Problem 64 - Elementary and Intermediate Algebra

Question

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function \(g\) using translations and/or a reflection. See Examples 7 and \(8 .\) $$ g(x)=-|x|-4 $$

Step-by-Step Solution

Answer

The graph of \( g(x) = -|x| - 4 \) is an upside-down V-shape shifted 4 units down, with vertex at (0,-4).

Step 1: Identify the Parent Function

The function given is \(g(x) = -|x| - 4\). First, we identify the parent function. The parent function in this case is \(f(x) = |x|\), which represents the absolute value function.

Step 2: Sketch the Parent Function

Sketch the graph of the parent function \(f(x) = |x|\). This graph is a V-shaped graph with its vertex at the origin \((0, 0)\) and opens upwards, having straight lines passing through the points \((1, 1)\) and \((-1, 1)\).

Step 3: Reflect the Parent Function Across the x-axis

To graph \( -|x| \), reflect the graph of \( |x| \) across the x-axis. This flips the V-shape upside down, resulting in a graph that opens downwards with its vertex still at the origin.

Step 4: Translate the Graph Vertically

The transformation \( -|x| - 4\) involves translating the graph of \( -|x| \) vertically downward by 4 units. Take the vertex from \((0, 0)\) to \((0, -4)\) and move all points of the graph down by 4 units.

Step 5: Draw the Final Graph

Combine these transformations to draw the final graph of \( g(x) = -|x| - 4 \). It is a downward-opening V-shaped graph, centered at \((0, -4)\). Each arm of the V extends at a 45-degree angle downwards.

Key Concepts

Understanding Parent Functions

In mathematics, a parent function is the simplest form of a set of functions that share common characteristics. Think of it as the most basic version of functions in a particular family. A common example of a parent function is the absolute value function, represented as \( f(x) = |x| \). When you graph \( |x| \), it forms a V-shape with its lowest point, called the vertex, located at the origin \( (0, 0) \).

• The graph of the absolute value function is symmetric along the y-axis.
• It passes through points like \( (1, 1) \) and \( (-1, 1) \).
• This V-shape visually represents that the output is always non-negative, no matter the input value of \( x \).

Parent functions play a crucial role because they serve as the starting point for understanding more complex graphs, allowing us to see how different transformations shift or reshape them.

Reflection Across the X-Axis

A reflection across the x-axis is one of the basic transformations you can apply to parent functions. To reflect a function across the x-axis, you multiply each output by \(-1\).

• For \( f(x) = |x| \), reflecting it across the x-axis gives you \( -|x| \).
• The resulting graph flips the V-shape upside down.
• Thus, the vertex remains at the origin \((0, 0)\), but the V now opens downward.

Reflections like this are useful for understanding inverse relationships and symmetry in functions. In a graph, it helps to visualize how each value of the function is affected by multiplying with \(-1\). It effectively shows the negative equivalent of the original values, like seeing a mirror image below the x-axis.

Exploring Vertical Translation

Vertical translation is a transformation that shifts the entire graph of a function up or down. This is achieved by adding or subtracting a constant value from the function. In our example with \( g(x) = -|x| - 4 \), the graph of the reflection \( -|x| \) is translated vertically downward by 4 units.

• Each point on the graph is moved directly down by the same distance.
• The vertex shifts from \((0, 0)\) to \((0, -4)\). It is now centered at a new lower point along the y-axis.
• Vertical translation affects the range of function values, adjusting where they begin and end relative to the y-axis.

Understanding vertical translation helps in graphing functions quickly. It's particularly helpful in determining the new position of the graph after it has been moved along the y-axis, making it easier to analyze and predict the behavior of functions.