Problem 44 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=2(x-3)^{2}+2 $$

Step-by-Step Solution

Answer

The vertex of the function is \((3, 2)\).

Step 1: Identify the Vertex Form

The given function is already in the vertex form of a quadratic, which is \( f(x) = a(x-h)^2 + k \). Here, \( a = 2 \), \( h = 3 \), and \( k = 2 \).

Step 2: Determine the Vertex

The vertex of a quadratic function in the form \( a(x-h)^2 + k \) is \( (h, k) \). So, substituting \( h = 3 \) and \( k = 2 \), the vertex is \( (3, 2) \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a valuable tool for identifying key features of a parabola quickly and efficiently. It is expressed as \( f(x) = a(x-h)^2 + k \). This format highlights the vertex of the parabola, represented by \((h, k)\), making it straightforward to locate the vertex on a graph.
In the vertex form, the parameter \(a\) indicates the direction and the width of the parabola's opening. If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, it opens downwards. The absolute value of \(a\) affects the narrowness or wideness of the parabola—larger values result in a narrower parabola, while smaller values produce a wider one.
In our example, the function \( f(x) = 2(x-3)^2 + 2 \) is already in vertex form. From the expression, \( a = 2 \), \( h = 3 \), and \( k = 2 \). This directly tells us about the shape, position, and direction of the parabola without needing any further calculation.

Vertex of a Parabola

The vertex of a parabola is a crucial point that acts as the maximum or minimum of a quadratic function, depending on its orientation. In the vertex form \( f(x) = a(x-h)^2 + k \), the vertex is clearly represented by the point \((h, k)\). This makes it incredibly easy to pinpoint the vertex just by identifying \(h\) and \(k\) from the equation.
In the function \( f(x) = 2(x-3)^2 + 2 \), the vertex is found at \((3, 2)\). This vertex represents the minimum point of the parabola because \( a = 2 \) is positive, indicating an upward opening of the graph.
The vertex not only identifies the turning point of the graph but also provides the line of symmetry for the parabola. This axis of symmetry can be used to mirror points on the parabola, which is helpful when plotting it on a graph.

Graphing Quadratic Functions

Graphing quadratic functions becomes easier once the function is in vertex form. The vertex \((h, k)\) provides a starting point for plotting the function on a coordinate plane. Knowing the value of \(a\) helps in understanding how the parabola will be shaped and oriented.
To graph the function \( f(x) = 2(x-3)^2 + 2 \), follow these steps:

• Identify the vertex \((3, 2)\) and plot it on the graph.
• Determine the parabola's direction: since \( a = 2 \), the parabola opens upwards.
• Consider the value of \(a\), which suggests the parabola is relatively narrow due to the larger value.
• Sketch the axis of symmetry at \( x = 3 \), and use it to plot additional points by choosing x-values and calculating their respective y-values.
• Connect the points in a smooth, u-shaped curve to complete the parabola.

By understanding these aspects, you can easily graph any quadratic function in vertex form, creating a visual representation that mirrors the mathematical expression.