Problem 73 - Elementary Algebra

Question

Determine the vertex. $$ y=-(x-5) 2+3 $$

Step-by-Step Solution

Answer

The vertex is (5, 3).

Step 1: Identify the standard form

The given quadratic equation is \(y = -(x-5)^2 + 3\). This equation is in the standard form for a vertex equation, which is generally \(y = a(x-h)^2 + k\), where \( (h, k) \) represents the vertex of the parabola. In this equation, \(a = -1\), \(h = 5\), and \(k = 3\).

Step 2: Determine the vertex

Now that the equation is in standard vertex form, we can directly read out the vertex. From the form \(y = a(x-h)^2 + k\), the vertex \((h, k)\) is \((5, 3)\). Thus, the vertex of the parabola represented by \(y = -(x-5)^2 + 3\) is at \((5, 3)\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are mathematical expressions of the form \[ ax^2 + bx + c = 0 \] where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations are essential because they represent a parabola when graphed. The curve of the parabola opens upwards if 'a' is positive and downwards if 'a' is negative.
Quadratic equations are important in various fields such as physics, engineering, and finance because they describe phenomena involving squared relationships.
Here are a few key points about quadratic equations:

• They always graph to form a parabolic curve.
• The solutions of the equation (or roots) determine where the parabola intersects the x-axis.
• They can be solved using factoring, completing the square, or the quadratic formula.

By understanding how to manipulate and solve these equations, you can determine vital characteristics of the parabola like its vertex, axis of symmetry, and direction of opening.

The Importance of Standard Form

The standard form of a quadratic equation is \[ ax^2 + bx + c = 0 \] however, there's a specific form often referred to as "vertex form" which is \[ y = a(x-h)^2 + k \].
This vertex form is particularly helpful when we need to quickly identify the vertex of the parabola. In this form, the vertex can be found without any calculations as it is simply \((h, k)\), where 'h' and 'k' are directly taken from the equation.
Understanding these forms is crucial for:

• Quickly identifying the vertex of a parabola.
• Understanding how changes in 'a', 'h', and 'k' affect the parabola's shape and position.
• Transforming equations back and forth between forms depending on the information required.

The vertex form makes it very convenient to graph a parabola accurately, predict its behavior, and solve real-world problems that model parabolic shapes.

Finding the Parabola Vertex

The vertex of a parabola is a significant point as it represents the maximum or minimum value of the quadratic function, depending on the direction in which the parabola opens. For a quadratic equation written in the vertex form \[ y = a(x-h)^2 + k \], the vertex is simply \((h, k)\).
In our example equation given as \( y = -(x-5)^2 + 3 \), we identify:

• 'a' is -1 indicating the parabola opens downwards.
• 'h' is 5, signifying the horizontal shift.
• 'k' is 3, indicating the vertical shift.

Thus, the vertex of this parabola is at the point \((5, 3)\).
To find the vertex when not given in vertex form directly, you could convert the equation from standard form to vertex form, or use the vertex formula \( h = -\frac{b}{2a}\) and solve for 'k' using the original equation.
Identifying the vertex helps in graphing the parabola and understanding the basic characteristics like its peak or valley, and whether it represents a maximum or minimum of the set of values.