Problem 21 - Beginning and Intermediate Algebra

Question

Given a quadratic equation of the form \(x=a(y-k)^{2}+h,\) answer the following. What is the vertex?

Step-by-Step Solution

Answer

The vertex of a quadratic equation in the form \(x = a(y-k)^2 + h\) is at the point (h, k).

Step 1: Identify the equation in standard form

The given equation is of the form \(x = a(y-k)^2 + h\).

Step 2: Find h and k values

In the given equation, the values of h and k are not specified. But in general, for any quadratic equation in the given form, the vertex will be at the point (h, k).

Step 3: Present the vertex

Since the given equation is in the form \(x = a(y-k)^2 + h\), the vertex is located at the point (h, k).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a powerful tool when analyzing quadratic functions.
The general formula is given by \[ x = a(y-k)^2 + h \] where

• \( a \) determines the opening direction and the width of the parabola.
• \( (h, k) \) represents the vertex of the parabola.

The vertex form highlights the vertex location at point \((h, k)\). Because it reveals the vertex directly, this form is beneficial when graphing parabolas. Moreover, comparing the vertex form with other forms like standard form or factored form helps in understanding different aspects of the parabola's shape and position.
An important note is that any quadratic equation can be rewritten in vertex form using the method of completing the square. This conversion helps identify the vertex and provides a visual insight into the parabola behavior.

Parabola Vertex

The parabola vertex is a critical point in the study of quadratic functions.
It is the peak or trough of the curve, indicating where the curve changes direction. The vertex provides essential information about the graph of the quadratic equation.
There are two types of parabolas:

• Opens Upward: The vertex is the minimum point.
• Opens Downward: The vertex is the maximum point.

The vertex can be easily identified when a quadratic equation is in vertex form \((x = a(y-k)^2 + h)\) at the point \((h, k)\).
Knowing the vertex helps in sketching the graph more accurately and understanding the parabola's behavior. For functions in other forms, finding the vertex often requires algebraic manipulation like completing the square or using derivatives to determine turning points. Regardless of the method used, finding the vertex is integral to solving many quadratic function problems.

Standard Form of Quadratic Equation

The standard form of a quadratic equation provides a different perspective from which to understand quadratic functions and is typically written as \[ ax^2 + bx + c = 0 \] where

• \( a \), \( b \), and \( c \) are constants.
• \( a eq 0 \) to ensure the equation is quadratic.

The standard form is useful for performing operations like expanding the equation or applying the quadratic formula to find roots.
One of its main uses is identifying the intercept points, which can be useful to find where the parabola crosses the x-axis. Another is that it allows for the calculation of the vertex using the formula derived: \[ h = -\frac{b}{2a} \] and substitute back into the original equation to find \( k \).
Even though the standard form doesn't explicitly show the vertex, it provides fundamental insights into the quadratic equation's behavior, enabling conversions to vertex or factored forms for more detailed analysis. This versatility makes it an essential tool in solving quadratic problems.