Problem 19 - Calculus

Question

19 Suppose \(y=m x+b\) for negative \(x\) and \(y=M x+B\) for \(x \geqslant 0 .\) The graphs meet if _________ The two slopes are____________. The slope at \(x=0\) is __________( what is possible? \()\)

Step-by-Step Solution

Answer

The graphs meet when \( b = B \); the slopes are \( m \) and \( M \); the slope at \( x=0 \) can be undefined or a point of non-smooth transition.

Step 1: Understanding the Problem

The problem involves two linear equations. The first equation is \( y = mx + b \) for negative \( x \), and the second equation is \( y = Mx + B \) for non-negative \( x \). Our task is to determine the conditions under which these two graphs meet and explore details about the slopes.

Step 2: Finding the Intersection Point

For the graphs of the equations \( y = mx + b \) and \( y = Mx + B \) to meet, they must have a common point. This means their \( y \)-values must be equal at the same \( x \) value. The potential point of intersection is at \( x = 0 \). At this point, both equations must satisfy the equality \( b = B \).

Step 3: Analyzing the Slopes

The slopes of the equations are given by \( m \) and \( M \) respectively. Even though they are different for \( x < 0 \) and \( x \geq 0 \), at the meeting point, the behavior of the graph may change depending on these slopes.

Step 4: Calculating the Slope at \( x=0 \)

At \( x=0 \), the slope can be considered as a transition between the two equations. Therefore, the slope at this point would generally not be defined as a single value unless \( m = M \). If the slopes are different, \( x = 0 \) could mark a point of non-smooth transition.

Key Concepts

Intersection Point

An intersection point between two graphs is where both equations share the same coordinates, implying that their respective outputs are equal at this specific input. In the given problem, we have the equations:

• \(y = mx + b\) for \(x < 0\)
• \(y = Mx + B\) for \(x \geq 0\)

For these equations to intersect, their \(y\)-values must equal each other at one particular \(x\) value. Here, the logical consideration is the point where these separate domains meet, which is at \(x = 0\). Thus, their intersection point is at \((0, b)\), assuming that \(b = B\). This is a crucial part of solving piecewise functions, where the transition point often serves as the possible point of intersection unless specified otherwise.

Slope

The slope of a line represents its steepness and direction. It is generally given as the coefficient of \(x\) when a line is in the form \(y = mx + b\). For the given lines:

• The slope for \(y = mx + b\) (for negative \(x\)) is \(m\).
• The slope for \(y = Mx + B\) (for \(x \geq 0\)) is \(M\).

Slopes help to determine how quickly \(y\) values change as compared to \(x\). In understanding the problem, recognizing that \(m\) and \(M\) are not necessarily equal is important. This difference signifies a potential change in direction or a sharp turn at \(x = 0\) if \(m eq M\). If \(m = M\), the graph transitions smoothly without any visible corners or breaks.

Piecewise Functions

Piecewise functions are defined by different expressions over various intervals. In this exercise, we see a piecewise scenario with:

• \(y = mx + b\) when \(x < 0\)
• \(y = Mx + B\) when \(x \geq 0\)

Understanding the application of piecewise functions is crucial. They are used to describe a system that behaves differently based on the input value. When working with piecewise functions, ensuring continuity between the different pieces is key, especially where the function's domain splits. Here, by ensuring \(b = B\) at \(x = 0\), the y-values of the functions remain continuous.

Coordinate Geometry

Coordinate geometry is the study of geometric figures through the coordinate plane, using algebraic formulas and graphs. It helps us visualize equations and solutions. In this problem, coordinate geometry involves plotting:

• The line segment \(y = mx + b\) for \(x < 0\)
• The line segment \(y = Mx + B\) for \(x \geq 0\)

Visualizing where these line segments meet helps to understand the intersection point. Coordinates \((x, y)\) are plotted to see where transformations or overlaps in the figure occur. Within coordinate geometry, having clarity on how equations are represented graphically enables solving and verifying conditions such as the point when \(b = B\) as stated, ensuring both equations indeed meet at the point \((0, b)\).