Problem 29 - 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 \(f\). \(g (x) = \frac{2}{3}x^2 + 4\)

Step-by-Step Solution

Answer

The parent function is \(f(x) = x^2\). The sequence of transformations from \(f(x) = x^2\) to \(g(x) = \frac{2}{3}x^2 + 4\) includes a vertical compression by a factor of \(\frac{2}{3}\) and a vertical shift upward by \(4\) units. The function \(g(x)\) can be expressed in terms of \(f(x)\) as \(g(x) = \frac{2}{3}f(x) + 4.\)

Step 1: Identify the parent function

First, look at the given function and compare it with common parent functions. The given function \(g (x) = \frac{2}{3}x^2 + 4\) shares the \(x^2\) term, which is similar with the parent quadratic function \(f(x) = x^2\). Thus, the parent function f is \(f(x) = x^2\).

Step 2: Describe sequence of transformations

Next, examine each component of the given function \(g\). The coefficient of \(x^2\) is \(\frac{2}{3}\), which suggests vertical compression by a factor of \(\frac{2}{3}\). In addition, the constant \(4\) indicates vertical shift up by four units. Hence, the sequence of transformations from \(f\) to \(g\) involves vertical compression by a factor of \(\frac{2}{3}\) followed by vertical shift up by four units.

Step 3: Write g in terms of f

The function \(g\) can be rewritten in terms of \(f\) by replacing \(x^2\) in \(f(x)\) with the transformations applied. In this case, \(x^2\) is scaled by \(\frac{2}{3}\) and then shifted up by \(4\). Hence, \(g(x)\) can be rewritten as \(g(x) = \frac{2}{3}f(x) + 4.\)

Step 4: Sketch the graph of g

For creating the sketch of \(g(x)\), Start with the parent graph of \(f(x) = x^2\). Then, compress it vertically by a factor of \(\frac{2}{3}\) and shift it upwards by \(4\) units. The sketch of the graph should resemble the parabolic shape of \(f(x) = x^2\), but it will be narrower due to the compression and located higher due to the upward shift.

Key Concepts

Quadratic Function

A quadratic function is fundamental in mathematics. It is represented by the formula \( f(x) = ax^2 + bx + c \). The simplest form of a quadratic function, known as the parent function, is \( f(x) = x^2 \). This form is a standard parabola that opens upwards and has its vertex at the origin (0,0).
Quadratic functions are characterized by their "U"-shaped graphs, called parabolas. The direction of the parabola (upwards or downwards) depends on the sign of the coefficient \( a \) in the formula:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In the exercise, the function \( g(x) = \frac{2}{3}x^2 + 4 \) is derived from the parent quadratic function \( f(x) = x^2 \). Here, the coefficient of \( x^2 \) is positive, so the graph of \( g(x) \) will open upwards, just like \( f(x) \). Understanding this basic structure is crucial before exploring transformations applied to the quadratic function.

Function Transformations

Function transformations allow us to manipulate and adjust graphs to suit different mathematical needs and to explore the behavior of functions. These transformations include shifts, stretches, compressions, and reflections.
The function \( g(x) = \frac{2}{3}x^2 + 4 \) undergoes two main transformations from its parent function \( f(x) = x^2 \):

• Vertical Compression: The factor \( \frac{2}{3} \) applied to \( x^2 \) results in a vertical compression of the graph. This means that each point on the graph is squeezed closer to the x-axis by a factor of \( \frac{2}{3} \).
• Vertical Shift: The constant \( +4 \) indicates that the entire graph shifts vertically upwards by 4 units. This moves the vertex of the parabola from the origin to (0,4).

Transformations significantly modify the appearance and position of the original graph while maintaining the general shape of the parent function. They are vital in modeling real-world situations where adjustments are necessary to fit specific conditions.

Graphing Functions

Graphing functions, particularly quadratic ones, involves plotting points based on the transformed equation and recognizing how these adjustments alter the graph.
To graph \( g(x) = \frac{2}{3}x^2 + 4 \), start by drawing the parent function \( f(x) = x^2 \) as a reference. This graph is a symmetrical parabola centered at the origin.
Next, apply the transformations:

• Vertical Compression: Multiply each \( y \)-value of \( f(x) \) by \( \frac{2}{3} \). This change narrows the graph – it becomes less steep than the parent function.
• Vertical Shift: Add 4 to each \( y \)-value to move the entire graph upwards.

The result is a narrower graph that sits above the original, with its vertex at (0,4). Graphing in this step-by-step manner helps in understanding the impact of each transformation and visualizing the resulting function accurately. Keep in mind that the essential shape and symmetry of a parabola are preserved throughout these transformations.