Problem 11 - College Algebra with Modeling and Visualization

Question

Graph by hand. (a) Find the \(x\) -intercept. (b) Determine where the graph is increasing and where it is decreasing. $$ y=|2 x-3| $$

Step-by-Step Solution

Answer

The x-intercept is at \( (\frac{3}{2}, 0) \). The graph is increasing for \( x \geq \frac{3}{2} \) and decreasing for \( x < \frac{3}{2} \).

Step 1: Find the x-intercept

To find the \( x \)-intercept of the function \( y = |2x - 3| \), set \( y = 0 \). Solve for \( x \): \[ |2x - 3| = 0 \] This equation means that \( 2x - 3 = 0 \). Solving for \( x \), we get: \[ 2x = 3 \] \[ x = \frac{3}{2} \] Therefore, the \( x \)-intercept is \( (\frac{3}{2}, 0) \).

Step 2: Analyze the piecewise function

Write the absolute value function as a piecewise function. This will help to determine where the graph is increasing or decreasing: \[ y = \begin{cases} 2x - 3, & \text{if } 2x - 3 \geq 0 \ -(2x - 3), & \text{if } 2x - 3 < 0 \end{cases} \] Simplifying the conditions, for \( x \geq \frac{3}{2} \), the function is \( y = 2x - 3 \), and for \( x < \frac{3}{2} \), the function is \( y = -(2x - 3) = -2x + 3 \).

Step 3: Determine where the graph is increasing

For \( y = 2x - 3 \) when \( x \geq \frac{3}{2} \), the slope \( (m = 2) \) is positive, meaning the function is increasing in this interval. Thus, the graph is increasing for \( x \geq \frac{3}{2} \).

Step 4: Determine where the graph is decreasing

For \( y = -2x + 3 \) when \( x < \frac{3}{2} \), the slope \( (m = -2) \) is negative, meaning the function is decreasing in this interval. Thus, the graph is decreasing for \( x < \frac{3}{2} \).

Key Concepts

x-intercept

The concept of an \(x\)-intercept is crucial when graphing functions. It refers to the point(s) where a graph crosses the \(x\)-axis. In other terms, it's where the output \(y\) of the function equals zero. Understanding how to find the \(x\)-intercept helps to get a clear picture of the graph when starting with an equation.

To find the \(x\)-intercept of an absolute value function like \(y = |2x - 3|\), you need to set \(y\) to zero and solve the equation. This changes the absolute value expression to zero, so you solve \(|2x - 3| = 0\).

• Remove the absolute value to get \(2x - 3 = 0\).
• Solve for \(x\) by adding 3 to both sides: \(2x = 3\).
• Divide by 2 to isolate \(x\), giving \(x = \frac{3}{2}\).

This means the \(x\)-intercept is the point \((\frac{3}{2}, 0)\), where the graph touches the \(x\)-axis. Recognizing the \(x\)-intercept is fundamental in accurately plotting any graph especially when dealing with absolute value functions.

piecewise function

Piecewise functions are employed to define functions stranded in different conditions, which is particularly useful in handling absolute value expressions. When graphing \(y = |2x - 3|\), it's beneficial to convert it into a piecewise form to better understand its behavior. By breaking it down, we can see how the function behaves differently across various intervals.

For \(y = |2x - 3|\), the piecewise decomposition looks like this:

• When \(2x - 3 \geq 0\), it simplifies as \(y = 2x - 3\). This applies when \(x \geq \frac{3}{2}\).
• When \(2x - 3 < 0\), the function takes the form \(y = - (2x - 3)\) or \(y = -2x + 3\). This is valid for \(x < \frac{3}{2}\).

Breaking down into piecewise segments allows us to handle the positive and negative segments of the function separately, providing clarity especially when analyzing where the function increases or decreases.

increasing and decreasing functions

Understanding when a function is increasing or decreasing is key in graphing and analyzing its behavior. A function is said to be increasing in a segment if, as \(x\) increases, \(y\) also increases. Conversely, it's decreasing if \(y\) decreases as \(x\) increases.

For the absolute value function \(y = |2x - 3|\):

• For the segment \(y = 2x - 3\) where \(x \geq \frac{3}{2}\), the slope \(m = 2\) is positive, meaning the graph is increasing in this part. The line rises as \(x\) increases beyond \(\frac{3}{2}\).
• In the segment \(y = -2x + 3\) for \(x < \frac{3}{2}\), the slope \(m = -2\) is negative, indicating decrease. Here, the graph falls as \(x\) increases up to \(\frac{3}{2}\).

Recognizing these behaviors helps in sketching accurate graphs, especially for piecewise functions like those involving absolute values.