Problem 95 - A Graphical Approach to College Algebra

Question

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt[3]{27 x+54}-5$$

Step-by-Step Solution

Answer

Compress, shift left 2 units, shift down 5 units.

Step 1: Identify the Base Function

The given function is related to the cube root function, which has the form \( y = \sqrt[3]{x} \). We'll use transformations to modify this base function to match the given function \( y = \sqrt[3]{27x+54} - 5 \).

Step 2: Handle the Horizontal Stretch/Compression

The expression inside the cube root, \( 27x \), indicates a horizontal compression. Normally, \( x \) would be transformed by a factor of \( \frac{1}{a} \) if the form is \( \sqrt[3]{ax} \). Here, \( a = 27 \), so the transformation compresses the graph by a factor of \( \frac{1}{27} \).

Step 3: Handle the Horizontal Shift

The expression inside the cube root function is \( 27x + 54 \). This can be rewritten as \( 27(x + 2) \), indicating a horizontal shift. The graph is shifted 2 units to the left because of the \( x + 2 \) term inside the cube root.

Step 4: Handle the Vertical Shift

The transformation outside the cube root is \( -5 \), which shifts the graph vertically by 5 units down. This is a straightforward vertical shift affecting the output of the function.

Step 5: Combine All Transformations

By combining all the identified transformations, the graph of \( y = \sqrt[3]{27x+54} - 5 \) can be obtained by first compressing the standard cube root graph by a factor of 27, shifting it 2 units to the left, and then moving it 5 units down.

Key Concepts

Horizontal Compression

Horizontal compression modifies the width of a graph by squeezing it closer to the y-axis. This occurs when you multiply the variable inside a function by a factor greater than 1. In this exercise, the term inside the cube root, \( 27x \), creates a horizontal compression.

• The basic cube root function is \( y = \sqrt[3]{x} \).
• In \( y = \sqrt[3]{27x} \), the 27 compresses the graph horizontally by a factor of 1/27.

The larger the factor inside the function, the tighter the graph squeezes together. Think of it like pinching a piece of paper from the sides; the edges come closer, making it narrow.

Horizontal Shift

Horizontal shifts move the graph left or right along the x-axis. The direction of the shift depends on the sign inside the function. This function, \( y = \sqrt[3]{27(x + 2)} \), includes a shift.

• The \( x + 2 \) means the graph shifts left 2 units.

Remember, if the formula inside seems backwards (like \( x + 2 \)), it shifts in the opposite direction of what you might initially think. Outside of math terms, imagine walking two steps to the left after squeezing sideways through a door. That's the role horizontal shift plays here.

Vertical Shift

Vertical shifts move a graph up or down along the y-axis. When you add or subtract a constant from the function, you're shifting vertically. Here, the \( -5 \) in \( y = \sqrt[3]{27x+54} - 5 \) shifts the graph downward.

• This is straightforward: subtracting 5 moves it down by 5 units.

Envision this like adjusting a guitar's pitch; you simply tune it lower. It's an easy way to change the graph's height without affecting its shape.

Cube Root Function

The cube root function is a fundamental transformation in algebra. It differs from quadratic or square root functions as it handles all real numbers. The standard cube root function is \( y = \sqrt[3]{x} \).

• Its graph is symmetric in relation to the origin.
• It crosses the origin at (0, 0).

The cube root function increases more gradually than quadratic functions. It adjusts gently along the x-axis, accommodating negative and positive inputs smoothly. Imagine it like a gentle wave crossing through zero, with transformations like those in this exercise compressing, shifting, and altering its path.