Question

The vertex of the parabola \((x-b)^{2}=4 b(y-b)\) is ........ (a) (b,0) (b) \((0, b)\) (c) \((0,0)\) (d) \((\mathrm{b}, \mathrm{b})\)

Short answer

The vertex of the given parabola \((x-b)^2 = 4b(y-b)\) is (b, b), which corresponds to option (d).

Step-by-step solution

Identify the vertex form of the given parabolic equation

We have the given equation of the parabola as \((x-b)^2 = 4b(y-b)\). Comparing it with the standard vertex form of a parabola, \((x-h)^2 = 4a(y-k)\), we can match the corresponding values.

Compare the given equation to the vertex form equation

We have: \((x-b)^2 = 4b(y-b)\) \((x-h)^2 = 4a(y-k)\) Comparing the two equations, we get: h = b k = b a = b Thus, the vertex of the parabola is at the point (h, k) which is (b, b).

Determine the correct answer

According to our finding, the vertex of the given parabola is (b, b). Comparing this with the given options, we see that the vertex of the parabola is the point (b, b) which corresponds to option (d). Therefore, the correct answer is (d) \((b, b)\).

Key concepts

Vertex Form of Parabola

The vertex form of a parabola is a particular way of expressing the quadratic equation of a parabola to make it easy to identify one vital feature—the vertex. The vertex is the point where the parabola changes direction, typically the highest or lowest point on the graph. In vertex form, a parabola's equation is written as \[(x-h)^2 = 4a(y-k)\]Here,

• \( (h, k) \) represents the vertex of the parabola.
• \( a \) gives us information about the parabola's "width" and direction.

In this form, the vertex \( (h, k) \) is directly readable from the equation. You can see the relationship between horizontal and vertical translations right away."For example, if the equation is \[(x - 2)^2 = 8(y + 3)\]The vertex can be identified to be \( (2, -3) \). Recognizing the vertex form directly from an equation allows you to quickly graph the parabola and understand its key features.

Standard Form of a Parabola

Parabolas can also be expressed in what is known as the standard form. The standard form is usually given as:\[y = ax^2 + bx + c\]This form is useful for determining the direction of the parabola and finding the roots of the equation by factoring, if possible, or by using the quadratic formula. In the standard form:

• \( a \) indicates if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).
• \( b \) affects the position of the vertex.
• \( c \) is the y-intercept.

While the standard form is extremely useful for other calculations, converting it to vertex form can make tasks like sketching the parabola on a graph easier, because it gives you the vertex outright. Knowing both forms allows you to analyze a parabola from different perspectives, whether you're doing algebraic manipulations or visual graphing.

Matching Vertex Form

Converting a quadratic equation to its vertex form is a valuable skill as it helps to easily identify the vertex and sketch the graph. To rewrite the standard form \( y = ax^2 + bx + c \) into vertex form:1. Identify or compute \( h \) and \( k \), the coordinates of the vertex.2. Use completing the square method for the x-terms.3. Rearrange the equation so that it matches \((x-h)^2 = 4a(y-k)\).For the given equation in this exercise \[(x-b)^2=4b(y-b)\],comparing it against your known vertex form \((x-h)^2=4a(y-k)\)it is easy to match and deduce:

• \( h = b \)
• \( k = b \)
• \( a = b \)

Thus, reaffirming the vertex at \((b, b)\). The primary task when dealing with different forms of equations is matching elements wherever possible so that the resulting equation provides as much information about the graph as possible.

Coordinate Geometry Basics

Coordinate geometry combines algebra with geometry to locate points on a plane, such as the key vertex of a parabola. At its heart, coordinate geometry involves using the Cartesian coordinate system, where points are defined by ordered pairs \( (x, y) \).

• To understand the position of the vertex on a plane, simply evaluate \( h \) and \( k \), from the vertex form or deduced comparisons.
• This vertex position clarifies the parabola's orientation and assists in plotting.
• Graphs of equations are the foundation upon which complex mathematical relationships are visualized.

By applying coordinate geometry, you gain a holistic view of how algebraic equations translate into graphical forms. This empowers analyzing individual elements like slopes, intercepts, and, essential for parabolas, the vertex. Visualizing these features through graphing is often key to understanding their relationships in real-world applications. With practice, this becomes a vital tool for students looking to excel in mathematics.