Problem 32 - 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 \(f\). \(g (x) = -(x+10)^2 + 5\)

Step-by-Step Solution

Answer

The quadratic function \(f(x) = x^2\) is the parent function with transformations: horizontally shifted to the left by 10 units, vertically reflected and shifted upwards by 5 units. Plotting \(g(x)\) starts from vertex (-10,5) with the parabola opening downwards. The function \(g(x)\) in terms of \(f(x)\) can be expressed as \(g(x) = -f(x+10) + 5\).

Step 1: Identify the Parent Function \(f\)

The given function \(g (x) = -(x+10)^2 + 5\) resembles the general form of a quadratic function \(f(x) = x^2\). Hence, \(f(x) = x^2\) is the parent function.

Step 2: Describe the Transformations

In the given function, the value inside the parenthesis \(x+10\), causes a horizontal shift of 10 units to the left. The negative sign outside the square function causes a vertical reflection (upside down). Adding 5 to the function shifts the graph 5 units up vertically. Therefore, the transformations can be defined as: a horizontal shift to the left by 10 units, a vertical reflection, and a vertical shift upwards by 5 units.

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

Plot the vertex point at (-10,5), this is the point of the minimum value of the function. As the parabola is vertically reflected, it opens downwards. Sketch the parabola using these details with axis of symmetry at \(x=-10\). For a more accurate graph, additional points could be plotted by substituting different values for \(x\) into \(g(x)\).

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

To express \(g\) in terms of \(f\), isolate the parent function \(f(x) = x^2\) in the equation for \(g\). Therefore, \(g(x) = -f(x+10) + 5\).

Key Concepts

Parent Function

The term 'parent function' refers to the simplest form of a function family, which serves as the template for more complex functions related by transformations. For example, the parent function for the family of quadratic functions is \( f(x) = x^2 \). It has a basic 'U' shaped graph called a parabola, which opens upwards and has its vertex at the origin (0,0). The importance of recognizing the parent function lies in predicting the general shape and the orientation of the graph of related functions, like \( g(x) \).

Understanding the parent function helps students to identify the necessary transformations that will lead to the function \( g(x) \) under consideration. For instance, in our exercise, \( g(x) = -(x+10)^2 + 5 \) can be seen as a transformation of the parent function \( f(x) = x^2 \).

Graph of a Function

The 'graph of a function' visually represents the set of all possible points \( (x, y) \) where \( y \) corresponds to \( f(x) \) in a two-dimensional coordinate system. The graph of a parent function like \( f(x) = x^2 \) is a basic parabola. When graphing a function like \( g(x) \) resulting from transformations, you'll notice shifts and changes in the shape or orientation compared to its parent.

These transformations make graphing an essential tool for understanding the behavior of functions. When sketching \( g(x) \), it's crucial to accurately plot key points, such as the vertex and intercepts, and to consider how transformations such as reflections and shifts will alter these points from the parent function's graph.

Function Notation

Function notation uses the symbol \( f(x) \) to denote a function that maps any value \( x \) to its corresponding output value. The notation emphasizes the relationship between \( x \) and \( f \) without expressing the form of the function. When a function, such as \( g(x) \) in our exercise, is expressed in terms of another function, like \( f(x) = x^2 \) for the quadratic parent function, we indicate the connection between these functions using composition.

In the exercise, we saw this expressed as \( g(x) = -f(x+10) + 5 \), indicating that \( g(x) \) consists of the parent function \( f(x) \) with subsequent transformations applied. This notation is succinct and reveals the underlying structure of \( g(x) \) in relation to \( f(x)\).

Horizontal and Vertical Shifts

Shifts are a type of transformation that move the graph of a function without changing its shape. 'Horizontal shifts' occur when a function is adjusted along the x-axis. For example, a function \( f(x) \) that becomes \( f(x - h) \) is shifted \( h \) units to the right, while \( f(x + h) \) is shifted \( h \) units to the left.

In contrast, 'vertical shifts' occur when a constant is added or subtracted from the function, affecting its y-axis alignment. Adding \( k \) to \( f(x) \) results in an upward shift by \( k \) units to get \( f(x) + k \) and subtracting \( k \) moves it downward to form \( f(x) - k \). The exercise involved both types of shifts: the function \( g(x)\) is the parent function shifted 10 units to the left (horizontal) and 5 units up (vertical).