Problem 19 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=\frac{1}{2}|x|$$

Step-by-Step Solution

Answer

Plot points for the vertex and additional symmetric points, then connect them in a V shape.

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{2}|x| \). This is an absolute value function, which means it will produce a V-shaped graph that opens upwards. The coefficient \( \frac{1}{2} \) affects the steepness or slope of the arms of the V.

Step 2: Identify Key Features

The vertex of the function is at the origin (0,0) because there are no horizontal or vertical shifts. Since the coefficient \( \frac{1}{2} \) is a positive fraction, the arms of the V are less steep than those of \( |x| \). The graph is symmetric around the y-axis.

Step 3: Calculate Points

To help plot the graph, calculate a few points. Evaluating the function at some key values, we find: \( f(0) = 0 \), \( f(2) = 1 \), \( f(-2) = 1 \), \( f(4) = 2 \), and \( f(-4) = 2 \). These points will guide the shape of the graph.

Step 4: Draw the Graph

Plot the calculated points on a coordinate plane: (0,0), (2,1), (4,2), (-2,1), and (-4,2). Draw straight lines connecting these points, creating a V shape opening upwards. Make sure the arms of the V are aligned symmetrically with respect to the y-axis.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. An absolute value, represented as \(|x|\), means the distance of a number from zero on the number line, irrespective of the direction. Therefore, it is always non-negative. In the context of functions, the expression \(|x|\) forms a distinctive V-shaped graph. This happens because, for all negative inputs, the output is positive, mirroring the positive value for positive inputs.For the function \(f(x) = \frac{1}{2}|x|\), the factor \(\frac{1}{2}\) before the absolute value alters the graph's steepness. Generally, in any absolute value function of the form \(f(x) = a|x|\), the parameter \(a\) dictates how stretched or compressed the V-shape will be.

• If \(a\) is greater than 1, the V will be narrower or steeper.
• If \(a\) is between 0 and 1, like \(\frac{1}{2}\) in this case, it results in a wider and less steep graph.
• If \(a\) is negative, the V opens downward instead of upward.

In the given function \(f(x) = \frac{1}{2}|x|\), the graph is less steep, broadening the V as compared to the standard \(|x|\) graph.

Coordinate Plane

The coordinate plane is crucial for graphing functions, as it allows us to visualize relationships between variables. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. These axes help identify the position of any point using a pair of numerical coordinates \((x, y)\).For the given function \(f(x) = \frac{1}{2}|x|\), plotting points on the coordinate plane is the first step in graphing. By determining various x-values and computing the corresponding y-values, such as \(f(0) = 0\), \(f(2) = 1\), and \(f(-2) = 1\), we get distinct points that guide us in sketching the graph. Once these points are plotted on the plane:

• They should be connected to visualize the V-shape.
• This shape opens upward, given the function's positive coefficient \(\frac{1}{2}\).
• The wider V-shape indicates the effect of the smaller-than-one coefficient.

The coordinate plane provides a clear and intuitive way to comprehend the function's behavior across different values.

Function Symmetry

Symmetry in functions is a characteristic that makes the graph visually appealing and easier to understand. A function is symmetric about an axis if its graph is mirrored on each side of that axis. For the function \(f(x) = \frac{1}{2}|x|\), symmetry plays a vital role in its graph.This specific function is symmetric concerning the y-axis. This means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\). Therefore, it doesn't matter whether x is positive or negative; the outcome for the y-value remains the same.This symmetry implies that once one side of the V is known, the other side can be deduced without additional calculations, due to the mirror-like reflection along the y-axis. This feature:

• Eases the graph plotting process, ensuring both sides of the V-shape are drawn correctly.
• Confirms that the graph will look identical on either side of the y-axis, reinforcing the overall function's predictability.

Understanding symmetry also helps to quickly identify key behavior patterns of the function's graph, making the visualization process simpler and more efficient.