Problem 49 - Algebra and Trigonometry

Question

49–52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8,8]\) by \([-2,8]\) $$ \begin{array}{ll}{\text { (a) } y=\sqrt[4]{x}} & {\text { (b) } y=\sqrt[4]{x+5}} \\ {\text { (c) } y=2 \sqrt[4]{x+5}} & {\text { (d) } y=4+2 \sqrt[4]{x+5}}\end{array} $$

Step-by-Step Solution

Answer

Each subsequent graph is a transformation of the previous, involving translations and stretches.

Step 1: Graph the function in part (a)

Graph the function \( y = \sqrt[4]{x} \) using the viewing rectangle \([-8, 8]\) by \([-2, 8]\) on the coordinate plane. This curve should start from the origin and increase slowly, showing a typical fourth root curve to the right.

Step 2: Graph the function in part (b)

For the function \( y = \sqrt[4]{x+5} \), plot it in the same viewing rectangle. This graph is a horizontal translation 5 units to the left of the graph in part (a). The starting point now is at \( x = -5 \).

Step 3: Graph the function in part (c)

Graph \( y = 2\sqrt[4]{x+5} \). This is a vertical stretch of the graph in part (b) by a factor of 2. The curve will rise quicker but starts from the same point \( x = -5 \).

Step 4: Graph the function in part (d)

For \( y = 4 + 2 \sqrt[4]{x+5} \), graph this as a vertical translation of part (c). The entire graph of part (c) is shifted 4 units up, starting now from \( y = 4 \) at \( x = -5 \).

Key Concepts

Fourth root function

The fourth root function is a type of radical function represented by \( y = \sqrt[4]{x} \). This means that we are finding a number which, when multiplied by itself four times, equals \( x \). Fourth root functions generally exhibit a particular curve that begins at the origin and increases slowly if you consider positive values of \( x \).

• For values of \( x \) greater than zero, \( y \) increases, showing a gentle upward curve.
• The function is defined only for non-negative numbers since even roots of negative numbers are not real in a basic level context.

Fourth root graphs are typically mundane in the way they look like they crawl along the x-axis at first, gradually steepening as \( x \) increases. Understanding this characteristic is essential when you transform these functions algebraically or graphically.

Horizontal translation

Horizontal translation involves shifting a graph left or right along the x-axis. For the function \( y = \sqrt[4]{x+5} \), a horizontal translation is applied.

• If \( h \) is positive, the function \( y = \sqrt[4]{x-h} \) shifts \( h \) units to the right.
• In our case, \( y = \sqrt[4]{x+5} \) shifts the graph 5 units to the left.

You can imagine grasping the graph and simply moving it without altering its shape. This manipulation on the x-axis can significantly affect where the function starts (here at \( x = -5 \)), without altering its general shape. Understanding this allows for better control and prediction of changes in function behavior.

Vertical stretch

A vertical stretch happens when we multiply a function by a factor greater than one. This affects how quickly the values of \( y \) increase as \( x \) moves away from the origin.For the function \( y = 2\sqrt[4]{x+5} \), the original graph from \( y = \sqrt[4]{x+5} \) is vertically stretched by a factor of 2.

• Instead of plodding neatly upwards, the curve rises more sharply.
• This makes the function appear taller and steeper, amplifying its impact.

Visualizing vertical stretches helps in comparing and contrasting different functions on the same axes, allowing a clearer view of their relative changes and structures.

Vertical translation

Vertical translation is a shift of a graph up or down along the y-axis. This alters the starting height of the function without changing its shape.For \( y = 4 + 2\sqrt[4]{x+5} \), the graph from the previous step is lifted 4 units upwards.

• The function no longer starts at \( y = 0 \), but at \( y = 4 \) when \( x = -5 \).
• This move does not change the rate at which \( y \) increases as \( x \) does, yet repositions the entire curve.

Grasping vertical translation can particularly aid in real-world scenarios where initial values of a function need adjustments. It's like elevating a curve but maintaining its inherent nature.