Problem 15 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ H(x)=|2 x| $$

Step-by-Step Solution

Answer

The graph is a V-shape opening upwards centered at the origin.

Step 1: Understand the Absolute Value Function

The function given is \( H(x) = |2x| \), which is an absolute value function. An absolute value function always outputs a non-negative value, meaning \( H(x) \) takes the positive value of \( 2x \), whether \( x \) is positive or negative. Therefore, the graph will be V-shaped, symmetric, and open upwards with the vertex at the origin (0,0).

Step 2: Choose a Range of x-values

Select a range of \( x \) values around the vertex. For simplicity, choose integer values, including negative, zero, and positive. Let's use \( x = -3, -2, -1, 0, 1, 2, 3 \).

Step 3: Create a Table of Values

Calculate \( H(x) = |2x| \) for each chosen \( x \) value, forming a table. For example, when \( x = -3 \), \( H(-3) = |2 imes -3| = 6 \). Compute similar calculations for the other \( x \) values:- \( x = -3, H(x) = 6 \)- \( x = -2, H(x) = 4 \)- \( x = -1, H(x) = 2 \)- \( x = 0, H(x) = 0 \)- \( x = 1, H(x) = 2 \)- \( x = 2, H(x) = 4 \)- \( x = 3, H(x) = 6 \).

Step 4: Plot the Points on a Graph

Using the table of values, plot each point on the coordinate plane. Plot \((x, H(x))\) pairs: \((-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4), (3, 6)\).

Step 5: Draw the Graph

Connect the plotted points with straight lines to visually represent the absolute value function \( H(x) = |2x| \). The lines will meet at the vertex, forming a 'V' shape.

Key Concepts

Graphing Functions

When learning about functions, "graphing" is an essential skill that helps visually represent how a function behaves. Functions like \( H(x) = |2x| \) show a connection between input values (\( x \)) and output values (\( H(x) \)). Graphing allows us to see these relationships clearly. We begin by creating a "table of values," consisting of several \( x \)-values and their respective \( H(x) \) outputs calculated using the function rule. Once we have these points, we use them to plot on a graph.
In graphing, consistency is key. Start by marking values on both horizontal (x-axis) and vertical (y-axis) lines accurately. The points derived from our table will guide the shape of the graph. For absolute value functions, you will particularly expect a specific shape from the graphing process.

Table of Values

Creating a "table of values" is a fundamental step for graphing functions. It allows us to see specific outputs derived from the function rule for select \( x \)-values. Let's break this down:

• Choose several \( x \)-values: both negative and positive, with zero in between.
• Compute \( H(x) \) for each of these \( x \) values using the function \( H(x) = |2x| \).

This process will yield a clear set of coordinate points. For example, if \( x = -3 \), then \( H(-3) = |2 \times -3| = 6 \). For \( x = 2 \), \( H(2) = |2 \times 2| = 4 \), and so forth.
These coordinates, such as \((-3, 6)\) or \((0, 0)\), form the backbone of our graph. They dictate where to plot points on a coordinate plane for the visual representation.

Coordinate Plane

The coordinate plane is a fundamental component in graphing any function. It consists of two perpendicular lines:

• The horizontal line — x-axis
• The vertical line — y-axis

The origin, where these axes intersect, is the point \((0, 0)\).
To graph a function like \( H(x) = |2x| \), plot each pair of \( (x, H(x)) \) values on this plane. Remember:

• Each point is determined by its x-value and corresponding y-value \(H(x)\)
• Place each point accurately according to the table of values

Once all points are plotted, you can see the structure of the graph. This system helps distinguish rising, descending, or steady trends in the functions we're studying.

V-shaped Graph

Absolute value functions, like \( H(x) = |2x| \), naturally form a 'V-shaped' graph. This distinctive shape results from how the function outputs are computed:

• The absolute value ensures all outputs are non-negative
• The function reflects any negative input values positively

The vertex, located at the origin \((0, 0)\), is a key feature of this graph. It acts as the point where the V opens and the lines on either side meet.
In this scenario, the arms of the V (both upward-sloping from the vertex) indicate the linear increase of the function on either side of the x-axis. This symmetry and simplicity make the graph of \( H(x) = |2x| \) both intuitive and visually distinct. Each plotted point not only contributes to forming this V-shape, but it also helps confirm the characteristics unique to absolute value functions.