Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ F(t)=-|t+3| $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand the definitions of even and odd functions

A function \( f(t) \) is even if \( f(-t) = f(t) \) for all \( t \) in its domain. It is odd if \( f(-t) = -f(t) \). If neither condition holds, the function is neither even nor odd.

Step 2: Evaluate \( F(-t) \)

Compute \( F(-t) \) by substituting \( -t \) into the function. So \( F(-t) = -|-t + 3| = -|-(t-3)| = -|t-3| \).

Step 3: Compare \( F(-t) \) with \( F(t) \)

Now compare the expressions: \( F(t) = -|t+3| \) and \( F(-t) = -|t-3| \). Since \( F(t) \) and \( F(-t) \) are not equal and \( F(-t) \) is not equal to \(-F(t)\), the function is neither even nor odd.

Step 4: Sketch the graph of \( F(t) = -|t+3| \)

To sketch \( F(t) = -|t+3| \), note that it is a reflection of the graph of \( |t+3| \) across the x-axis. The vertex of \( |t+3| \) is at \((-3, 0)\), so the graph of \( F(t) \) will have a vertex at \((-3, 0)\) as well, with the arms of the 'V' opening downwards, creating an inverted V-shape with its lowest point at \((-3, 0)\).

Key Concepts

Function Symmetry

In mathematics, understanding the symmetry of functions is crucial. Symmetry allows us to predict and understand the behavior of a function without having to compute every value. There are two main types of function symmetry: even and odd.

• Even Functions: A function is even if the equation \( f(-t) = f(t) \) holds true for all \( t \) in its domain. Graphically, this means that the function is symmetric with respect to the y-axis. For instance, if a point \((a, b) \) is on the graph, then \((-a, b) \) will also be on the graph.
• Odd Functions: A function is considered odd if \( f(-t) = -f(t) \). This means the graph is symmetric about the origin. If \((a, b) \) is on the graph of an odd function, then \((-a, -b) \) will also appear on the graph.

When neither symmetry applies, the function is classified as neither even nor odd. For the function \( F(t) = -|t+3| \), we can determine that it is neither. By checking \( F(-t) \), we find \( F(-t) = -|t-3| \), which is neither equal to \( F(t) \) nor to \(-F(t) \). Understanding these properties can greatly ease the process of graph sketching and function analysis.

Graph Sketching

Sketching the graph of a function allows us to visualize its behavior across the domain. Let's break down sketching the graph of the function \( F(t) = -|t+3| \) step by step.
Given the absolute value function \( |t+3| \), its graph forms a 'V' shape with a vertex where the function switches from increasing to decreasing or vice versa. The vertex for this part of the expression \( |t+3| \) is at \((-3, 0) \). This means for \( F(t) = -|t+3| \), the vertex is also at \((-3, 0) \) but reflected across the x-axis. This reflection changes the direction of the 'arms' of the graph.

• The arms of the 'V' now open downwards rather than upwards, forming an inverted 'V' shape.
• The vertex \((-3, 0) \) is the lowest point of this graph because of the negative sign in front of the absolute value, which inverts it.

Thus, by identifying the transformations applied to the parent absolute value function, you can effectively create a sketch of the function \( F(t) = -|t+3| \). This graph will showcase a downward opening 'V', helping us understand the function's rate of change and intercepts.

Absolute Value Functions

Absolute value functions are characterized by their peculiar shape and transformations, which make them quite intriguing to study. They're typically in the form \( |t+b| \), producing a 'V' shape in their graphs.
The critical part of understanding absolute value functions is recognizing how particular transformations alter their graphs:

• The expression \( |t+3| \) shifts the basic absolute value function \( |t| \) 3 units to the left. This shift is due to the 'positive 3' being added inside the absolute value.
• When a negative sign is applied outside, as in \( -|t+3| \), the graph is reflected across the x-axis. This reflection changes the 'V' from pointing upwards to pointing downwards.
• The vertex or the turning point of \( |t+3| \) is at \((-3,0)\), because that's where \( t+3 = 0 \), meaning it can equally impact both sides of the origin.

Understanding these shifts and reflections allows us to predict the behavior of absolute value graphs, making it easier to sketch and analyze functions like \( F(t) = -|t+3| \), and see their impact on the overall shape.