Problem 12 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ g(x)=|x| \text { for }-2 \leq x \leq 3 $$

Step-by-Step Solution

Answer

The graph is a V-shape from (-2, 2) through (0, 0) to (3, 3).

Step 1: Understand Absolute Value Function

The absolute value function \( g(x) = |x| \) results in a V-shaped graph. For any real number \( x \), \( |x| \) is non-negative and \( g(x) = |x| = x \) if \( x \geq 0 \), and \( g(x) = -x \) if \( x < 0 \). This means for positive \( x \), the graph is a line with slope 1, and for negative \( x \), it is a line with slope -1.

Step 2: Define the Domain

We need to sketch the graph for \( -2 \leq x \leq 3 \). This means we only consider the absolute value function within this interval. Our graph will range from \( x = -2 \) to \( x = 3 \). This helps in limiting the portion of the graph we focus on.

Step 3: Calculate Key Points

Identify key values of \( x \) within the specified domain and find the corresponding \( g(x) \) values.- When \( x = -2 \), \( g(-2) = |-2| = 2 \)- When \( x = 0 \), \( g(0) = |0| = 0 \)- When \( x = 3 \), \( g(3) = |3| = 3 \)These points allow us to sketch the structure of the graph accurately.

Step 4: Sketch the Graph

Draw the graph using the key points from Step 3. - Start at point \((-2, 2)\).- Draw a straight line to point \((0, 0)\) which forms the left side of the V.- From \((0, 0)\), draw another straight line to \((3, 3)\) representing the right side of the V.- These lines should be solid to indicate they are part of the graph within the given domain.

Key Concepts

Graph Sketching

Graph sketching transforms mathematical functions into visual representations, making them easier to interpret. For the absolute value function like \( g(x) = |x| \), the graph takes a distinct V shape. The peak of the V occurs at the origin, where \( x = 0 \), because the absolute value of zero is zero. As part of sketching, note that one side of the V graph represents positive \( x \) values, and the other side represents negative \( x \) values.
Understanding the shape of the graph allows you to predict the behavior of the function rapidly.

• The left side of the V is a line with a negative slope.
• The right side of the V is a line with a positive slope.

This duality aligns perfectly with the conditions of the function \( |x| \), combining them into the recognizable V-shaped graph. Each section sketches out the gradient change and how it mirrors across the y-axis once it passes the origin. In our specific exercise, the graph extends only over the domain \( -2 \leq x \leq 3 \), reflecting that you don't always need the entire function range to understand its behavior.

Piecewise Functions

Absolute value functions are excellent examples of piecewise functions. They split into separate linear functions, each defined over distinct intervals. For \( g(x) = |x| \), the function transforms as follows:

• When \( x \geq 0 \), \( g(x) = x \), representing the positive linear portion of the function.
• When \( x < 0 \), \( g(x) = -x \), which forms the negative-sloped line.

Piecewise functions like \( |x| \) elegantly manage such shifts via conditionality. In one interval, you employ one formula, and in another, a different one. This creates a stepladder of formulas that depict every portion of the V-shaped graph. In essence, mastery of piecewise interpretation helps you grapple with more complex functions efficiently. It prepares you for understanding further advanced functions which also break into segments or sections.

Key Points Identification

Identifying and plotting key points is crucial when sketching graphs, as these points outline the important features of a function. For absolute value functions like \( g(x) = |x| \), key points guide the drawing of the V shape accurately. Start with notable points within the domain \( -2 \leq x \leq 3 \):

• At \( x = -2 \), \( g(x) = 2 \).
• At \( x = 0 \), \( g(x) = 0 \).
• At \( x = 3 \), \( g(x) = 3 \).

These calculations highlight the vertex and endpoints that define the 'legs' of the V. Placing these points accurately on a graph gives clarity in maintaining the true shape and slope characteristics of the absolute value function. Such plotting serves as the framework for visualizing even the most complex piecewise variations.