Problem 100 - College Algebra Essentials

Question

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$h(x)=-(x-2)^{3}$$

Step-by-Step Solution

Answer

The transformed graph of the given function \(h(x)=- (x-2)^3\) is obtained by shifting the graph of the parent function \(f(x)=x^3\) towards the right by 2 units and then reflecting it over the x-axis.

Step 1: Draw the original function

Start by sketching a quick graph of the parent function \(f(x)=x^3\). For this function, when \(x\) is less than zero then the value of \(y\) will be negative and when \(x\) is more than zero, the value of \(y\) will be positive. In particular, note down the points \((-2,-8)\), \((-1,-1)\), \((0,0)\), \((1,1)\) and \((2,8)\). The curve will pass through these points.

Step 2: Apply the horizontal shift

The function \(h(x)\) has \((x-2)\) in place of \(x\), which represents a horizontal shift of 2 units to the right. To achieve this, add 2 to the \(x\)-coordinate of every point in the original function. This means the points become \((0,-8)\), \((1,-1)\), \((2,0)\), \((3,1)\) and \((4,8)\). Plot these on the graph.

Step 3: Apply the vertical reflection

The presence of the '-' sign before \(x-2)^3\) represents a reflection over the x-axis. Reflect each of the points from step 2 over the x-axis, i.e., change the sign of the \(y\)-coordinate. The points now become \((0,8)\), \((1,1)\), \((2,0)\), \((3,-1)\) and \((4,-8)\). Plot these on your graph to obtain the graph of the given function.

Key Concepts

Cubic Function

A cubic function is defined by an equation of the form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants, and \(a eq 0\). The simplest cubic function is \(f(x) = x^3\). It is a fundamental example in algebra, showcasing its characteristic shape known as a cubic curve.

The graph of a cubic function exhibits both positive and negative values, reflecting its ability to cross the x-axis up to three times. This curve has a distinctive "S" shape, bending through the origin. Key points on the graph, such as \((-2, -8)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), and \((2, 8)\), help define the contour of the function.

Unlike linear or quadratic functions, cubic functions do not have a line of symmetry. This makes them unique, as the curve is neither symmetric around the y-axis (like parabolas) nor does it have a consistent slope (like lines). Understanding the basic shape of \(x^3\) is crucial for applying transformations effectively.

Transformations

Transformations alter the appearance of a graph without changing its fundamental nature. They allow us to move, reflect, stretch, or compress graphs in various ways. The function \(h(x) = -(x-2)^3\) is derived from \(f(x)=x^3\) through a series of transformations.

Key transformations include:

• Translation: Moves the entire graph horizontally or vertically.
• Reflection: Flips the graph over a line, such as the x-axis or y-axis.
• Stretching and Compressing: Alters the steepness or width of the graph.

In our example, the graph undergoes a horizontal shift to the right and a reflection over the x-axis. These transformations are applied systematically, first adjusting the position, and then reflecting to obtain the final graph.

Horizontal Shift

A horizontal shift moves a graph left or right along the x-axis without altering its shape. This transformation is achieved by modifying the function's formula, typically through replacing \(x\) with \(x-k\) (shift to the right) or \(x+k\) (shift to the left), where \(k\) is a constant.

In the function \(h(x) = -(x-2)^3\), the expression \(x-2\) indicates a shift of 2 units to the right. To implement this:

• Add 2 to each x-coordinate of the key points in the original cubic function \(f(x) = x^3\).
• The point \((-2, -8)\) shifts to \((0, -8)\).
• Similarly, adjust all other points accordingly to form new x-coordinates.

Plots these revised points to observe the horizontal shift on the graph. This step is crucial because it repositions the baseline of the cubic function.

Reflection Over the x-axis

A reflection over the x-axis flips the graph upside down. This means that for each point on the function, the y-coordinate is multiplied by -1.

For \(h(x) = -(x-2)^3\), the negative sign outside the cube operation creates this reflection. To apply this:

• Take each of the y-coordinates from the shifted function and change them into their negative counterparts.
• For instance, the point \((0, -8)\) becomes \((0, 8)\).
• Repeat the process for all key points.

These transformations result in a complete inversion of the graph's orientation. By reflecting over the x-axis, the graph's peaks become troughs and vice versa, providing a mirror image along the x-axis. This ensures the final function graph aligns with the transformed equation exactly.