if(showSteps){result+='
Steps:
1. Axis of Symmetry: x = -b/2a = -('+b+')/(2*'+a+') = '+vX.toFixed(2)+'
2. Vertex Y: Plug '+vX.toFixed(2)+' into equation.
3. Discriminant: b²-4ac = '+disc.toFixed(2)+'
';}ans.innerHTML=result;}else{var xInt=-c/a;ans.innerHTML='Slope (m): '+a.toFixed(2)+'
Y-intercept (b): '+c.toFixed(2)+'
X-intercept: ('+xInt.toFixed(2)+', 0)
Equation: y = '+a.toFixed(2)+'x + '+c.toFixed(2);}}

Using the Desmos Graphing Calculator Assistant

The desmos graphing calculator is a revolutionary tool for students, engineers, and mathematicians. While the actual Desmos platform provides a visual interface for plotting, our companion calculator helps you solve the specific numerical properties of functions before you plot them. This tool is designed to find vertices, roots, and intercepts for quadratic and linear equations—the same data points you would identify when clicking on "gray dots" in the Desmos interface.

To get the most out of your graphing experience, follow these steps:

Coefficient (a)
This represents the leading coefficient of your function. In a quadratic ($ax^2$), it determines the direction (up or down) and the width of the parabola.
Coefficient (b)
The linear coefficient which, along with 'a', helps locate the axis of symmetry and the vertex position.
Constant (c)
The constant value where the function crosses the y-axis (y-intercept).

How the Math Works

When using a desmos graphing calculator, the software automatically solves complex algebraic expressions. Behind the scenes, it uses standard algebraic formulas. For a quadratic function, the most important formula used by our tool is the Vertex Formula:

Vertex X (h) = -b / 2a

To find the roots (where the graph touches the horizontal x-axis), we use the Quadratic Formula:

x = [-b ± sqrt(b² – 4ac)] / 2a

  • Discriminant (b²-4ac): Determines if you have two, one, or zero real roots.
  • Vertex Y: Calculated by plugging the Vertex X back into the original function.
  • Slope: In linear mode, the coefficient 'a' acts as the rise over run (m).

Calculation Example

Example: Suppose you want to analyze the function $f(x) = 2x^2 + 4x – 6$ as you would in a desmos graphing calculator.

Step-by-step solution:

  1. Identify inputs: a = 2, b = 4, c = -6.
  2. Calculate Vertex X: $h = -4 / (2 * 2) = -1$.
  3. Calculate Vertex Y: $k = 2(-1)^2 + 4(-1) – 6 = 2 – 4 – 6 = -8$.
  4. Find Roots: Discriminant = $4^2 – 4(2)(-6) = 16 + 48 = 64$. Since 64 is positive, there are two roots.
  5. Result: Vertex is (-1, -8). Roots are x = 1 and x = -3.

Frequently Asked Questions

Why is the vertex important in Desmos?

The vertex represents the maximum or minimum point of a parabola. In Desmos, this is a key clickable point that helps define the range of the function and its behavior.

What does a negative discriminant mean?

If the discriminant is negative, the graph does not cross the x-axis. On the desmos graphing calculator, you would see the parabola floating entirely above or below the x-axis.

Can I use this for linear equations?

Yes, simply switch the calculation type to "Linear Properties". This will allow you to find the slope and intercepts for equations in the form $y = mx + b$.

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