Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describing Transformations \(g\) is related to one of the six parent functions on page 122 (a) Identify the parent function \(f .\) (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g\) by hand. (d) Use function notation to write \(g\) in terms of the parent function \(f\).$$g(x)=\frac{1}{x+8}-9$$.

Step-by-Step Solution

Answer

For \(g(x)=\frac{1}{x+8}-9\), the parent function is \(f(x) = \frac{1}{x}\). The function \(g(x)\) is transformed from the parent function by a sequence of shifting 8 units to the left and 9 units down. Hence, \(g(x)\) = \(f(x + 8) - 9\) where \(g(x)\) is the transformation of \(f(x)\).

Step 1: Identify the parent function

For \(g(x)=\frac{1}{x+8}-9\), the parent function is \(f(x) = \frac{1}{x}\). This is because \(g(x)\) is a transformation of this parent function.

Step 2: Describe the sequence of transformations

The given function \(g(x)\) can be obtained from the parent function \(f(x)\) through a series of transformations which include a horizontal shift of 8 units to the left and a vertical shift of 9 units down. This is derived from \(g(x)=f(x+8)-9\).

Step 3: Sketch the graph of \(g\)

To sketch the graph, we begin with the graph of the parent function \(f(x) = \frac{1}{x}\). This graph is a hyperbola with two asymptotes at \(x=0\) and \(y=0\). Next, we shift affected graph 8 units to the left and 9 units down and draw the new graph.

Step 4: Write \(g\) in terms of \(f\)

The original function \(f(x)\) is subject to transformations to get \(g(x)\). Hence, \(g\) can be written in terms of \(f\) as \(g(x) = f(x + 8) - 9\).

Key Concepts

Parent Functions

A parent function is the simplest form of a family of functions that share characteristics. Think of it as the 'blueprint' from which other more complex functions are derived through various transformations, such as stretching, shrinking, reflecting, and translating.

For instance, the parent function of a linear family is represented by the equation \( y = x \), which is a straight line passing through the origin. Other common parent functions include the quadratic function \( y = x^2 \), the absolute value function \( y = |x| \), and the reciprocal function \( y = \frac{1}{x} \), which is also the parent function addressed in the exercise.

Graph of a Function

The graph of a function is a visual representation of all the points that satisfy the equation of the function. Here, each point on the graph corresponds to an input-output pair. These points taken together form the shape of the function on a coordinate plane.

Understanding the graph helps to visualize the function's behavior, for example, identifying intervals of increase or decrease, the symmetry, and any asymptotic behavior. Properly graphing a function also involves indicating asymptotes, intercepts, and any significant points that characterize the function's shape.

Transformation of Functions

Changing a parent function in specific ways results in transformations of that function. There are four main types of transformations that affect the graph of a function:

• Translations - Shifts the graph horizontally or vertically
• Reflections - Flips the graph over a certain axis
• Stretches and Compressions - Changes the graph's steepness or shallowness
• Rotations - Pivots the graph around a certain point, although less common in Cartesian coordinates

Transformations can either be applied separately or combined to achieve the desired effect. In the context of the given exercise with the function \( g(x) \), the graph of the parent function \( f(x) = \frac{1}{x} \) is shifted 8 units to the left and 9 units down.

Hyperbola Graph

The graph of the reciprocal parent function \( f(x) = \frac{1}{x} \) is a hyperbola. A hyperbola consists of two separate curves, known as branches, that mirror each other across the graph's asymptotes. In this case, the asymptotes are the x-axis and the y-axis.

For the reciprocal function, as the x-values approach zero from either the positive or negative direction, the y-values increase or decrease without bound, moving farther away from the x-axis. Conversely, as the x-values move away from zero, the y-values approach zero, resulting in the distinct 'V' shapes of the hyperbola. When graphing transformations of a hyperbola, it's important to apply those shifts to the asymptotes as well to accurately depict the function's new placement on the coordinate plane.