How to Find the Equation of a Quadratic Function from a Graph

TLDR; Finding the equation of a quadratic function from a graph involves using the vertex form to determine the standard form and then converting it to the general form.

💡 Determining the Equation

The goal is to find the equation of a quadratic function from the graph and express it in general form.

The general form of a quadratic function is F(x) = ax^2 + bx + c.

The process starts by using the standard form, F(x) = a(x - h)^2 + k, where h and k represent the coordinates of the vertex.

📈 Identifying Coordinates

The vertex, with coordinates (h, k), and another point on the graph, with coordinates (x, y), are used to initiate the process.

For the given graph, the vertex has coordinates (-3, 4), and another point on the function has coordinates (-4, 2).

🔄 Substitution and Solving for 'a'

By substituting the known values into the standard form, an equation is formed with 'a' as the only unknown.

Solving for 'a' yields the value necessary to establish the function in standard form.

🔍 Converting to General Form

The function is then converted to general form by multiplying out, combining like terms, and arranging the terms in descending order.

This process involves squaring the quantity (x - h), distributing the coefficient, and combining like terms.

📝 Completing the Conversion

The final step involves multiplying out the squared quantity, distributing the coefficient, and combining like terms to obtain the function in general form.

The resulting general form of the quadratic function is F(x) = -2x^2 - 12x - 14.

📊 Final Result

The general form of the quadratic function, given by the graph, is F(x) = -2x^2 - 12x - 14.

This concludes the process of finding the equation of a quadratic function from the graph.