Problem 64 - 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 \(f\).$$g(x)=-3 \sqrt{x+1}-6$$.

Step-by-Step Solution

Answer

The parent function is \(f(x)=\sqrt{x}\). The function \(g\) is obtained after shifting the parent function one unit to left, stretching it vertically by a factor of 3, taking its reflection across the x-axis, and shifting it down by 6 units. The graph of g(x) starts from (-1,-6) and decreases steeply, and \(g(x)\) in terms of the parent function f can be formulated as \(g(x) = -3 f(x+1) - 6\).

Step 1: Identifying the Parent Function

The parent function \(f\) in this context is the simple square root function \(f(x)=\sqrt{x}\). This is because the overall form of \(g(x)\) fits the square root function shape, despite the additional transformations.

Step 2: Describing Transformation Sequence

Three transformations are applied to the parent function to produce \(g(x)\): 1. The function is horizontally translated one step to the left leading to \(f(x+1)\).2. It is then vertically stretched by a factor of 3 and reflected across the x-axis as indicated by the negative sign in front of 3 resulting in \(-3 f(x+1)\). 3. Lastly, it is vertically translated downward by 6 units, bringing us to the final form of the function \(-3 f(x+1) - 6\).

Step 3: Sketching the Graph

A graph must be plotted carefully to mirror the described transformations. Starting with the basic square root graph, first perform the horizontal shift to the left. The stretch, reflection and vertical shift can then be applied. The graph should start from the point (-1,-6) and decrease steeply due to the stretch factor of 3.

Step 4: Writing \(g\) in Terms of \(f\)

In function notation, taking the sequence of transformations into account, the function \(g\) in terms of the parent function \(f\) can be written as \(g(x)=-3 f(x+1) - 6\), where \(f(x)=\sqrt{x}\).

Key Concepts

Square Root Function

Understanding the square root function is critical for mastering various topics in algebra and calculus. It is represented mathematically as \( f(x) = \sqrt{x} \), with the simplest form starting at the origin (0,0) and extending to the right, passing through points like (1,1), (4,2), and (9,3), making a gradual, upward curve.

The domain of this function, which is the set of all possible x-values, only includes non-negative numbers, as the square root of a negative number is not a real number. Any transformations applied to this parent function are essential in understanding more complicated functions and are pivotal in graphing equations efficiently.

When you see a function like \( g(x) = -3 \sqrt{x+1} - 6 \), identifying that its 'parent' is the square root function is the first step to breaking down its behavior and how it’s graphed.

Graph Transformations

Graph transformations modify the parent function in various ways to create a new function. The transformations can include translations (shifts), reflections, stretches, and compressions. Each transformation changes the graph's shape, position, or orientation in a specific way.

In our example \( g(x) = -3 \sqrt{x+1} - 6 \), several transformations are applied to the parent square root function:

• A horizontal translation to the left by 1 unit, moving every point on the graph one unit to the left.
• A vertical stretch by a factor of 3, which multiplies the y-values of the graph by 3, making it steeper.
• A reflection across the x-axis, achieved by the negative sign, flips the function upside down.
• Finally, a vertical translation downward by 6 units, shifting the entire graph down on the y-axis.

Understanding these transformations allows you to describe and predict the outcome of complex function manipulations accurately.

Function Notation

Function notation is a concise way to define functions and operations applied to them, typically written as \( f(x) \), where 'f' denotes the function, and 'x' represents the input variable, or independent variable. It's a powerful tool for denoting transformations and algebraic manipulations in a structured manner.

For instance, given our parent function \( f(x) = \sqrt{x} \), we can describe the result of applying multiple transformations to obtain \( g(x) \) through function notation. In the final expression \( g(x) = -3 f(x+1) - 6 \), each part correlates to a specific transformation that has been applied to the parent function, providing clarity to the equation's meaning and easing the processes of both solving and graphing the function.