Problem 36 - Precalculus : Building Concepts and Connections

Question

Sketch the polynomial function using transformations. $$g(x)=-\frac{1}{2} x^{4}$$

Step-by-Step Solution

Answer

The graph of \( g(x) = -\frac{1}{2} x^{4} \) would mirror the graph of \( g(x) = x^{4} \) in the x-axis and be vertically stretched making it steeper compared to the original function.

Step 1: Sketch the initial function

First, sketch the graph of the original function \( g(x) = x^{4} \), which is a polynomial function with no transformations. This graph will start from zero and rise towards both ends.

Step 2: Apply Vertical Reflection

Since there is a negative sign on the function \( g(x) = -\frac{1}{2} x^{4} \), this signifies a vertical reflection, meaning the graph will be flipped over the x-axis. Where the graph of \( x^{4} \) was going up, the graph of \( -x^{4} \) will now be going down.

Step 3: Apply Vertical Stretch

The presence of \(-\frac{1}{2}\) also implies a vertical stretch. This means that for a given x-value, the y-value will be half of the y-value on the original graph. The graph appears to be pressed downwards as a result and the curve will be steeper than the original function.

Step 4: Final Combination of all transformations

Combine the transformations from steps 2 and 3. The graph of \( g(x) = -\frac{1}{2} x^{4} \) will mirror the graph of \( g(x) = x^{4} \) in the x-axis and be vertically stretched making it steeper compared to the original function.

Key Concepts

Graphing Polynomial Functions

Understanding how to graph polynomial functions requires an appreciation of some fundamental features of these functions. The function in the example, \( g(x) = x^4 \), belongs to the family of even-degree polynomials. Their distinct shape resembles a 'W' or 'M', and they share several characteristics. For instance, they have a minimum or maximum point at the origin and are symmetric about the y-axis.

When graphing an even-degree polynomial function like \( g(x) = x^4 \), you start by plotting the powers of x to get critical points. These points dictate the curvature of the graph. As the power increases, the graph becomes flatter at the origin and steeper as it moves away from the origin. In the absence of transformations, an even-degree polynomial function such as this will always rise to positive infinity on both ends, giving it a predictable 'end behavior'.

Graphing these functions manually begins with a table of values. Selecting x-values and then calculating the corresponding y-values will give you points through which the graph will pass. Carefully plotted points and a smooth curve that follows the calculated pattern will result in the accurate representation of the polynomial function.

Vertical Reflection

Vertical reflection is a transformation that alters the orientation of a graph across the x-axis. In the function \( g(x) = -\frac{1}{2} x^4 \), the negative sign in front of the function indicates that every point on the original graph will be reflected over the x-axis.

For any point with coordinates \( (x, y) \) on the original graph of \( x^4 \), its reflected image will have coordinates \( (x, -y) \). It's essential to note that this transformation doesn't affect the x-values; only the y-values are negated. As a result, the parts of the graph that were above the x-axis will now be below it, and vice versa. The overall shape remains consistent with the original, while the 'hills' and 'valleys' of the graph are inverted.

Vertical Stretch

A vertical stretch compresses or expands a graph in the y-direction. In our function \( g(x) = -\frac{1}{2} x^4 \), the coefficient of \( \frac{1}{2} \) signifies a vertical stretch by a factor of \frac{1}{2}. This transformation changes the steepness of the graph.

Let's imagine stretching or compressing the graph like it's made of rubber. A coefficient less than 1 (such as \( \frac{1}{2} \)) compresses the graph vertically, making it appear flatter. Conversely, a coefficient greater than 1 would stretch the graph away from the x-axis, making it taller or steeper. In the example provided, the vertical stretch causes the graph to widen, reducing the rate at which it rises or falls as it moves away from the origin. Effectively, this transformation affects the 'speed' or 'rate of change' of the y-values with respect to the x-values.