Problem 46 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=(x+1)^{2}+4 $$

Step-by-Step Solution

Answer

The vertex is \((-1, 4)\).

Step 1: Identify the function type

The given function is in the form \( f(x) = (x + h)^2 + k \), which is a standard form of a parabola: \( (x-h)^2 + k \). This form is known as the vertex form, \( y = (x-h)^2 + k \). In our function, \( h = -1 \) and \( k = 4 \).

Step 2: Determine the vertex

The vertex of a parabola expressed in the form \( y = (x-h)^2 + k \) is given by the point \( (h, k) \). For our function \( f(x) = (x+1)^2 + 4 \), the values are \( h = -1 \) and \( k = 4 \).

Step 3: Write down the vertex

Using the values \( h = -1 \) and \( k = 4 \) from the vertex form, we find the vertex to be \( (-1, 4) \). This is the point where the parabola either reaches its minimum or maximum (for a basic parabola, it is the minimum point if it opens upwards).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a special way of writing a parabola equations. Having the equation in vertex form makes identifying the vertex straightforward. This form is expressed as \( y = (x-h)^2 + k \), where \( h \) and \( k \) determine the vertex. This vertex represents the highest or lowest point on the parabola.From our given function, \( f(x) = (x+1)^2 + 4 \), we see it's already in vertex form. Here, \( h \) is \(-1\) and \( k \) is \(4\). So, the vertex of the parabola is at the point \((-1, 4)\). This simple transformation highlights how vertex form gives you instant access to the central characteristics of a parabola without needing complex calculations.

Parabola Graphing

Graphing a parabola when it is in vertex form \( y = (x-h)^2 + k \) is quite simple. The vertex \((h, k)\) serves as the main point on the graph. To plot:

• Mark the vertex on the coordinate plane.
• From the vertex, use symmetry. Parabolas are symmetrical around a vertical line through the vertex.
• Choose values to the right and left of the vertex and substitute them into the function to find corresponding \(y\) values.

These steps generate a clear U-shape for upward-opening parabolas, like in our example \( f(x) = (x+1)^2 + 4 \). Here, the parabola opens upward and the vertex is at \((-1, 4)\). Understanding this makes drawing and interpreting parabolas a much more visual and intuitive activity.

Quadratic Functions

Quadratic functions form the basis for parabolas and are represented as \( ax^2 + bx + c \) in their standard form. However, transformations into different forms like vertex form make certain tasks, like graphing or finding the vertex, easier.In the given exercise, the quadratic function is initially presented in vertex form \((x-h)^2 + k\). This showcases how the same quadratic function can be expressed in different forms, each useful in determining diverse characteristics of the graph, such as the vertex or interception points.These functions are crucial in many fields, representing real-world phenomena like projectile motion. Understanding how to manipulate and interpret these forms allows us to solve problems ranging from simple math exercises to complex physics models.