Problem 51 - 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 \([-4,6]\) by \([-4,4]\) $$ \begin{array}{ll}{\text { (a) } y=x^{6}} & {\text { (b) } y=\frac{1}{3} x^{6}} \\\ {\text { (c) } y=-\frac{1}{3} x^{6}} & {\text { (d) } y=-\frac{1}{3}(x-4)^{6}}\end{array} $$

Step-by-Step Solution

Answer

Each graph is a scaled, reflected, or shifted version of \( y = x^6 \).

Step 1: Graph Function (a)

Graph the function \( y = x^6 \) using the viewing rectangle \([-4, 6]\) for the \(x\)-axis and \([-4, 4]\) for the \(y\)-axis. This graph is a symmetric U-shaped curve because it is an even degree polynomial. The vertex of this curve is at the origin \((0,0)\).

Step 2: Graph Function (b)

Graph the function \( y = \frac{1}{3} x^6 \). This graph will have the same basic shape as \( y = x^6 \) but will be wider since it is vertically compressed by a factor of \frac{1}{3}. The vertex remains at the origin \((0,0)\).

Step 3: Graph Function (c)

Graph the function \( y = -\frac{1}{3} x^6 \). This graph is the reflection of \( y = \frac{1}{3} x^6 \) across the \(x\)-axis. It is an inverted U-shape with a vertex at the origin \((0,0)\).

Step 4: Graph Function (d)

Graph the function \( y = -\frac{1}{3}(x-4)^6 \). This function is a horizontally shifted version of \( y = -\frac{1}{3} x^6 \). The graph shifts to the right by 4 units, so the vertex is now at \((4,0)\). It maintains the same inverted U-shape.

Key Concepts

Even Degree Polynomials

Even degree polynomials, such as those involving the expression \( y = x^6 \), are characterized by their symmetrical, parabola-like shapes when graphed. The symmetry arises because the power of the polynomial is even, causing both ends of the graph to point in the same direction. In the equation \( y = x^6 \), the graph forms a U-shaped curve that is symmetric about the y-axis, and the vertex of this curve is positioned at the origin \((0,0)\). By contrast, odd degree polynomials look different as they display a varying pattern of direction on their ends.Here are some general properties of even degree polynomial functions:

• The leading coefficient affects how wide or narrow the graph appears.
• All graphs open either upwards or downwards completely, offering a neat visual symmetry.
• The vertex is often located at a significant point, typically the origin, unless the graph is transformed by shifts or reflections.

Vertical Compression

Vertical compression is a transformation that affects the steepness or width of a graph. Essentially, it makes the graph appear wider or shallower relative to the original curve. For example, in \( y = \frac{1}{3}x^6 \), the factor \( \frac{1}{3} \) compresses the graph, making it less steep compared to \( y = x^6 \). A smaller leading coefficient results in a graph that stretches away from the y-axis, thereby producing the effect of vertical compression.To interpret vertical compression do note:

• The value of the coefficient dictates the extent of the compression - the closer it is to zero, the broader the graph becomes.
• The vertex remains unaffected, staying at the same position even if the shape of the graph changes.
• It’s important to compare the compressed function with the original one to visually see the difference.

Horizontal Shifts

Horizontal shifts in polynomial functions involve the lateral movement of the graph along the x-axis. This transformation alters the graph’s position without changing its shape.Consider the function \( y = -\frac{1}{3}(x-4)^6 \). The term \( (x-4) \) indicates a horizontal shift to the right by 4 units. It's because the graph of the original function \( y = -\frac{1}{3}x^6 \) relocates its vertex from \((0,0)\) to \((4,0)\).Key points to remember about horizontal shifts are:

• A positive shift means moving to the right, whereas a negative result to a shift to the left.
• The shape remains constant, retaining all characteristics of the original graph.
• Use the shift formula \( x-h \), where \( h \) indicates how many units to move.

Function Transformations

Function transformations cover a wide range of adjustments that can be applied to a function’s graph, including vertical compressions, horizontal shifts, and reflections. These transformations modify the visual representation while preserving the fundamental nature of the function.Understanding function transformations is crucial in graphing polynomial functions:

• Vertical transformations involve stretching or compressing the graph, making it taller or flatter.
• Horizontal shifts change only the position, moving the graph along the x-axis.
• Reflections, like in \( y = -\frac{1}{3}x^6 \), flip the graph over a particular axis, usually resulting in an inverted shape.

By combining these transformations, you can adjust any graph to better fit the context or specific requirements, aiding in visual analysis and understanding.