Problem 69 - Calculus Single Variable

Question

In each of Exercises \(69-72,\) find a function \(f\) whose graph is the given curve \(\mathcal{C}\). \(\mathcal{C}\) is obtained by translating the graph of \(y=x^{2}\) to the right by 3 units.

Step-by-Step Solution

Answer

The function is \(f(x) = (x - 3)^2\).

Step 1: Understand the Original Function

The original function is given as \( y = x^2 \). This is a standard parabola opening upwards with its vertex at the origin (0,0).

Step 2: Determine the Effect of Translation

A translation of a function to the right by \(a\) units is done by replacing \(x\) with \(x - a\) in the function. In this case, since the curve \(\mathcal{C}\) is obtained by translating the graph of \(y = x^2\) to the right by 3 units, we replace \(x\) with \(x-3\).

Step 3: Apply the Translation

Substitute \(x - 3\) into the original function \(y = x^2\):\[ y = (x - 3)^2 \]This represents the function \(f\) whose graph is the curve \(\mathcal{C}\), after translating the original parabola 3 units to the right.

Key Concepts

Understanding the Parabola

A parabola is a U-shaped curve that you encounter quite often in mathematics, especially in quadratic functions. It has distinct characteristics:

• Symmetrical: It mirrors itself on either side of its vertical axis.
• Vertex: Its highest or lowest point, depending on the orientation.
• Direction: Opens either upwards or downwards.

In our context, the standard parabola is represented by the equation \( y = x^2 \), which opens upwards. This type of parabola has its vertex at the origin, which means the lowest point of this U-shape is at \((0, 0)\).
This basic form is central to understanding transformations like translations or shifts in position. Each transformation alters how we perceive the graph, but not its general U-shape.

Exploring Quadratic Functions

Quadratic functions play a crucial role in mathematics, and they are expressed in the form \( y = ax^2 + bx + c \). Here's a simplified breakdown:

• The term \( ax^2 \) denotes that the highest degree of \( x \) is 2, indicating a parabolic graph.
• If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• The coefficients \( b \) and \( c \) determine the parabola's position and shape.

In our exercise, the starting function is as simple as it gets with \( y = x^2 \), which means \( a = 1 \), and both \( b \) and \( c \) are zero. Thus, the graph is a basic parabola with no horizontal or vertical shifts involved initially.
As transformations occur, these coefficients change, altering the graph's position and the value of its vertex.

Understanding Graph Transformation

Graph transformations involve movements or changes to the graph of a function, impacting its shape, position, or orientation. For quadratic functions like parabolas, the most common transformations are translations, reflections, and stretches/shrinks.
Translation is particularly straightforward:

• Horizontal Translation: Shifts the graph left or right. This is achieved by replacing \( x \) with \( x-h \). A positive \( h \) shifts right, negative shifts left.
• Vertical Translation: Shifts the graph up or down by replacing \( y \) with \( y-k \).

In the given exercise, a horizontal translation is performed by 3 units to the right. You achieve this by replacing each \( x \) in the function with \( x - 3 \).
Consequently, \( y = x^2 \) becomes \( y = (x - 3)^2 \). This tells the whole story: the parabola keeps its shape but moves horizontally to a new central position at \((3, 0)\), the new vertex.