Problem 11 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=(x-1)^{2} $$

Step-by-Step Solution

Answer

The vertex of the parabola is (1, 0).

Step 1: Identify the Standard Form

The given function is already in the standard form of a parabola: f(x) = (x - h)^2 + k where h and k are the coordinates of the vertex.

Step 2: Extract the Vertex Coordinates

Compare the given equation f(x) = (x - 1)^2 with the standard form f(x) = (x - h)^2 + k to find h and k . Here, h is 1 and k is 0 because there is no k term present (which implies k = 0).

Step 3: Write the Vertex

The vertex is the point (h, k) . Therefore, the vertex of the parabola is (1, 0).

Key Concepts

Standard Form of a Parabola

Understanding the standard form of a parabola is crucial for working with these types of functions. The standard form is expressed as:
f(x) = (x - h)^2 + k
In this form, the letters h and k represent constants that will locate the vertex of the parabola. Specifically, h is the x-coordinate, and k is the y-coordinate of the vertex. Recognizing this form will help you correctly identify the elements of the equation needed to extract the vertex.

Vertex Coordinates

The vertex of a parabola is a specific point that represents the maximum or minimum value of the quadratic function, depending on its orientation. In the standard form of a parabola f(x) = (x - h)^2 + k, the coordinates of the vertex are given directly by the constants h and k. To find these coordinates:
1. Identify the value of h inside the parenthesis within the squared term (x - h).
2. Identify the value of k as the constant term added or subtracted outside the squared term.
For example, in the function f(x) = (x - 1)^2, there is no constant term added or subtracted outside the square, so k = 0. Thus, the vertex coordinates are (h, k) = (1, 0).

Extracting Vertex

Extracting the vertex from a given parabolic function involves comparing it with the standard form as discussed earlier. Here’s a step-by-step way to extract the vertex:
1. **Compare the function to the standard form:** Take the given function and align it visually or mentally with f(x) = (x - h)^2 + k.
2. **Identify h and k:** From the comparison, determine the values of h and k. For f(x) = (x - 1)^2, you would identify h = 1 and k = 0.
3. **Write the vertex:** Form the vertex coordinates as (h, k). Therefore, the vertex of the function f(x) = (x - 1)^2 is at (1, 0). This process helps in finding the highest or lowest point on the parabola where the function turns.