Problem 42 - Algebra 2

Question

Graph each pair of functions. Find the approximate point(s) of intersection. \(y=-\frac{1}{x-3}-6, y=6.2\)

Step-by-Step Solution

Answer

Referencing the graph, at \(x = 3.03\), \(y = 6.2\). Thus the approximate point of intersection is \((3.03, 6.2)\).

Step 1: Graph the first function

Start by graphing the function \(y=-\frac{1}{x-3}-6\). This function is a transformation of the function \(y=-\frac{1}{x}\), which is a hyperbola of two decreasing sections. In this transformation, the hyperbola is translated 3 units to the right and 6 units down.

Step 2: Graph the second function

Once the first function has been graphed, graph the second function \(y=6.2\). This is basically a horizontal straight line at \(y=6.2\).

Step 3: Locate the point of Intersection

After graphing the functions, the next step is to locate the point of intersection. This point would be where the horizontal line \(y=6.2\) crosses the curve of function \(y=-\frac{1}{x-3}-6\). A detailed examination would reveal the approximate \(x\) value at the point of intersection.

Step 4: Use the second equation to get the y-coordinate

The point of intersection will be represented as \((x,6.2)\), as the y-coordinate of the point of intersection comes directly from the equation of the straight line.

Key Concepts

Intersection Points

When graphing multiple functions on the same coordinate plane, the intersection point(s) are where the graphs of these functions meet or cross each other. An intersection point between two functions is significant because it represents a solution where both function equations are valid simultaneously.

• For the exercise given, to find the intersection points, you graph both functions on the same axes.
• The intersection happens where the graph of the hyperbola and the horizontal line meet. This point represents the x-value that satisfies both equations.
• To determine this intersection point, carefully examine the graphs, looking for the crossover.

After locating the x-coordinate on the graph, verify this against the equations to ensure accuracy.

Hyperbola

The hyperbola is a type of curve on a plane, identified in algebra by equations like \(y = -\frac{1}{x}\). Hyperbolas have two branches, which tend to move towards infinity in opposite directions.

• In this instance, the hyperbola \(y = -\frac{1}{x-3} - 6\) has undergone transformations from the basic form of a hyperbola.
• Key characteristics include: asymptotes, which are lines the hyperbola approaches but never touches; and a vertex, where the curve changes direction.
• Here, we shift the standard hyperbola 3 units right and 6 units down from its origin.

The transformation changes the hyperbola’s position, which is vital for locating intersection points with other functions.

Horizontal Line

A horizontal line is a straight line on the graph parallel to the x-axis. The line equation is simple: \(y = c\), where all y-values on the line are the same.

• In this scenario, the line \(y = 6.2\) represents the constant y-value for all points along this line.
• When graphing a horizontal line, it cuts across the y-axis at the value given, creating a stable reference for finding intersections.

The simplicity of horizontal lines makes them easy to analyze when finding where they intersect with more complex curved graphs.

Transformation of Functions

Transformations shift or change graphs in predictable ways, allowing the original shape of the graph to remain consistent while its position changes. Translations, reflections, stretches, and compressions are types of transformations.

• For the function \(y=-\frac{1}{x-3}-6\), there is a transformation from the basic hyperbola \(y=-\frac{1}{x}\) by shifting 3 units to the right and 6 units downward.
• This translates the curve without altering its fundamental shape or properties.
• Understanding transformations helps predict how the graph of a function will behave on a coordinate plane.

These transformations are essential in finding accurate points of intersection with other functions, as they help visualize where graphs will intersect.