Problem 12 - 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. $$ y=\left|\frac{1}{2} x+1\right| $$

Step-by-Step Solution

Answer

The x-intercept is at \((-2,0)\); the graph decreases for \(x < -2\) and increases for \(x > -2\).

Step 1: Identify the Function Type

The function given is \( y = \left|\frac{1}{2} x + 1\right| \). This is an absolute value function. The effect of the absolute value is that it reflects any negative output to be positive, causing a V-shaped graph.

Step 2: Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\).\[0 = \left|\frac{1}{2} x + 1\right|\]This results in two possible equations based on the property of absolute value:\[\frac{1}{2} x + 1 = 0\quad \text{or} \quad \frac{1}{2} x + 1 = 0\]Solving this:\(\frac{1}{2} x + 1 = 0 \rightarrow x = -2\)Thus, the \(x\)-intercept is \((-2, 0)\).

Step 3: Determine the Vertex

The absolute value function \(y = \left|\frac{1}{2} x + 1\right|\) has its vertex where the expression inside the absolute value is zero. From Step 2, this occurs when \(x = -2\), which means the vertex is \((-2,0)\). The graph will have a V-shape centered at this point.

Step 4: Determine Where the Graph is Increasing

To determine where the graph is increasing, consider when the slope of the inner function, \(\frac{1}{2}x + 1\), is positive due to the absolute value reversing negative slopes:For \(x > -2\), the function \(\frac{1}{2}x + 1 > 0\) remains untouched, and the graph is increasing.

Step 5: Determine Where the Graph is Decreasing

For \(x < -2\), the graph of \(\left|\frac{1}{2} x + 1\right|\) reflects the linear part \(\frac{1}{2} x + 1\), translating it as \(-\left(\frac{1}{2}x + 1\right)\) due to the absolute value. The graph decreases before reaching the vertex at \(x = -2\).

Step 6: Graph the Function by Hand

Plot the vertex \((-2,0)\) on a coordinate plane. For \(x < -2\), draw a line with a negative slope (descending to the left), representing \(-\left(\frac{1}{2}x + 1\right)\). For \(x > -2\), draw a line with a positive slope (ascending to the right), representing \(\frac{1}{2}x + 1\). Ensure the graph forms a V shape.

Key Concepts

x-intercept

In mathematics, the x-intercept of a graph is the point where the graph crosses the x-axis. This is the value of \(x\) when \(y = 0\). For the absolute value function given by \(y = \left|\frac{1}{2} x + 1\right|\), finding the x-intercept helps us understand one of the fundamental points through which the graph passes.

To determine the x-intercept for this function, we set \(y = 0\) and solve for \(x\): \[0 = \left|\frac{1}{2} x + 1\right|\] Because the absolute value can equal zero when its contents are zero, we set the equation inside the absolute value to zero: \[\frac{1}{2} x + 1 = 0\] By solving it, we find: \[\frac{1}{2} x = -1 \rightarrow x = -2\] Thus, the x-intercept is at the point \((-2, 0)\). This single intersection point is crucial as it also coincides with the vertex of the V-shape graph of the absolute value function. It's important to recognize this x-intercept as a keystone in sketching the overall graph.

graphing

Graphing an absolute value function like \(y = \left|\frac{1}{2}x + 1\right|\) involves understanding its V-shape structure. It consists of two linear pieces meeting at a vertex.

• The vertex here is a pivotal point found at \((-2, 0)\), where the expression inside the absolute value is zero.
• To the left of the vertex (\(x < -2\)), the graph decreases, reflecting the line \(-\left(\frac{1}{2}x + 1\right)\).
• To the right of the vertex (\(x > -2\)), the graph increases following the line \(\frac{1}{2}x + 1\).

Drawing the graph by hand can be straightforward: plot the vertex, then sketch the lines on either side. When graphing:

• Past the vertex, imagine an equal distance below and above the vertex slope, showing a clear V shape.
• Use consistent units along the axes for accurate representation.
• The shift in behavior at \(x = -2\) defines the transition between the decreasing and increasing sections of the function.

Understanding this piecewise nature of the absolute value function helps in graphing it successfully and recognizes how shifts and shapes occur in different function transformations.

increasing and decreasing intervals

In considering the increasing and decreasing intervals of the function \(y = \left|\frac{1}{2}x + 1\right|\), it is vital to break down how the absolute value impacts the graph.

• **Increasing Interval:** This occurs when \(x > -2\). Beyond the vertex at \(x = -2\), the graph follows the slope \(\frac{1}{2}x + 1\), where the function behaves like a straightforward linear equation. As \(x\) increases, so does \(y\), meaning the function is increasing.
• **Decreasing Interval:** Before the vertex, where \(x < -2\), the graph showcases the reflection of the linear function due to the absolute value, specifically \(-\left(\frac{1}{2}x + 1\right)\). Here, as \(x\) decreases, the \(y\) also decreases as it approaches the vertex.

This behavior perfectly demonstrates the symmetry and structure of absolute value functions, focusing on the vertex as the dividing line between these two intervals. Recognizing these intervals provides insights not only on graph behavior but also on the function's foundational properties. Through these, students gain deeper appreciation and understanding of the nature and application of absolute value equations.