Problem 48 - A Graphical Approach to College Algebra

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x|-3$$

Step-by-Step Solution

Answer

Shift the graph of \( y = |x| \) downward by 3 units to sketch \( y = |x| - 3 \).

Step 1: Identify the Basic Function

The given function is related to one of the basic functions, which is the absolute value function. The basic function here is \( y = |x| \).

Step 2: Understand Translation

The function \( y = |x| - 3 \) involves a vertical translation of the basic graph \( y = |x| \). Specifically, it is shifted downward by 3 units.

Step 3: Graph the Basic Function

Start by sketching the graph of the basic function \( y = |x| \), which is a V-shaped graph with its vertex at the origin (0,0) and opening upwards.

Step 4: Apply Vertical Translation

Translate the graph of \( y = |x| \) downward by 3 units. This means you will move every point on the graph down 3 units. The new vertex of the graph will be at (0, -3).

Step 5: Sketch the Final Graph

Draw the new graph after the translation. It will look similar to the graph of \( y = |x| \), but with the vertex now at (0, -3) and still having the V shape.

Key Concepts

Understanding the Absolute Value Function

The absolute value function is an essential building block in mathematics, especially when dealing with graph transformations. At its core, this function is represented by the expression \( y = |x| \), which creates a distinctive V-shaped graph.
- **V-Shape**: The graph consists of two linear pieces that meet at the vertex (the "V" point). It heads upwards symmetrically to the left and right from the origin (0,0).- **Behavior**: For any input \( x \), the output \( y \) is always non-negative. This is because the absolute value of a number is its distance from zero, meaning it strips away any negative signs.
The basic absolute value graph is pivotal for observing how changes to the function, like shifts and other transformations, affect its shape and position on the Cartesian plane.
Knowing the general shape and position of the absolute value graph helps in understanding more complex transformations.

Exploring Vertical Translation

Vertical translation involves shifting a graph up or down along the y-axis without altering its original shape. In our function \( y = |x| - 3 \), the "minus 3" tells us that we need to perform a vertical translation.
- **Downward Shift**: The negative sign indicates that every point on the graph of \( y = |x| \) should be moved 3 units downward.- **Effect on Vertex**: Originally, the vertex of \( y = |x| \) is at (0,0). With a downward shift of 3 units, the new vertex will end up at (0,-3).
Understanding the mechanics of vertical translations is crucial in graph transformations, providing a straightforward method to visualize how an entire graph can be repositioned vertically, thus altering its placement on the graph without changing its inherent structural identity.

The Basics of Graph Sketching

Graph sketching is an invaluable skill in mathematics, especially when it comes to visualizing functions and their transformations. When sketching the graph of \( y = |x| \) and its transformations, here's what you should keep in mind:

• Start by sketching the basic graph, which is \( y = |x| \). Mark the vertex at (0,0) and ensure the V shape is clear, with lines extending symmetrically upwards.
• Apply any transformations systematically. For \( y = |x| - 3 \), this involves shifting every point on the graph down by 3 units. Mark the new vertex position at (0,-3).
• Check symmetry after transformations to ensure consistency with the original graph shape.

Graph sketching by hand requires attention to detail and a clear understanding of the transformations applied. By breaking the process into manageable steps, you can accurately present how graphs translate across the plane, a skill beneficial not only in academics but also in practical applications of mathematics.