Problem 45 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=3(x-4)^{2}+1 $$

Step-by-Step Solution

Answer

The vertex of the function is \((4, 1)\).

Step 1: Identify the Vertex Form of the Quadratic Function

The given function is already in the vertex form, which is \( f(x) = a(x-h)^2 + k \). Here, \( a = 3 \), \( h = 4 \), and \( k = 1 \).

Step 2: Determine the Vertex

In the vertex form \( f(x) = a(x-h)^2 + k \), the vertex of the parabola is \((h, k)\). Substitute \( h = 4 \) and \( k = 1 \) into this formula to find the vertex coordinates.

Key Concepts

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is one of the most useful ways to express a quadratic function. It helps us easily find the vertex of the parabola, which is a key feature of the graph. The general form of the vertex form is \( f(x) = a(x-h)^2 + k \). Here, \( a \), \( h \), and \( k \) are constants that transform the parabola:

• \( a \) determines the width and direction of the parabola. A positive \( a \) opens upwards, while a negative \( a \) opens downwards.
• \( h \) is the x-coordinate of the vertex and shifts the parabola left or right.
• \( k \) is the y-coordinate of the vertex and shifts the parabola up or down.

Using the vertex form, identifying the vertex of the quadratic function becomes straightforward. In the example \( f(x) = 3(x-4)^2 + 1 \), we recognize immediately that the vertex is at \((4, 1)\). Breaking down mathematical problems in this way can simplify identifying critical points on a graph like the vertex.

Parabola

A parabola is the U-shaped curve that represents the graph of a quadratic function. It is symmetrical and can open upwards or downwards depending on the sign of the coefficient \( a \) in the quadratic equation. Key features of a parabola include:

• **Vertex:** The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. It serves as the center of symmetry.
• **Axis of Symmetry:** A vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For a function \( f(x) = a(x-h)^2 + k \), the axis of symmetry is given by \( x = h \).
• **Direction:** Determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.

Understanding the shape and properties of a parabola allows us to make predictions about the behavior of quadratic functions. In any quadratic function expressed in vertex form, identifying the vertex provides major insight into where the parabola is positioned on the coordinate plane.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, often written in the standard form \( ax^2 + bx + c \) or the vertex form \( a(x-h)^2 + k \). These functions are fundamental in algebra and appear in various real-world applications. Characteristics include:

• **Graph Shape:** The graph of a quadratic function is always a parabola.
• **Solutions:** Depending on the discriminant \( b^2 - 4ac \), a quadratic equation can have 0, 1, or 2 real roots. These are the x-intercepts of the graph.
• **Vertex and Axis of Symmetry:** Clearly found using the vertex form, especially useful in graphing and understanding the functions.

In our exercise, understanding and using the vertex form directly reveals the vertex, helping us quickly graph or predict the quadratic function's behavior without additional calculations. Quadratic functions are essential mathematical tools, forming the basis for a wide range of analyses in physics, engineering, economics, and beyond.