Problem 64 - College Algebra with Modeling and Visualization

Question

Assume that each function is continuous. Do not use a graphing calculator. Sketch a graph of a linear function \(f\) that intersects a constant function \(g\) exactly once.

Step-by-Step Solution

Answer

Draw a horizontal line for \(g(x)\) and a slanted line for \(f(x)\) that intersects once.

Step 1: Understand the Functions

The problem involves two types of functions: a linear function \(f(x) = mx + b\) and a constant function \(g(x) = c\). The linear function is defined as any straight line, and the constant function is a horizontal line on the graph.

Step 2: Identify Intersection Condition

These functions intersect when they have the same value for at least one \(x\)-coordinate. This means \(f(x) = g(x)\), or \(mx + b = c\). Since we want exactly one intersection, the line must cross the constant line once.

Step 3: Arrange Function Equation for Intersection

Set \(mx + b = c\). To find the point \(x\) where they intersect, rearrange the equation: \[ mx + b = c \]\[ mx = c - b \]\[ x = \frac{c-b}{m} \]This gives the exact \(x\)-coordinate where the intersection occurs.

Step 4: Sketch the Intersection in a Graph

Draw a horizontal line, representing \(g(x) = c\). Then, draw a straight line with slope \(m eq 0\) that crosses this horizontal line exactly once. The intersection point is \(\left(\frac{c-b}{m}, c\right)\). This ensures that they intersect exactly once.

Key Concepts

Linear Function

A linear function is a function defined by an equation of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear functions are represented graphically as straight lines. The slope \(m\) determines how steep the line is:

• If \(m > 0\), the line rises as it moves from left to right.
• If \(m < 0\), the line falls as it moves from left to right.
• If \(m = 0\), the line is horizontal.

Linear functions have several important properties, such as constant rate of change and no curvature. Every pair of distinct points on the graph of a linear function is connected by a line segment, making it predictable and easy to draw. Understanding linear functions is crucial in identifying where they intersect with other types of functions, like constant functions.

Constant Function

A constant function is a function with the form \(g(x) = c\), where \(c\) is a fixed value. The graph of a constant function is a horizontal line across the coordinate plane.

• Every point on this line has the same y-coordinate, \(c\).
• No matter what x-value you choose, the output will always be \(c\).

Constant functions are simple yet powerful in analysis, as they provide a consistent benchmark or reference on a graph. They are used to determine intersections solely based on when other functions reach this fixed value. For example, a linear function intersecting a constant function means finding where their outputs equal \(c\). Understanding constant functions is key to setting up problems that require identifying intersections with more complex function types.

Graph Sketching

Graph sketching is a valuable skill in mathematics, allowing one to visually interpret functions and their intersections. When sketching a graph of functions like a linear and constant function intersection:

• Start by identifying key components, such as slopes and intercepts.
• Draw the constant function as a horizontal line at \(y = c\).
• Next, graph the linear function \(f(x) = mx + b\). Choose two points using easy x-values (e.g., \(x = 0\) for the y-intercept).
• Find the intersection point: set \(f(x) = g(x)\) to find \(x = \frac{c-b}{m}\), and plot the calculated point.
• Ensure that the linear line only intersects the horizontal line once for a single intersection.

By understanding these steps, graph sketching becomes a clear and systematic method to find and illustrate function intersections accurately, bringing abstract algebraic solutions to life visually.