Problem 40 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=|4-x| $$

Step-by-Step Solution

Answer

The x-intercept is (4, 0), the y-intercept is (0, 4), and the graph is not symmetric.

Step 1: Understanding the Function

The given function is in the form of an absolute value: \( y = |4-x| \). This expression means that the output \( y \) is always non-negative since it represents the distance of \( 4-x \) from zero on the number line.

Step 2: Making a Table of Values

To understand the behavior of the function, let's make a table of values by choosing a few values of \( x \). | \( x \) | \( y = |4-x| \) ||--------|-----------------|| 0 | 4 || 2 | 2 || 4 | 0 || 6 | 2 || 8 | 4 |These points will help us sketch the graph.

Step 3: Plotting the Graph

Using the table of values, we can plot the corresponding points on a coordinate plane: (0, 4), (2, 2), (4, 0), (6, 2), and (8, 4). Connect these points with straight lines, knowing that the graph of an absolute value function is typically 'V' shaped with its vertex at the point where the expression inside the absolute value equals zero.

Step 4: Finding x-intercept

An \( x \)-intercept occurs where \( y = 0 \). Set the function's equation to zero: \[ |4-x| = 0 \]This implies:\[ 4-x = 0 \]Solving for \( x \) gives \( x = 4 \). So, the \( x \)-intercept is the point (4, 0).

Step 5: Finding y-intercept

A \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function:\[ y = |4-0| = 4 \]So, the \( y \)-intercept is the point (0, 4).

Step 6: Testing for Symmetry

To test for symmetry, we consider reflection over the y-axis. Check if the function satisfies \( f(-x) = f(x) \):\[ f(x) = |4-x| \quad ext{and} \quad f(-x) = |4+x| \]Since \(|4-x|\) is not equivalent to \(|4+x|\), the function is not symmetric about the y-axis. Check origin symmetry with \(-f(x) = f(x)\). The function does not satisfy this, either, confirming no symmetry about the origin.

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics, which helps to visualize how a function behaves. In the case of an absolute value function like \( y = |4-x| \), the graph typically forms a "V" shape. It has a vertex, which is the point where the expression inside the absolute value equals zero.
To begin graphing this function, one should first create a table of values. This involves choosing a variety of \( x \) values and calculating the corresponding \( y \) values. For example:

• For \( x = 0 \), \( y = |4-0| = 4 \).
• For \( x = 2 \), \( y = |4-2| = 2 \).
• For \( x = 4 \), \( y = |4-4| = 0 \).
• For \( x = 6 \), \( y = |4-6| = 2 \).
• For \( x = 8 \), \( y = |4-8| = 4 \).

These points, when plotted, will help form the graph. Remember, the shape of the graph is a result of the absolute value, which makes every output non-negative.

X-Intercepts

The x-intercepts of a function are the points where the graph crosses the x-axis. These are crucial for understanding where the function equals zero. For the function\( y = |4-x| \), the x-intercept occurs where \( y = 0 \). This means solving the equation:\[ |4-x| = 0 \]The absolute value equation implies:\[ 4-x = 0 \]Solving for \( x \) results in \( x = 4 \).
Therefore, the x-intercept of this function is at the point \( (4, 0) \). This tells us where the graph of the function touches or crosses the x-axis, making it a vital part of the graph's structure.

Y-Intercepts

Y-intercepts are where the graph intersects the y-axis, providing a point where the value of the function can be directly visualized when \( x = 0 \). For \( y = |4-x| \), to find the y-intercept, we substitute \( x = 0 \) into the function:\[ y = |4-0| = 4 \]
Thus, the y-intercept is the point \( (0, 4) \). This tells us that when \( x \) is zero, the output or \( y \)-value is 4. Finding these intercepts simplifies understanding where the function begins its graph on the corresponding axes.

Symmetry Testing

Symmetry testing involves checking if a function is symmetric regarding either the y-axis, the x-axis, or the origin. A function is y-axis symmetric if replacing \( x \) with \( -x \) results in an unchanged function. For our function, \( y = |4-x| \): \[ f(x) = |4-x| \quad \text{and} \quad f(-x) = |4+x| \]
Since \( |4-x| eq |4+x| \), there is no symmetry about the y-axis. As for origin symmetry, its condition is where \(-f(x) = f(x)\). Checking, we find it does not satisfy this condition either:\[ -|4-x| eq |4-x| \]
Thus, the function is not symmetric about the origin or the y-axis. Identifying symmetry aids in predicting the behavior of the graph across various sections with minimal calculations.