Problem 21 - Precalculus with Limits

Question

In Exercises 19-22, graphically estimate the \( x \)- and \( y \)-intercepts of the graph. Verify your results algebraically. \( y = |x + 2| \)

Step-by-Step Solution

Answer

The x-intercept of the function \(y = |x + 2|\) is -2 and the y-intercept is 2. These results were verified both graphically and algebraically.

Step 1: Graphical representation

Draw the graph of the function \(y = |x + 2|\). The graph displays a V-shaped curve. The pivot point of the curve (also known as the vertex) is the point where the function crosses the y-axis. This point estimates the y-intercept. Also, note where the graph meets the x-axis, this is your estimate for the x-intercept.

Step 2: Graphical estimation of intercepts

From the graph, it can be estimated that the curve intercepts the x-axis at x = -2, which is the x-intercept. Then, we can see that the curve meets the y-axis when the function value is positive. Hence, the y-intercept is 2.

Step 3: Algebraic verification of x-intercept

To calculate the x-intercept algebraically, substitute \(y = 0\) in the function and solve for \(x\). So, \(0 = |x + 2|\). Which implies x = -2 when the modulus operator is removed. Thus, x-intercept is -2.

Step 4: Algebraic verification of y-intercept

To calculate the y-intercept algebraically, substitute \(x = 0\) in the function, so \(y = |0 + 2|\), which results in \(y = 2\). Thus, y-intercept is 2.

Key Concepts

Graphical Estimation

Graphical estimation is a handy approach to predict values of certain points on a graph without performing detailed calculations. When looking at the graph of a function, like the V-shaped curve of the absolute value function \( y = |x + 2| \), you can estimate where the graph crosses the x-axis and the y-axis—these are known as intercepts.
To make a graphical estimation:

• First, identify where the graph intersects the x-axis. This point is typically where the graph outputs zero (for the function \( y \)).
• Second, look at where the graph crosses the y-axis. This is found by determining the point where the input value \( x \) is zero.

These graphical estimates help in creating initial predictions before delving into algebraic solutions for more precise results. This method offers a quick insight, especially when dealing with simple functions like linear or absolute value functions.

Absolute Value Functions

Absolute value functions are a fundamental concept in algebra, often resulting in a distinctive V-shaped graph. An absolute value function, such as \( y = |x + 2| \), includes the absolute value operator which serves to make any input positive by converting negatives into positives.
The key characteristics of absolute value functions include:

• **Vertex**: The 'corner' or pivot of the V-shaped graph, marking a point of symmetry. For \( y = |x + 2| \), this vertex is at (-2, 0).
• **Non-Negative Output**: Since the absolute value cannot be negative, the graph stays entirely in the upper half of a coordinate grid, unless it's been shifted downwards.
• **Symmetry**: The graph is symmetrical around a vertical line passing through the vertex.

Recognizing these qualities is crucial for quickly plotting the graph of an absolute value function and understanding its behavior.

Intercepts of a Function

Intercepts in a function's graph provide valuable information about its behavior at specific points. For any function, intercepts refer to the points where the graph crosses the axes.
There are two main types of intercepts:

• **x-intercept**: This is the point where the graph crosses the x-axis. Here, the function's output (\( y \)) is zero, which allows us to solve for \( x \) by setting the equation to zero. For \( y = |x + 2| \), solving \( 0 = |x + 2| \) leads to two possible solutions due to the absolute value, but results in a single x-intercept at \( x = -2 \).
• **y-intercept**: This occurs where the graph meets the y-axis, and is found by substituting \( x = 0 \) into the function. For the function \( y = |x + 2| \), substituting gives \( y = 2 \), making the y-intercept at the point (0, 2).

Understanding intercepts makes it easier to quickly sketch a function and analyze how it might interact with real-world data. They are the cornerstone for identifying initial points on a graph before detailed evaluations.