Quadratic Equations from Graphs

What does the equation y = a(x - h)^2 + k represent in a quadratic function?

What information is needed to find the equation of a quadratic function in standard form?

Given the vertex (3, -4) and the point (4, -2), what is the value of 'a' in the vertex form equation?

If the equation of a quadratic function is y = 3x^2 - 5x - 2, what is the value of c?

Which of the following statements is true regarding finding the equation of a quadratic function?

Vertex Form of a Quadratic Function

Finding the Equation using Vertex Form

Standard Form of a Quadratic Function

Finding the Equation using Standard Form

Solving for Coefficients

Finding the Equation of a Quadratic Function from a Graph

Vertex Form: y = a(x - h)^2 + k, where (h, k) represents the vertex.
Standard Form: y = ax^2 + bx + c

Using the Vertex Form

To find the equation using the vertex form, you need the vertex and one other point on the graph.
Example: Given the vertex (3, -4) and the point (4, -2), substitute the values into the vertex form:
- y = a(x - 3)^2 - 4
- Substitute (x, y) with (4, -2) to solve for 'a'.
- Solve for 'a' = 2
- The equation is y = 2(x - 3)^2 - 4.

Using the Standard Form

To find the equation using the standard form, you need three points.
Example: Given the points (-1, 6), (0, -2), and (2, 0), use the following steps:
- Identify the y-intercept: The point (0, -2) gives you the value of c.
  - c = -2
- Use the other two points: Substitute the values into the standard form and solve for 'a' and 'b' using systems of equations.
  - Solve for a and b using the remaining points.
  - This gives you a = 3 and b = -5.
  - The equation is y = 3x^2 - 5x - 2.

Key Points

If the vertex is known, using the vertex form is easier.
If the vertex is unknown, use the standard form and three points.
Understanding how to correctly substitute values and accurately solve for the coefficients (a, b, c) is vital.
Practice with various examples to improve your understanding of finding the equation of a quadratic function from its graph.

This quiz will assess your understanding of finding the equation of a quadratic function from its graph in both vertex and standard form. You will practice using vertex coordinates and points to derive the respective equations. Test your skills with practical examples and solutions provided!