Problem 47 - Elementary and Intermediate Algebra

Question

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function \(g\) using a translation and give its domain and range. See Examples 5 and \(6 .\) $$ g(x)=(x+4)^{2} $$

Step-by-Step Solution

Answer

The graph of \( g(x) = (x+4)^2 \) is a parabola 4 units left of \( f(x) = x^2 \). Domain: \((-\infty, \infty)\), Range: \([0, \infty)\).

Step 1: Understand the Parent Function

The function given for translation is \( g(x) = (x+4)^2 \). The parent function is \( f(x) = x^2 \), which is a basic parabola open upwards and symmetric about the y-axis.

Step 2: Identify the Transformation

The function \( g(x) = (x+4)^2 \) is a horizontal translation of the parent function \( f(x) = x^2 \) by 4 units to the left. This is because adding 4 inside the function with the variable shifts the graph left by 4 units.

Step 3: Sketch the Graph

Begin by sketching the graph of the parent function \( f(x) = x^2 \), which is a parabola centered at the origin. Then translate each point of \( f(x) \) 4 units left to get the graph of \( g(x) = (x+4)^2 \). The vertex now is at (-4, 0) instead of (0, 0).

Step 4: Determine the Domain

The domain of a function is the set of all possible input values. For \( g(x) = (x+4)^2 \), like \( f(x) = x^2 \), there is no restriction on \( x \). Thus, the domain of \( g(x) \) is all real numbers, \( (-\infty, \infty) \).

Step 5: Determine the Range

The range of a function is the set of all possible output values. The smallest value that \( g(x) = (x+4)^2 \) can take is 0 (at \( x = -4 \)), and it increases without bound as \( x \) moves away from -4 in either direction. Thus, the range is \( [0, \infty) \).

Key Concepts

Quadratic Functions

Quadratic functions are one of the most fundamental types of mathematical functions you'll encounter. They're characterized by the polynomial expression of the form \( f(x) = ax^2 + bx + c \). A special case is when the function is simply \( f(x) = x^2 \), which gives us a basic parabola. This parabola has some key features:

• It opens upwards if the coefficient \( a \) is positive, and downwards if \( a \) is negative.
• The vertex is the point where the parabola turns; for \( f(x) = x^2 \), the vertex is positioned at the origin (0, 0).
• The axis of symmetry is a vertical line that goes through the vertex, here it is the line \( x = 0 \).
• The graph is symmetrical about this axis, meaning the left and right sides of the parabola are mirror images.

Understanding these features helps tremendously when graphing the function or applying transformations.

Domain and Range

When discussing functions, particularly quadratic functions like \( g(x) = (x+4)^2 \), it's crucial to understand the concepts of domain and range.

The **domain** refers to all possible input values (\( x \)-values) that the function can accept. For most quadratic functions, including \( f(x) = x^2 \) and its siblings like \( g(x) = (x+4)^2 \), the domain is all real numbers. There's no restriction, so we express this as \( (-\infty, \infty) \).

On the other hand, the **range** involves the output values (\( y \)-values) that the function can produce. For a quadratic function that opens upwards like \( g(x) = (x+4)^2 \), the smallest value y can take is at the vertex. As the vertex of \( g(x) \) is at \(-4, 0\), the smallest output value is 0. This means the range of the function is \( [0, \infty) \).

These concepts are vital for understanding where the graph of the function lies on the coordinate plane.

Graph Translation

Graph translation involves shifting a graph horizontally or vertically without altering its shape or orientation. When you have a quadratic function like \( g(x) = (x+4)^2 \), it involves a horizontal translation of its parent function \( f(x) = x^2 \).

This particular translation shifts the graph 4 units to the left.

• The \(+4\) inside the parentheses indicates a **leftward shift**. It's a bit contrary to what you'd initially think since \(+\) usually suggests moving to the right.
• For instance, the vertex, originally at (0,0) for \( f(x) = x^2 \), moves to \(-4,0\) for \( g(x) = (x+4)^2 \).
• This movement affects the position of every point on the graph but does not change its orientation or symmetry.

Understanding graph translation is crucial because it shows how transformations can alter the appearance of a function's graph without changing its general properties.