Problem 36 - Algebra Form and Function

Question

The graphs of two linear functions have the same slope, but different \(x\) -intercepts. Can they have the same \(y\) intercept?

Step-by-Step Solution

Answer

Answer: No, because if they also have the same y-intercept, their formulas would be identical, and they would represent the same line. Since they have different x-intercepts, their y-intercepts must be different.

Step 1: Recall the slope-intercept form

The slope-intercept form of a linear function is given by the equation \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Step 2: Set up the two linear functions

Let's consider the two linear functions \(f_1(x) = m_1x + b_1\) and \(f_2(x) = m_2x + b_2\). We are given that their slopes are equal, so \(m_1 = m_2 = m\). Also, they have different x-intercepts, which means they intersect the x-axis at different points.

Step 3: Analyze the x-intercepts and y-intercepts

For each linear function, we can determine their x- and y-intercepts using their equations. The x-intercepts occur when \(y = 0\), and the y-intercepts occur when \(x = 0\). Thus, for each function, we have: 1. \(f_1(x)\): x-intercept: \(0 = mx + b_1\), y-intercept: \(y = m(0) + b_1\) (so \(b_1\) is the y-intercept). 2. \(f_2(x)\): x-intercept: \(0 = mx + b_2\), y-intercept: \(y = m(0) + b_2\) (so \(b_2\) is the y-intercept).

Step 4: Determine if the y-intercepts can be equal

We have established that the functions have the same slope. If they also have the same y-intercept, their formulas would be identical, and they would represent the same line. However, we know they have different x-intercepts, so the y-intercepts must be different. In other words, the two linear functions with the same slope but different x-intercepts cannot have the same y-intercept.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a pivotal concept when dealing with linear functions. It's expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) denotes the y-intercept.
This formula gives us powerful insight into the behavior of a linear line just by glancing at it. Since the slope is defined by \(m\), it tells us how steep a line is. For example, a larger \(m\) results in a steeper incline. Meanwhile, \(b\), the y-intercept, indicates where the line crosses the y-axis.
This form is incredibly useful for graphing lines and understanding their relationship to variables. To determine the shape or direction of a line, we simply substitute different values of \(x\) to solve for \(y\), plotting them until the line is fully represented on the graph.

X-Intercept

The x-intercept of a linear function is the point where the line crosses the x-axis. At this point, the value of \(y\) is zero because the line lies flat on the horizontal axis.
To find the x-intercept in the slope-intercept form equation \(y = mx + b\), substitute \(y = 0\) and solve for \(x\). This gives the equation \(0 = mx + b\), resulting in \(x = -\frac{b}{m}\).
Understanding x-intercepts is crucial because it helps in plotting the line accurately. Different x-intercepts are one reason why two lines, even with identical slopes, won't overlap unless they are the same line. For this reason, knowing how to compute the x-intercept enhances problem-solving in linear equations.

Y-Intercept

The y-intercept occurs where a line crosses the y-axis, making the value of \(x\) zero at this point. Every linear function has a y-intercept unless the line is perfectly vertical.
In the slope-intercept form \(y = mx + b\), finding the y-intercept is straightforward: simply observe the \(b\) value. This part of the equation is independent of \(x\) and remains constant.
This knowledge is essential because the y-intercept serves as a starting point for graphing a line. It helps determine one definite point through which the line will pass, making it easier to then use the slope \(m\) to plot the remaining points. Understanding the y-intercept clarifies how a line behaves across different graphs and datasets, ensuring comprehension of linear models.