Problem 53 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Use technology to rewrite the function \(g(x)=\frac{(97.6)(0.024)+x(0.003)}{12.2+x}\) in the form \(f(x)=\frac{a}{x-h}+k\). Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\).

Step-by-Step Solution

Answer

The function \(g(x)=\frac{(97.6)(0.024)+x(0.003)}{12.2+x}\) can be rewritten in the form \(f(x)=\frac{2.3424}{x+12.2}+0.003\). The graph of \(g\) is a horizontal shift of the graph of \(f(x)=\frac{a}{x}\) to the left by \(12.2\) units and a vertical shift upward by \(0.003\) units.

Step 1: Simplify the Function

Start by simplifying the function \(g(x)\) using standard mathematical operations: \(g(x)=\frac{(97.6)(0.024)+x(0.003)}{12.2+x}=\frac{2.3424+x(0.003)}{12.2+x}\)

Step 2: Isolate the Variable

We will isolate the x term in the numerator by splitting the fraction: \(g(x) = \frac{2.3424}{12.2+x} + \frac{x(0.003)}{12.2+x}\)

Step 3: Redefine Constants

Typically, \(f(x) = \frac{a}{x-h} + k\) is in terms of \(x-h\) in the denominator. To match this form, perform the following operations: redefine \(a\) to equal 2.3424, \(b\) (the coefficient of x in the second term) to equal \(0.003\), and \(h\) to equal -12.2, giving: \(g(x) = \frac{2.3424}{x+12.2} + \frac{x(0.003)}{x+12.2}\)

Step 4: Factor to Obtain Final Form

Now, factor the common denominator to reveal the desired form. Simplifying gives: \(g(x) = \frac{a}{x-h} + b\), or \(f(x) = \frac{2.3424}{x+12.2} + 0.003\). This is equivalent to the form \(f(x) = \frac{a}{x - h} + k\) where \(k = b = 0.003, a = 2.3424 \), and \(h = -12.2\).

Step 5: Graphical Transformation Analysis

The versions of \(g(x)\) afterwards are a horizontal shift of \(f(x)\) to the left by \(h = 12.2\) units and a vertical shift upward by \(k = 0.003\) units, so the graph of \(g(x)\) can be described as the graph of \(f(x)=\frac{a}{x}\), shifted left by \(12.2\) units and shifted upwards by \(0.003\) units.

Key Concepts

Rational Functions

A rational function is a type of function that is expressed as the ratio of two polynomials. The general form of a rational function is \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not equal to zero.
Rational functions are defined for all values of \( x \) except where the denominator equals zero, as division by zero is undefined. These points result in vertical asymptotes on the graph of the function.
The function \( g(x) = \frac{(97.6)(0.024) + x(0.003)}{12.2 + x} \) is an example, where the numerator and the denominator are polynomials of first degree. When understanding rational functions, one often looks at transformations which include shifts and asymptotic behaviors.

Graph Transformations

Graph transformations involve changing the position or the shape of a function's graph by manipulating it in various ways. For rational functions, these transformations often include translations, reflections, stretching, or compression.
To consider the transformation of \( g(x) \) into \( f(x) = \frac{a}{x - h} + k \), you should identify how the original graph shifts and deforms compared to the standard form of rational functions. Transformations facilitate understanding how the function behaves across its domain and how changes affect its range.

• Vertical shifts happen when each point on the graph moves up or down.
• Horizontal shifts occur when points move left or right.
• Depending on the coefficients, the graph can be stretched or compressed vertically or horizontally.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the fundamental processes applied to manipulate functions. In the context of rewriting functions, these operations are used to simplify, rearrange, and solve equations.
In the given exercise, operations are applied to simplify and reshape \( g(x) \) into \( f(x) = \frac{a}{x - h} + k \).
By expanding and collecting terms, one can isolate expressions that conform to the desired function format. These operations reveal underlying connections between the new and transformed functions, showing relationships such as shifts and asymptotes. This represents both computational skill and insight into algebraic manipulation.

Horizontal and Vertical Shifts

Shifting a graph horizontally and vertically involves moving every point of the graph the same distance in these directions.
For a function of the form \( f(x) = \frac{a}{x - h} + k \), \( h \) controls the horizontal shift, while \( k \) controls the vertical shift.
In the exercise, \( g(x) \) is shifted horizontally to the left by 12.2 units and vertically upwards by 0.003 units. Here’s how it works:

• **Horizontal Shift**: The presence of \( (x - h) \) in the denominator signifies movement along the x-axis. It's important to note that \( x + h \) in the final form indicates a shift to the left by \( h \) units.
• **Vertical Shift**: The term \( + k \) results in moving upwards by \( k \) units. This is represented in the alteration from the base function \( \frac{a}{x} \) form.

This transformation results in no change in the overall shape of the graph but merely its position either along or perpendicular to the axes.