Problem 69 - Algebra and Trigonometry

Question

\(69-72\) . 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

Graphs (b), (c), and (d) are transformations of graph (a). (b) shifts horizontally, (c) stretches vertically, and (d) also shifts vertically.

Step 1: Understand the Function

The function in part (a) is a fourth root function, expressed as \(y = \sqrt[4]{x}\). The fourth root function typically starts at the origin (0,0) and increases to the right in a curve.

Step 2: Plot the Basic Graph

Sketch the graph of \(y = \sqrt[4]{x}\) within the viewing rectangle \([-8, 8]\) by \([-2, 8]\). This curve will only exist for non-negative \(x\) values, starting at the origin.

Step 3: Graph Transformation for (b)

The function in part (b) is \(y = \sqrt[4]{x+5}\). This is a horizontal shift of the graph from part (a) to the left by 5 units. Plot this graph within the same viewing rectangle.

Step 4: Graph Transformation for (c)

The function in part (c) is \(y = 2\sqrt[4]{x+5}\). This graph results from a vertical stretch by a factor of 2 of the graph from part (b). Plot this transformed curve in the rectangle.

Step 5: Graph Transformation for (d)

The function in part (d) is \(y = 4 + 2\sqrt[4]{x+5}\). This graph is a vertical translation of the function in part (c) up by 4 units. Draw this new graph in the provided rectangle.

Step 6: Analyze the Graph Changes

Each function is a transformation of the function in part (a). The function in (b) is a horizontal shift; (c) is both a horizontal shift and vertical stretch; (d) combines a shift, stretch, and a vertical translation upward.

Key Concepts

Fourth Root Function

The fourth root function involves finding the value that, when raised to the power of four, gives the original number. In mathematical terms, the fourth root of a number \( x \) is represented as \( \sqrt[4]{x} \). This function has a specific pattern on its graph, starting from the origin \((0,0)\) and moving upwards to the right.
This behavior occurs because the output of the function is only defined for non-negative values of \( x \) (i.e., \( x \geq 0 \)).
The graph for this function lies in the first quadrant, as it mirrors the function's domain.

• The curve is gently increasing, indicating that the rise in y-values is relatively moderate.
• As x-values increase, the rate of change in y-values slightly diminishes.

Understanding this base graph is crucial as it sets the stage for recognizing and executing transformations.

Horizontal Shift

A horizontal shift occurs when a graph is moved left or right along the x-axis. For the fourth root function, this means modifying the function with a term inside the root, such as \(y = \sqrt[4]{x + 5}\).
This specific modification results in a shift to the left by 5 units.

• The term \(+5\) indicates that every point on the original graph \( y = \sqrt[4]{x} \) moves to the left by 5 units.
• This shift does not affect the shape of the graph, only its position on the x-axis.

If marked by a negative on the inside, such as \(-5\), the graph would instead shift to the right.

Vertical Stretch

A vertical stretch involves multiplying the entire function by a constant greater than one. This stretch makes the graph taller. For \( y = 2 \sqrt[4]{x+5} \), the factor 2 applies a vertical stretch to the function.

• The distances between points on the graph and the x-axis double, which means for every x-value, the output y-value is twice as large compared to the original function without the stretch.
• This transformation changes the steepness of the graph but retains its directional flow and symmetry.

The impact of a vertical stretch is easily observed in the elongation along the y-axis, increasing the vertical distance between y-values for a given x.

Vertical Translation

Vertical translation consists of shifting the graph up or down along the y-axis. In the function \( y = 4 + 2 \sqrt[4]{x+5} \), the graph is shifted upward by 4 units.

• The \(+4\) alters the starting point for the transformed graph, moving each point on the previous graph \(y = 2\sqrt[4]{x+5}\) 4 units higher.
• This change uniformly translates the entire graph, maintaining its shape and horizontal position while lifting it along the y-axis.

By integrating both a vertical stretch and translation, the resulting function portrays different heights and placements compared to the initial fourth root curve, demonstrating how both position and scale are modified in function transformations.