To identify the standard form of a quadratic equation from a graph, follow these steps:

1. Recognize the general form: A quadratic equation in standard form is expressed as: $y = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. The graph of a quadratic equation is a parabola.

2. Identify the key points on the graph:

   • Vertex: The highest or lowest point of the parabola (depending on whether it opens up or down).
   • Y-intercept: The point where the graph crosses the y-axis. This is the value of $c$ in the standard form.

3. Check the direction of the parabola:

   • If the parabola opens upwards, $a > 0$ (the coefficient of $x^2$ is positive).
   • If it opens downwards, $a < 0$ (the coefficient of $x^2$ is negative).

4. Find the y-intercept: The y-intercept is where the graph crosses the y-axis (where $x = 0$). In the equation $y = ax^2 + bx + c$, the value of $c$ is the y-intercept.

5. Estimate or calculate the coefficients:

   • The coefficient $a$ determines how "wide" or "narrow" the parabola is. A larger $a$ makes it narrow, and a smaller $a$ makes it wide.
   • The coefficients $a$, $b$, and $c$ can be determined by analyzing the symmetry and specific points on the parabola, such as the vertex and the y-intercept.

Example:

If the graph shows a parabola with a vertex at $(2, -3)$ and a y-intercept at $(0, 5)$, we know:

• The equation will be in the form $y = a(x - 2)^2 - 3$ (vertex form), and by expanding this, you can find the standard form.

Do you need details on how to expand from vertex form or other steps?

Here are 5 related questions for further learning:

1. How do you find the vertex of a parabola from the graph?
2. What is the difference between standard form and vertex form of a quadratic equation?
3. How does changing the value of $a$ affect the graph of a quadratic function?
4. How do you determine the axis of symmetry from the graph of a parabola?
5. How can you calculate the roots of a quadratic equation from its graph?

Tip: Always note the y-intercept from the graph first to directly identify the constant term $c$.