Problem 16 - The Calculus with Analytic Geometry

Question

The graph of \(y=4-|x|\) and the \(x\) axis from \(x=-4\) to \(x=4\) form a triangle. Use the method of this section to find the area of this triangle.

Step-by-Step Solution

Answer

The area of the triangle is 16 square units.

Step 1 - Identify Key Points

Identify the points where the graph intersects the x-axis and y-axis. For the given function, the intersections with the x-axis occur when: \[ 0 = 4 - |x| \] This gives \[ |x| = 4 \] which implies \[ x = 4 \] and \[ x = -4 \].Since the y-intercept is at \[ x = 0 \], \[ y = 4 - |0| = 4 \]. Thus, the key points are \[ (-4, 0) \], \[ (4, 0) \], and \[ (0, 4) \].

Step 2 - Determine the Base and Height

To find the area of the triangle, identify its base and height. The distance between \[ (-4, 0) \] and \[ (4, 0) \] defines the base: \[ 4 - (-4) = 8 \]. The height is the y-coordinate of the vertex at \[ (0, 4) \]. Height = 4.

Step 3 - Calculate the Area

Use the formula for the area of a triangle: \[ Area = \frac{1}{2} \times base \times height \]. Substitute the identified base and height: \[ Area = \frac{1}{2} \times 8 \times 4 \]. This yields: \[ Area = 16 \].

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( |x| \), represents the distance of a number from zero on the number line. It is always non-negative. For any real number \( x \), the absolute value is defined as:

• \( |x| = x \) if \( x \geq 0 \).
• \( |x| = -x \) if \( x < 0 \).

In the given function \( y = 4 - |x| \), the graph is a V-shaped curve symmetrical around the y-axis. When \( x = 0 \), the function reaches its maximum value at \( y = 4 \). As \( |x| \) increases, \( y \) decreases linearly, creating two linear segments.

Graph Intersections

Graph intersections are crucial to understanding and analyzing the behavior of functions. For the function \( y = 4 - |x| \), we need to find where it intersects the x-axis and y-axis.
For the x-axis intersections, set \( y = 0 \):

• Solve \( 0 = 4 - |x| \)
• Thus, \( |x| = 4 \)
• Providing intersections at \( x = 4 \) and \( x = -4 \)

For the y-axis intersection, set \( x = 0 \) and solve for \( y \):

• \( y = 4 - |0| = 4 \)
• Intersection at point \( (0, 4) \)

These intersection points form the vertices of the triangle involved in our area calculation.

Triangle Area Calculation

To find the area of a triangle, we can use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In our problem:

• Base: Distance between points \( (-4, 0) \) and \( (4, 0) \)
• Height: Y-coordinate of vertex at \( (0, 4) \)
• Base = 4 - (-4) = 8 units
• Height = 4 units
• Area = \[ \frac{1}{2} \times 8 \times 4 = 16 \text{ square units} \]

The final area of the triangle is 16 square units.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, is a mathematical discipline that uses algebraic equations to describe geometric properties. Key elements include:

• Using coordinates (x, y) to represent points on a plane
• Equations describe lines, curves, and other shapes
• Distance, midpoints, and area calculations

In this exercise, analytic geometry allowed us to:

• Identify intersection points of the function with axes
• Determine the triangle's vertices and calculate base and height
• Apply algebraic methods to find the triangle's area

Understanding these concepts is fundamental to solving geometry problems with algebraic methods.