Problem 9 - 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. $$y=|x+1|$$

Step-by-Step Solution

Answer

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

Step 1: Find the x-intercept

The x-intercept occurs where the graph crosses the x-axis, which means the y-value is 0. Set the equation equal to 0 and solve for x: \( y = |x+1| = 0 \).Since the absolute value is 0 when the expression inside it is 0, solve:\( x + 1 = 0 \) Thus, \( x = -1 \).Therefore, the x-intercept is \((-1, 0)\).

Step 2: Analyze the structure of the absolute value function

The function given is \(y = |x + 1|\). The graph of an absolute value function is V-shaped and is symmetric about the vertical line through the vertex. For the function \(|x + 1|\), the vertex is at \((-1, 0)\) since it's the point where the expression inside the absolute value equals 0.

Step 3: Determine where the graph is increasing and decreasing

An absolute value function \(y = |x + a|\) decreases to the left of the vertex and increases to the right of the vertex. Here, the vertex \((-1, 0)\) divides the graph into two parts:- The graph is **decreasing** on the interval \((-\infty, -1)\).- The graph is **increasing** on the interval \((-1, \infty)\).

Step 4: Graph the function

Now, graph the function \(y = |x+1|\) based on the information:1. Plot the x-intercept \((-1, 0)\) which is also the vertex.2. Draw the left branch of the V-shape decreasing towards the vertex, covering \(x < -1\).3. Extend the right branch of the V-shape increasing away from the vertex, covering \(x > -1\).4. Ensure symmetry about the line \(x = -1\).

Key Concepts

x-intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. This means the y-coordinate at this point is equal to zero. To find the x-intercept of an absolute value function, the expression inside the absolute value must be set to zero, because it is only at this point that the output of the function becomes zero.
For the function given, \(y = |x + 1|\), we start by setting the equation to zero:

• 0 = |x + 1|
• This implies x + 1 = 0
• Solving for \(x\), we find \(x = -1\).

Thus, the x-intercept is at the point \((-1, 0)\). Understanding this helps in plotting the graph of the function precisely at its fundamental crossing point with the x-axis.

increasing and decreasing intervals

When analyzing the increasing and decreasing behavior of a graph, especially for an absolute value function, it's significant to note the patterns that arise from its V-shape. For the function \(y = |x + 1|\), the vertex at \((-1, 0)\) acts as a pivotal point that divides the graph into two distinct intervals.

• The graph is **decreasing** on the interval \((-\infty, -1)\)\. This is because as \(x\) approaches \(-1\) from the left, the values of \(y\) reduce until they reach zero.
• On the other hand, the graph is **increasing** on the interval \((-1, \infty)\)\. Here, as \(x\) moves away from \(-1\) to the right, the values of \(y\) grow.

The vertex, acting as the central point, clearly delineates these intervals and aids in understanding the direction of the graph's motion across the x-axis.

vertex of a function

The concept of a vertex is pivotal in understanding and graphing absolute value functions. The vertex is essentially the point at which the graph changes direction. For the function \(y = |x + 1|\), the vertex is found where the expression within the absolute value is zero.

• Solve for \(x + 1 = 0\).
• The solution gives \(x = -1\), and the corresponding vertex is \((-1, 0)\).

The vertex \((-1, 0)\) is crucial because:

• It is the lowest point on the graph of \(y = |x+1|\), as the graph opens upwards resembling a 'V'.
• This point serves as the axis of symmetry, bisecting the graph into equal parts - one decreasing towards and one increasing away from this point.

Understanding the vertex gives clarity in sketching the graph, marking the intersection of its increasing and decreasing behaviors.