Problem 50 - Algebra A Combined Function

Question

Write the equation of the parabola that has the same shape as \(f(x)=5 x^{2}\) but with the given vertex. Call each function \(g(x) .\) $$ (4,-1) $$

Step-by-Step Solution

Answer

The equation is \( g(x) = 5(x-4)^2 - 1 \).

Step 1: Identify the standard form of the parabola

The standard form of a parabolic equation is given by \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, your task is to create a parabola with the given vertex \((4, -1)\) and the same 'shape' as \( f(x) = 5x^2 \).

Step 2: Determine the value of 'a'

The 'a' value in the vertex form \( y = a(x-h)^2 + k \) determines the shape of the parabola. In the given function \( f(x) = 5x^2 \), the coefficient of \( x^2 \) is 5. Thus, the 'a' in our new function \( g(x) \) will also be 5 to ensure the same shape.

Step 3: Substitute the vertex into the vertex form equation

Substitute \( a = 5 \) and the vertex \((h, k) = (4, -1)\) into the vertex form equation: \[ g(x) = 5(x-4)^2 - 1 \]. This equation represents the parabola with vertex \((4, -1)\) and the same width and direction as \( f(x) = 5x^2 \).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a useful way to represent quadratic functions, especially when you are interested in the vertex's coordinates. It is written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. By using this form, you can easily read off the vertex and understand how changes to the equation affect the graph of the function.

In this form, the parameter \( a \) controls the parabola's direction and width. If \( a \) is positive, the parabola opens upward; if it is negative, it opens downward. The absolute value of \( a \) determines how "steep" or "wide" the parabola is. A larger value of \( a \) will create a steeper graph, while a smaller value will create a wider graph.

Using the vertex form is particularly useful for graphing parabolas and solving problems where the focus is on the vertex’s position.

Vertex of a Parabola

The vertex of a parabola is a critical point on the graph of a quadratic function. It represents either the highest or lowest point of the parabola, depending on its orientation. The coordinates of the vertex, denoted \((h, k)\), provide valuable information about the function's graph.

Here are some key points about the vertex:

• If the parabola opens upward, the vertex is the minimum point.
• If the parabola opens downward, the vertex is the maximum point.
• The vertex is the point where the axis of symmetry of the parabola intersects the graph.
• In vertex form \( y = a(x-h)^2 + k \), \( h \) and \( k \) are directly the coordinates of the vertex.

In our example, the vertex given is \((4, -1)\), meaning this is the critical point where the parabola achieves its minimum value if \( a \) is positive.

Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form \( ax^2 + bx + c \), or, as shown earlier, in the vertex form \( y = a(x-h)^2 + k \). These functions are called quadratic because of the "quadratic" or square term, \( x^2 \).

Here are the main characteristics of quadratic functions:

• They graph to form a U-shaped curve called a parabola.
• The direction of the parabola (upward or downward) is determined by the sign of the coefficient \( a \).
• The vertex of the parabola provides key information about the function's graph, such as the maximum or minimum point.
• Quadratic functions have a line of symmetry that passes through the vertex, dividing the parabola into two mirror-image halves.

In the context of our original problem, understanding these components allows us to take a given quadratic and manipulate its form by changing the vertex while preserving its overall shape by keeping the same coefficient \( a \). This allows the graph to translate without changing its fundamental characteristics, like width and orientation.