Problem 25 - Calculus Volume 1

Question

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) $$ f(x)=2|x| $$

Step-by-Step Solution

Answer

The graph is a V-shape that is symmetric about the y-axis with vertices at each table point.

Step 1: Determine the function to be evaluated

The function given is \( f(x) = 2|x| \). This function involves applying the absolute value to \( x \), followed by multiplying by 2. The absolute value function outputs positive values regardless of the sign of \( x \).

Step 2: Create a table of values

We will evaluate \( f(x) \) at each of the specified \( x \) values: -3, -2, -1, 0, 1, 2, 3. Calculate \( f(x) \) for each \( x \) and fill the table with corresponding y-values.

Step 3: Calculate \( f(x) \) for each value of \( x \)

- For \( x = -3 \), \( f(x) = 2|-3| = 6 \)- For \( x = -2 \), \( f(x) = 2|-2| = 4 \)- For \( x = -1 \), \( f(x) = 2|-1| = 2 \)- For \( x = 0 \), \( f(x) = 2|0| = 0 \)- For \( x = 1 \), \( f(x) = 2|1| = 2 \)- For \( x = 2 \), \( f(x) = 2|2| = 4 \)- For \( x = 3 \), \( f(x) = 2|3| = 6 \)

Step 4: Complete and organize the table

Fill in the table with the calculated values:\[\begin{array}{c|c}x & f(x) \\hline-3 & 6 \-2 & 4 \-1 & 2 \0 & 0 \1 & 2 \2 & 4 \3 & 6 \\end{array}\]

Step 5: Sketch the graph based on the table

Using the table from Step 4, plot the points on a coordinate plane. The points are \((-3,6), (-2,4), (-1,2), (0,0), (1,2), (2,4), (3,6)\). Connect the points with straight lines, keeping in mind the graph is symmetric about the y-axis due to the absolute value.

Key Concepts

Absolute Value

The concept of absolute value is crucial in mathematics. It measures the distance of a number from zero on a number line, regardless of direction. For a number \( x \), its absolute value is written as \( |x| \). Here are some simple properties of absolute values:

• Absolute values are never negative, meaning \( |x| \geq 0 \) for any \( x \).
• If \( x \) is positive or zero, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \).

When evaluating a function like \( f(x) = 2|x| \), each \( x \) value is made non-negative by the absolute value, and then multiplied by 2. This ensures that the outcome, \( f(x) \), reflects the size, but not the sign, of the original \( x \). In our specific exercise, this means for any input \( x \), the output will be doubled after the positive conversion, maintaining the shape dictated by the absolute value.

Graphing Functions

Graphing functions is a fundamental skill in mathematics. It helps visualize how a function behaves across a range of values, showing us trends and patterns. For the function \( f(x) = 2|x| \), graphing involves several steps:

• Create a table of values by selecting a set of \( x \) values, such as \(-3, -2, -1, 0, 1, 2, 3\).
• Calculate the corresponding \( f(x) \) for each of these \( x \) values as detailed in the step-by-step solution.
• Plot these \( (x, f(x)) \) pairs on a graph.

For our piecewise function, the graph involves plotting points that will reflect symmetry around the y-axis because of the absolute value. Each segment of our graph will form a V-shape, a characteristic feature of absolute value functions which shows how it increases linearly on both sides of the y-axis.

Coordinate Plane

The coordinate plane is where we visualize mathematical relationships between variables. It's a two-dimensional surface formed by two perpendicular lines: the horizontal x-axis and the vertical y-axis. Here's how it works:

• Each point in the plane is identified by a pair of numbers, \((x, y)\), corresponding to its horizontal and vertical positions.
• The x-axis and the y-axis intersect at the origin, \((0,0)\).
• Plotting points involves locating each \( (x, f(x)) \) on the plane, aligning with its cross-section on the x-axis and y-axis.

In our example with the function \( f(x) = 2|x| \), after plotting the calculated points such as \((-3,6)\) and \((3,6)\), you connect these dots to form the graph. The use of the coordinate plane serves as a visual interpretation of how \( f(x) \) changes with \( x \), offering insights such as symmetry and slope.

Evaluating Functions

Evaluating functions involves finding the output \( f(x) \) for specific input values of \( x \). This is critical for understanding how different inputs affect the behavior of the function. Let's highlight essential steps for evaluation:

• Identify the function rule given. In our exercise, it is \( f(x) = 2|x| \).
• Substitute each chosen \( x \) value into the function to compute \( f(x) \).
• Consider the nature of the function's operations, such as absolute value, to accurately evaluate each input.

The exercise demonstrates this by using \( x = -3 \), finding \( f(-3) = 2|-3| = 6 \). Each \( x \) yields a different \( f(x) \) based on the magnitude of \( x \), doubling its absolute value. This method allows you to build a table of ordered pairs \( (x, f(x)) \), paving the way for graph construction and analysis.