Problem 31 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=|x-3|\\\&g(x)=-|x-3|\end{aligned}$$

Step-by-Step Solution

Answer

In summary, the graph of the given functions \(f(x) = |x-3|\) and \(g(x) = -|x-3|\) can be obtained using transformation techniques. Start by sketching the parent function \(y = |x|\). Then, for \(f(x)\), apply a horizontal shift of 3 units to the right, resulting in a v-shaped graph opening upwards with its vertex at \((3, 0)\). For \(g(x)\), apply both a horizontal shift of 3 units to the right and a vertical reflection about the x-axis. This results in a v-shaped graph opening downwards with the vertex at \((3, 0)\). Plot both graphs on the same coordinate plane to see their intersection at \((3, 0)\).

Step 1: Graph the parent function \(y = |x|\)

Begin by sketching the graph of the parent function, \(y = |x|\). This is a simple v-shaped graph, which has its vertex at the origin and opens upwards.

Step 2: Apply transformations for \(f(x) = |x-3|\)

Now we need to graph \(f(x) = |x-3|\). This function can be obtained from the parent function \(y = |x|\) by applying a horizontal shift of 3 units to the right. This means that the new vertex of the graph of \(f(x)\) will be at the point \((3, 0)\). The graph of \(f(x)\) will also be a v-shaped graph that opens upwards, with the vertex at \((3, 0)\).

Step 3: Apply transformations for \(g(x) = -|x-3|\)

To graph \(g(x) = -|x-3|\), we apply two transformations to the parent function \(y = |x|\). First, we apply the same horizontal shift of 3 units to the right, which moves the vertex to the point \((3, 0)\). Then, we apply a vertical reflection about the x-axis since the entire function is multiplied by -1. This causes the graph of\(g(x)\) to open downwards, with the vertex still at \((3, 0)\).

Step 4: Combine all the graphs into one coordinate plane

Finally, plot both the graphs of \(f(x) = |x-3|\) and \(g(x) = -|x-3|\) on the same coordinate plane. You should see two v-shaped graphs intersecting at the point \((3, 0)\). The graph of \(f(x)\) opens upwards, and the graph of \(g(x)\) opens downwards.

Key Concepts

Graph Transformations

When we talk about graph transformations, we're referring to the shifts and reflections applied to a parent function to alter its appearance. Transformations can change the position, shape, and orientation of the graph. Generically, these include:

• Horizontal shifts, where the graph moves left or right.
• Vertical shifts, where the graph moves up or down.
• Reflections, where the graph flips over an axis.
• Stretches and compressions, which change the graph's size.

For absolute value functions, these transformations create significant visual changes, turning them into more complex forms from their basic 'V' shape. Understanding each transformation individually helps in sketching transformed graphs accurately.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right. For the function \( f(x) = |x-3| \), a horizontal shift has occurred. The parent function is \( y = |x| \), which has its vertex at the origin \((0, 0)\).

By writing \( f(x) = |x-3| \), a horizontal shift of 3 units to the right is evident. This shift occurs because we replace \(x\) with \(x-3\). Thus, instead of the vertex being at \((0, 0)\), it moves to \((3, 0)\).

This straightforward calculation makes drawing and understanding the graph's new position far simpler.

Vertical Reflection

A vertical reflection flips a graph over the x-axis. This transformation can be seen in the function \( g(x) = -|x-3| \).

The negative sign in front of the absolute value indicates a reflection. The parent function \( y = |x| \) opens upwards, like a regular 'V'. When multiplied by -1, the graph turns to face downwards, reversing its direction. The graph still retains its vertex at \((3, 0)\) due to the horizontal shift discussed earlier.

Visualizing this helps in understanding how simple algebraic changes affect a function's graph dramatically.

V-shaped Graphs

V-shaped graphs are characteristic of absolute value functions like \( y = |x| \). These graphs are symmetric and feature a vertex, from which two linear parts extend.

The 'V' shape arises because the absolute value of a number is always non-negative, causing the graph to bounce back upward at points where regular functions might dip below the x-axis.

For the given functions, \( f(x) = |x-3| \) and \( g(x) = -|x-3| \), both maintain the 'V' shape but with their own unique orientations based on added transformations. Understanding this fundamental form makes interpreting transformations much simpler.