How do you find the 'a' of a vertex form equation?

To find the value of 'a' in a vertex form equation, follow these steps:

Understanding Vertex Form

The vertex form of a quadratic equation is expressed as:

   y = a(x - h)^2 + k

Where:

• (h, k) represents the coordinates of the vertex of the parabola.
• a determines the direction and stretch of the parabola.

Steps to Find 'a'

Here's a step-by-step guide to calculate the 'a' value, based on the provided references:

1. Identify the Vertex:

   • According to reference point 2, begin by finding the vertex (h, k) of the parabola. This is the highest or lowest point of the graph, as indicated in reference point 3.

2. Choose Another Point:

   • As per reference point 4, select any point on the graph that is not the vertex. Label this point as (x, y).

3. Plug into the Vertex Form:

   • Substitute the values of the vertex (h, k) and the chosen point (x, y) into the vertex form equation:

      y = a(x - h)^2 + k

4. Solve for 'a':

   • Now, you have an equation with one unknown, 'a'. Solve this equation to find the value of 'a'.

Example

Let's say you have a vertex at (2, 3) and another point on the graph at (4, 11).

1. Vertex (h,k): (2,3)

2. Point (x,y): (4,11)

3. Plug in the values:

    11 = a(4 - 2)^2 + 3

4. Solve for 'a':

   • Simplify: 11 = a(2)^2 + 3
   • Further simplify: 11 = 4a + 3
   • Subtract 3 from both sides: 8 = 4a
   • Divide by 4: a = 2

   Therefore, the value of 'a' is 2.

Key Takeaways

• Finding 'a' requires using the vertex and one other point on the parabola.
• The 'a' value determines how the parabola opens (upward if positive, downward if negative) and its width (narrower if |a| > 1, wider if 0 < |a| < 1).