Understanding Vertex Form of a Quadratic Equation
The vertex form of a quadratic equation is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful for graphing and analyzing quadratic functions because it clearly indicates the vertex's position and the direction in which the parabola opens. The value of 'a' determines the width and direction of the parabola: if 'a' is positive, the parabola opens upward; if negative, it opens downward.
For example, if you have a vertex at (3, -2) and a stretch factor of 2, the equation becomes y = 2(x - 3)² - 2. This indicates the parabola opens upwards and is narrower than the standard parabola.
Identifying the Vertex from a Graph
To write a quadratic equation in vertex form from a graph, the first step is to identify the vertex of the parabola. The vertex is the point where the parabola changes direction, either reaching a maximum or minimum value. Look for the highest or lowest point on the graph.
For instance, if the vertex is located at (1, 4), you can note these coordinates as (h, k). This will be essential in forming the vertex equation later.
Determining the Direction of Opening
Next, observe the direction in which the parabola opens. This is crucial for determining the value of 'a'. If the parabola opens upwards, 'a' will be positive; if it opens downwards, 'a' will be negative. This can often be observed by looking at the vertex and the surrounding points on the graph.
For example, if the vertex is (1, 4) and the arms of the parabola extend upwards, 'a' should be a positive value, indicating that the parabola opens upwards.
Finding Another Point on the Parabola
To find the value of 'a', select another point on the parabola, preferably one that is easy to identify. This point should not be the vertex. For instance, if you choose the point (0, 5), you can use this point along with the vertex to find 'a'.
With the vertex at (1, 4) and the point (0, 5), you now have the necessary information to substitute into the vertex form equation.
Substituting Values into the Vertex Form Equation
Now that you have the vertex (h, k) and another point (x, y), you can substitute these values into the vertex form equation: y = a(x - h)² + k.
Using our example, substituting the vertex (1, 4) and the point (0, 5) gives:
5 = a(0 - 1)² + 4. This simplifies to:
5 = a(1) + 4, leading to 5 = a + 4.
Solving for 'a'
To find 'a', rearrange the equation obtained from the substitution. Continuing from the previous example:
5 = a + 4
Subtract 4 from both sides:
1 = a
Thus, 'a' equals 1. This indicates a standard width for the parabola, as it is not stretched or compressed.
Writing the Final Equation
With the values of 'a', 'h', and 'k' determined, you can now write the final equation in vertex form. From our example, we have:
y = 1(x - 1)² + 4, which simplifies to y = (x - 1)² + 4.
This equation represents a parabola with its vertex at (1, 4) and opens upwards.
Practical Applications of Vertex Form
Understanding how to write a quadratic equation in vertex form from a graph can have several practical applications. In fields such as physics, engineering, and economics, quadratic equations are used to model various phenomena, including projectile motion and profit maximization.
For example, if a business wants to determine the optimal price for a product to maximize profit, it might model its profit function as a quadratic equation. By identifying the vertex, the business can find the price point that yields the highest profit.
