Problem 56 - Algebra 2

Question

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{25 x-100}-1\)

Step-by-Step Solution

Answer

The function \(y = \sqrt{25x - 100} - 1\) can be derived from the parent function \(y = \sqrt{x}\) by applying a horizontal compression by factor 25, a horizontal shift to the right by 4 units, and a vertical shift down by 1 unit.

Step 1: Identify the parent function and the transformations

The parent function is \(y = \sqrt{x}\). The given function \(y = \sqrt{25 x - 100} - 1\) can be written in the form \(y = a \sqrt{b(x - h)} + k\), where: - a is the vertical stretch or compression factor and is 1 in this case (no vertical stretch or compression)- b is the horizontal stretch or compression factor and it is 25.- h is the horizontal shift. Since the form is \(b(x - h)\), and in given function, the term inside the square root is 25x - 100, we can rewrite it as \(25(x - 4)\), so h = 4 (meaning the function will be shifted 4 units to the right).- k is the vertical shift, -1 (meaning the function will be shifted 1 unit down).

Step 2: Graph the transformed function

Start with the parent function \(y = \sqrt{x}\), which starts at the origin (0,0) and increases as you move to the right. Now apply the transformations to this graph:- Compress it horizontally by a factor of 25. This means the graph will be 'narrower'.- Move the graph 4 units to the right (due to the +4 inside the square root).- Move the graph 1 unit down (due to the -1 outside the square root).

Key Concepts

Parent Function

In the realm of function transformations, identifying the parent function is a crucial first step. The parent function is the simplest form of a function without any transformations applied to it. In this exercise, the parent function is the square root function, represented as \(y = \sqrt{x}\). This function graphically resembles a smooth curve starting at the origin—(0, 0)—and extending to the right.
The parent function provides a base shape and behavior that we will manipulate using various transformations like shifts and stretches. Understanding how the parent function behaves initially will help you predict how transformations will affect the graph's appearance.

Square Root Function

The square root function is characterized by its unique curve, forming one half of a parabola on its side. Mathematically expressed as \(y = \sqrt{x}\), this function only exists where \(x \geq 0\), as square roots of negative numbers are not real numbers within this context.
Graphically, the function starts at point (0,0) and rises gradually as x values increase. This slow and steady rise reflects the nature of square roots and how they increase at a decreasing rate.

• The square root function is continuous for \(x \geq 0\).
• Its domain is all real numbers greater than or equal to zero.
• The range is also all real numbers greater than or equal to zero.

The square root function is vital to understand because it acts as the foundation for any transformation we apply in function editing.

Horizontal Shift

A horizontal shift changes the position of the graph of a function along the x-axis. It includes moving the graph left or right without altering its shape. In our specific function \(y = \sqrt{25(x - 4)} - 1\), the horizontal shift occurs due to the expression \((x-4)\) inside the square root.
This indicates a shift to the right by 4 units. In general, for a function in the form \(y = \sqrt{b(x-h)}\):

• If \(h > 0\), the graph shifts \(h\) units to the right.
• If \(h < 0\), the graph shifts \(|h|\) units to the left.

This transformation enables us to move the graph along the horizontal plane, facilitating better visualization and positioning of the function within a coordinate system.

Vertical Shift

Vertical shifts allow functions to be moved up or down along the y-axis, maintaining the graph's shape but adjusting its height. In the given function \(y = \sqrt{25(x - 4)} - 1\), the \(-1\) causes a vertical shift downwards by 1 unit.
This is part of the transformation process where:

• If \(k > 0\), the graph shifts upwards.
• If \(k < 0\), the graph shifts downwards by \(|k|\) units.

Vertical shifts are helpful for adjusting the baseline of the function to match the desired or required visual layout. Understanding both horizontal and vertical shifts enhances how one manipulates the base position of any given graph.