Problem 51 - A Graphical Approach to College Algebra

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x+2)^{2}+3$$

Step-by-Step Solution

Answer

The graph of the function is a parabola with vertex at \((-2,3)\), opening upwards.

Step 1: Identify the Basic Function

The given function is \(y=(x+2)^2+3\). The basic function here is \(y=x^2\), which is a parabola opening upwards with the vertex at the origin \((0,0)\).

Step 2: Determine the Horizontal Translation

In the function \(y=(x+2)^2+3\), the term \((x+2)^2\) represents a horizontal translation. Specifically, the graph of \(y=x^2\) is translated 2 units to the left because of \(x+2\).

Step 3: Determine the Vertical Translation

The term \(+3\) outside the squared term indicates a vertical translation of the graph. The graph is translated 3 units upwards.

Step 4: Sketch the Translated Graph

Starting with the basic parabola \(y=x^2\), first shift the entire graph 2 units to the left. This moves the vertex from \((0,0)\) to \((-2,0)\). Then, shift the graph 3 units upwards, placing the vertex at \((-2,3)\). Draw the parabola opening upwards, maintaining its shape.

Key Concepts

Quadratic Functions

Quadratic functions are a specific type of polynomial function. They are defined by the general formula \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Importantly, \( a \) must not be zero because it ensures that the function is quadratic and not linear. The simplest form of a quadratic function is \( y = x^2 \).
This basic quadratic function, \( y = x^2 \), forms a U-shaped graph known as a parabola. Quadratic functions have several key features, including a vertex, axis of symmetry, and they open either upwards or downwards depending on the sign of \( a \).
For instance, with the basic function \( y = x^2 \):

• The vertex is at the origin (0,0).
• The parabola is symmetrical about the y-axis.
• As \( a \) is positive, the parabola opens upwards.

Understanding quadratic functions and their graphs is integral to comprehending transformations and translations.

Graph Sketching

Graph sketching of quadratic functions involves accurately drawing the shape of the parabolic curve on a coordinate plane. To sketch the graph of the function \( y = (x+2)^2 + 3 \): begin by identifying the basic parabola from the function \( y = x^2 \).
Start with the basic aspects of the parabola:

• The U-shape of the parabola opens upwards as the base function \( y = x^2 \) does.
• The vertex is the starting reference point, initially at the origin for \( y = x^2 \).

Next, apply the transformations:

• First, adjust for the horizontal translation by moving the vertex 2 units to the left. The new vertex is at (-2,0).
• Then, adjust for the vertical translation by moving the vertex 3 units upwards. This establishes the vertex at (-2,3).

Finally, use the symmetry of the parabola to sketch it around the vertex (-2,3). Ensure that the directional shape and size of the opening remain consistent with the parent function \( y = x^2 \). This careful stepwise approach helps visualize transformations without computational tools.

Translation of Functions

Translations in functions involve shifting the graph horizontally and/or vertically on the coordinate plane. They do not alter the shape of the graph, only its position.
In the case of the function \( y = (x+2)^2 + 3 \):

• The horizontal translation is determined by the expression \((x+2)\). It means moving the graph 2 units to the left, opposite to the sign inside the bracket.
• The vertical translation is indicated by the constant \(+3\) outside the squared term. This translates the graph 3 units up.

Support your understanding with these key points:

• Horizontal shifts affect the x-values in the function. A positive change means shifting left.
• Vertical shifts adjust the y-values, with positive values moving the graph upwards.

Translation is an essential concept in altering the position of graphs and allows us to manually sketch functions by systematically adjusting their positions on the plane. Practice visualizing these shifts to develop a deeper understanding of graph transformations.