Problem 47 - 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 \(g(x) .\) $$ (2,3) $$

Step-by-Step Solution

Answer

The equation is \(g(x) = 5(x-2)^2 + 3\).

Step 1: Identify the Vertex Form of a Parabola

The vertex form of a parabola is expressed as \( g(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola, and \(a\) is the same coefficient as in \(f(x)\) to maintain the shape.

Step 2: Use Vertex Information

Substitute the vertex \((2, 3)\) into the vertex form. The values \(h = 2\) and \(k = 3\) are substituted into the equation, so it becomes \( g(x) = a(x-2)^2 + 3 \).

Step 3: Maintain the Shape of the Parabola

Since the shape of the parabola must remain the same as \(f(x) = 5x^2\), the coefficient \(a\) remains \(5\). Substitute \(a = 5\) into the equation: \( g(x) = 5(x-2)^2 + 3 \).

Step 4: Present the Final Equation

The final equation, after substituting \(a\), \(h\), and \(k\), is \( g(x) = 5(x-2)^2 + 3 \).

Key Concepts

Vertex Form

The vertex form of a parabola provides a simplified way to write the equation of a quadratic function. This form is expressed as \( g(x) = a(x-h)^2 + k \). This notation clearly shows how the graph of the parabola shifts on a coordinate plane.

• \(a\) determines the parabola's vertical stretch or compression and the direction of its opening. A positive \(a\) opens upwards, while a negative \(a\) opens downwards.


• \((h, k)\) identifies the vertex, the highest or lowest point on the graph depending on how it opens.

Being aware of the vertex form can make graphing and understanding transformations much easier, since it directly provides both the vertex and stretch factor in a straightforward equation.

Vertex of a Parabola

The vertex of a parabola is a special point, commonly noted as \((h, k)\). This point marks the "tip" of the parabola, distinguishing between its upward or downward faces.

• In simple terms, if the parabola opens upwards, the vertex is the lowest point.


• If it opens downwards, the vertex is the highest point.

The vertex plays an important role in many applications, serving as a reference for maximum or minimum values of quadratic functions in various real-world scenarios. The vertex coordinates are key in transforming the standard form of a parabola into its vertex form, which makes understanding and manipulating parabolas much more accessible.

Quadratic Function

Quadratic functions are a type of polynomial function identifiable by their characteristic "U"-shaped graphs, called parabolas. These functions can be written in the form \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants.

• When \(a > 0\), the parabola opens upwards.


• When \(a < 0\), it opens downwards.

Each quadratic function naturally defines a symmetrical curve around a vertical line known as the axis of symmetry, which passes through the vertex. The "shape" and "location" of the graph are affected by the values of \(a\), \(b\), and \(c\), with the vertex form highlighting these shifts more directly. Understanding these principles helps in solving a variety of real-world and mathematical problems involving motion, architecture, and optimization.