Problem 22 - Algebra and Trigonometry

Question

(a) Sketch the graph of \(g(x)=\sqrt[3]{x}\) by plotting points. (b) Use the graph of \(g\) to sketch the graphs of the following functions. \(\begin{array}{ll}{\text { (i) } y=\sqrt[3]{x-2}} & {\text { (ii) } y=\sqrt[3]{x+2}+2} \\ {\text { (iii) } y=1-\sqrt[3]{x}} & {\text { (iv) } y=2 \sqrt[3]{x}}\end{array}\)

Step-by-Step Solution

Answer

All graphs are transformations of \( g(x) = \sqrt[3]{x} \) with various shifts and scalings.

Step 1: Plot the Graph of g(x)

Start by plotting the graph of the function \( g(x) = \sqrt[3]{x} \). Choose a set of points to evaluate, such as \( x = -8, -1, 0, 1, 8 \). Compute the corresponding \( g(x) \) values as follows: \( g(-8) = -2, \ g(-1) = -1, \ g(0) = 0, \ g(1) = 1, \ g(8) = 2 \). Plot these points: \((-8, -2), (-1, -1), (0, 0), (1, 1), (8, 2)\) and draw a smooth curve connecting them.

Step 2: Graph Transformation for y=\sqrt[3]{x-2}

Recognize that \( y = \sqrt[3]{x-2} \) is a horizontal shift of \( g(x) \) by 2 units to the right. Shift the points \((-8, -2), (-1, -1), (0, 0), (1, 1), (8, 2)\) to the right, resulting in the new points \((-6, -2), (1, -1), (2, 0), (3, 1), (10, 2)\), and redraw the curve.

Step 3: Graph Transformation for y=\sqrt[3]{x+2}+2

\( y = \sqrt[3]{x+2}+2 \) involves a horizontal shift to the left by 2 units and a vertical shift upward by 2 units. Shift the graph of \( g(x) \) left by 2 units, resulting in: \((-10, -2), (-3, -1), (-2, 0), (-1, 1), (6, 2)\), then shift these points up by 2 to obtain: \((-10, 0), (-3, 1), (-2, 2), (-1, 3), (6, 4)\), and draw this curve.

Step 4: Graph Transformation for y=1-\sqrt[3]{x}

The function \( y = 1 - \sqrt[3]{x} \) is derived by reflecting \( g(x) \) over the x-axis and shifting up by 1 unit. Reflect: \( (-8, 2), (-1, 1), (0, 0), (1, -1), (8, -2) \), then shift up: \( (-8, 3), (-1, 2), (0, 1), (1, 0), (8, -1) \). Plot these points and draw the curve.

Step 5: Graph Transformation for y=2\sqrt[3]{x}

For \( y = 2 \sqrt[3]{x} \), scale \( g(x) \) vertically by a factor of 2. Double the y-values: \( (-8, -4), (-1, -2), (0, 0), (1, 2), (8, 4) \). Plot these points and sketch the graph.

Key Concepts

Cube Root Function

A cube root function is a mathematical function of the form \( f(x) = \sqrt[3]{x} \), where the cube root of \( x \) is calculated. Unlike square roots, cube roots can yield both positive and negative results because every real number has a cube root in the real number system. This function has an interesting feature: it extends infinitely in both the positive and negative directions through the origin \(0,0\).

• It resembles an "S" curve through the origin.
• It is continuous and differentiable everywhere on the real number line.
• The curve is symmetric about the origin, which reflects its odd nature.

The domain of the cubic root function is all real numbers, \( (-\infty, +\infty) \), and so is the range. This characteristic offers great flexibility for modeling situations where negative inputs are possible.

Function Sketching

When sketching any function, first understand its basic characteristics. For the cube root function \( g(x) = \sqrt[3]{x} \), you start by selecting key values of \( x \) and determining their respective y-values. Plot these points on a graph, focusing on where the function changes rapidly, like around \( x = 0 \). Connect the points smoothly to capture the function's behavior faithfully.

• Choose points spanning a range of values, including negative, zero, and positive choices.
• Identify any symmetries or specific transformations to assist plotting.
• Keep the general shape of the curve in mind, especially the steep parts around the origin and the more gradual tails as \( x \) values become larger.

Visualizing these transformations helps in understanding how the graph changes for different values of the function. Practicing with different functions strengthens this vital skill in mathematics.

Horizontal Shift

Horizontal shifts affect where the graph of a function is positioned along the x-axis. If you have a cube root function like \( y = \sqrt[3]{x-2} \), it is equivalent to moving \( g(x) = \sqrt[3]{x} \) to the right by two units. For \( y = \sqrt[3]{x+2} + 2 \), the graph shifts left by two units.

• Remember: \( g(x-c) \) results in a shift to the right by \( c \) units.
• Conversely, \( g(x+c) \) will shift it to the left by \( c \) units.

To implement this transformation, adjust each x-coordinate of the plotted points accordingly, maintaining the original y-values. Then, redraw the curve, mindful of the shift, ensuring no change to the shape of the graph. Grasping horizontal shifts is crucial in graph transformations, as it relates directly to solving equations and modeling real-life processes.

Vertical Stretch

Vertical stretching involves modifying the \( y \)-coordinates of a function's graph, altering its steepness. Consider the function \( y = 2 \sqrt[3]{x} \); here, each \( y \)-value of the base function \( g(x) = \sqrt[3]{x} \) is multiplied by 2. This doubles the vertical distances from the x-axis, resulting in a steeper graph.

• A vertical stretch by a factor \( a \) is expressed as \( y = a \sqrt[3]{x} \).
• This multiplies each y-coordinate by \( a \), enhancing the slope.

It's important not to confuse a vertical stretch with a horizontal shift; the former influences the y-values rather than x-values. By mastering vertical stretches, one can more creatively adapt functions to various contexts, such as signal processing or engineering.