Problem 10 - 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. $$ y=|1-x| $$

Step-by-Step Solution

Answer

The x-intercept is at \((1, 0)\); the graph decreases for \(x < 1\) and increases for \(x > 1\).

Step 1: Interpret the Absolute Value Function

The given function is \( y = |1-x| \). This is a V-shaped graph where the vertex is formed at the point where the expression inside the absolute value equals zero. The function can be rewritten as \( y = |-(x-1)| = |x-1| \). This vertex occurs at \( x = 1 \).

Step 2: Find the x-intercept

The x-intercept occurs where the graph crosses the x-axis, which means \( y = 0 \). Set \( |1-x| = 0 \). Solving for \( x \), we have \( 1 - x = 0 \), thus \( x = 1 \). So the x-intercept is at the point \((1, 0)\).

Step 3: Analyze the Increasing and Decreasing Intervals

The graph of \( y = |1-x| \) is symmetric around the vertex at \( x = 1 \). To the left of \( x = 1 \) (when \( x < 1 \)), the graph is sloping downwards, indicating it is decreasing. To the right of \( x = 1 \) (when \( x > 1 \)), the graph is sloping upwards, indicating it is increasing.

Key Concepts

x-intercept

The x-intercept of a graph is the precise point where the graph meets the x-axis. At this intersection, the value of the function, or y-value, is zero. For our function given by \( y = |1-x| \), we want to find when \( y = 0 \). To do so, we set the equation \( |1-x| = 0 \).

Solving this, we have:

• \( 1 - x = 0 \)
• Therefore, \( x = 1 \).

Hence, our x-intercept is at the point \((1, 0)\). This means when \( x = 1 \), the graph crosses the x-axis. Knowing the x-intercept helps us in sketching the graph, as it gives a fixed point through which the V-shape of an absolute value function passes.

increasing and decreasing intervals

Increasing and decreasing intervals tell us where the graph is moving upwards or downwards as x changes. For absolute value functions, this is determined by looking at the vertex, since the graph changes its direction at that point. Here, the function is \( y = |1-x| \) and the vertex is at \( x = 1 \).

- **Decreasing Interval:**
For \( x < 1 \), the graph of \( y = |1-x| \) is sloping downwards as x approaches 1. Therefore, on the interval \((-\infty, 1)\), the function is decreasing.
- **Increasing Interval:**
Once x moves past 1, the graph starts moving upwards, indicating an increasing interval. This happens for \( x > 1 \), or on the interval \((1, \infty)\).

These intervals are crucial as they give us insight into the behavior of the function at various sections of the x-axis. Moreover, knowing where a function is increasing or decreasing helps in finding local extremities and understanding the function's overall trend.

vertex of absolute value function

The vertex of an absolute value function is the turning point, where the direction of the graph changes. This point is critical because it divides the graph into two symmetrical halves, with one section increasing and the other decreasing.

For \( y = |1-x| \), we first rewrite it as \( y = |-(x-1)| = |x-1| \). Here, the expression inside the absolute value is zero at \( x = 1 \), determining our vertex.

- **Vertex:** The vertex is at the point \((1, 0)\).
This vertex essentially tells us:

• The lowest value that the absolute value function can reach, here at \( y = 0 \).
• The point of symmetry around which the graph is mirrored, meaning any deviation from this point to the left will have the same shape and orientation as to the right, but in opposite direction.

Understanding the vertex position simplifies sketching the graph and predicting its shape. It is also the point where the function transitions from decreasing to increasing.