Problem 54 - Precalculus with Limits

Question

In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g\). (c) Sketch the graph of \(g\). (d) Use function notation to write \(g\) in terms of \(f\). \(g (x) = \sqrt{3x} + 1\)

Step-by-Step Solution

Answer

The parent function is \(f(x) = \sqrt{x}\). The transformations from \(f\) to \(g\) are a horizontal compression by a factor of 1/3 and a vertical shift upwards by 1 unit. The function \(g\) in terms of \(f\) is \(g(x) = f(3x) + 1\).

Step 1: Identify the parent function

The given function is \(g (x) = \sqrt{3x} + 1\). The parent function here is \(f(x) = \sqrt{x}\). This is because the structure of the given function follows the format of the square root function with a few alterations.

Step 2: Describe the transformations

Comparing the given function \(g (x) = \sqrt{3x} + 1\) with the parent function \(f(x) = \sqrt{x}\), it can be seen that two transformations have occurred. The presence of '3x' inside the square root indicates a horizontal compression by a factor of 1/3. The '+1' outside the square root indicates a vertical shift upwards by 1 unit.

Step 3: Sketch the graph

To sketch the graph of \(g(x) = \sqrt{3x} + 1\), start by plotting the parent function \(f(x) = \sqrt{x}\). Then, apply the horizontal compression by a factor of 1/3, which squeezes the graph towards the y-axis. After that, shift the graph upwards by 1 unit. The resulting graph represents \(g(x) = \sqrt{3x} + 1\).

Step 4: Write g in terms of f

Writing the given function in terms of the parent function, we get \(g(x) = f(3x) + 1\). This expression aligns with the transformations that have been identified before: '3x' for the horizontal compression and '+1' for the upwards shift.

Key Concepts

Square Root Function

The square root function, typically written as \(f(x) = \sqrt{x}\), is a fundamental parent function in algebra. It represents the principal square root of a non-negative variable \(x\). This function only takes non-negative inputs, as square roots of negative numbers are not real numbers within the scope of this particular function.

Graphically, the square root function forms a curve that starts from the origin (0,0) and gradually increases, slowing in growth as it moves to the right. It's important to note that this graph only exists in the first quadrant of a Cartesian coordinate system as it only handles positive square roots. The shape of the graph is a sideways parabola, exhibiting a slower rate of increase as the value of \(x\) gets larger.

Understanding the behavior of the square root parent function is critical because many transformations will be applied to this basic shape to achieve a variety of other functions, like the one seen in the exercise.

Horizontal Compression

A horizontal compression of a graph is a type of transformation that squeezes the graph of a function towards the y-axis. This type of transformation changes the width of the graph horizontally without affecting the height.

In the function notation \(g(x) = f(cx)\), where \(c > 1\), the graph of \(f(x)\) is compressed horizontally by a factor of \(1/c\). This is counterintuitive for some students because the factor inside the function is more than 1, yet it compresses the graph. Remembering that the compression is inverse to the factor is key - the larger the value of \(c\), the more compressed the graph will be. Paying attention to this detail can make understanding and applying graph transformations much easier.

Graph Transformations

Graph transformations involve changing the position and shape of the graph of a function. These transformations can be composed of translations (shifts), reflections, stretches, and compressions. Each transformation affects the graph in a specific way and can be applied in a sequence to move from the parent graph to a more complex one.

To correctly apply a sequence of transformations, one should follow the proper order:

• Apply horizontal transformations first (stretches, compressions, shifts).
• Follow with reflections across the axes if any.
• Finish with vertical transformations (stretches, compressions, shifts).

By following these steps, confusing mistakes in the order of operations can be avoided, and it becomes easier to visualize the effects of multiple transformations on the graph.

Function Notation

Function notation is a way to name and describe the relationship between variables in a concise mathematical form. It's written as \(f(x)\), which can be understood as 'the function \(f\) evaluated at the input \(x\)'. This notation provides a powerful and standardized method to communicate and work with functions.

When we reference transformations within function notation, such as \(g(x) = f(3x) + 1\), we're specifying the exact modifications made to the parent function to obtain the new function \(g\). In this notation, \(3x\) suggests a horizontal transformation, specifically a compression, and the \(+1\) indicates a vertical shift. Grasping this notation facilitates a deeper understanding of how different functions are related to each other and how to efficiently work with transformations.