Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{4}\) to sketch the graph of the function. $$f(x)=3-x^{4}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 3 - x^{4}\) is similar to the graph of \(y = x^{4}\), but it's shifted upwards by 3 units and reflected over the x-axis

Step 1: Graph original function

Start by graphing the function \(y = x^{4}\). This is a basic polynomial function, which looks like a sharp parabola. For small x-values, both positive and negative, y will be small, but will increase rapidly as \(|x|\) gets larger. The key feature of this graph is its bowl-like shape. It starts near the origin and expands upwards and outwards, but stays above the x-axis, reflecting its positive values.

Step 2: Apply vertical shift

The next step is to shift the graph of \(y = x^{4}\) upward by 3 units. This gives the graph of \(y = 3 + x^{4}\). This is because the value of '3' is being added to every y-coordinate of the original function.

Step 3: Apply reflection

The final step is to reflect this shifted graph over the x-axis. This gives the graph of \(y = 3 - x^{4}\). This is because the 'minus' sign in front of the \(x^{4}\) term means that every y-coordinate of the \(3 + x^{4}\) function is multiplied by -1.

Key Concepts

Polynomial Graph Transformations

Polynomial graph transformations involve changing the appearance of the graph of a polynomial function based on certain algebraic manipulations to its equation. These manipulations can include stretching or compressing the graph vertically or horizontally, reflecting it across an axis, translating (or shifting) it up, down, left, or right, and rotating about a point.

Understanding these transformations is crucial when graphing a new polynomial function based on a parent graph. For example, in the given exercise, the function to graph is related to the base function, y = x^4. The new function, f(x) = 3 - x^4, is derived by applying a combination of vertical shift and reflection across the x-axis to the original function's graph. By methodically applying each transformation in the correct order, one can accurately sketch the modified polynomial function.

Vertical Shifts in Graphs

Vertical shifts occur when every point on a graph is moved up or down by the same amount. This is a direct result of adding or subtracting a constant to the function's formula. In the context of the exercise, the function f(x) = x^4 undergoes a vertical shift upwards by 3 units, resulting in the new function y = 3 + x^4.

Visualizing a vertical shift is straightforward: imagine the entire graph being lifted or lowered along the y-axis without altering its shape. This shift does not affect the x-values, so the graph remains unaltered horizontally. It's important to note that a positive constant results in an upward shift, while a negative constant would lead to a downward shift. For the most effective learning, students should practice by taking a simple graph and applying vertical shifts of various magnitudes to see the resulting changes.

Reflection Across the X-axis

Reflecting a graph across the x-axis is a transformation that creates a mirror image of the original graph over the x-axis. Mathematically, this is achieved by multiplying the function's equation by -1. If you take the function y = 3 + x^4 from the vertical shift example and want to reflect it across the x-axis, you would change its equation to y = 3 - x^4.

This reflection will invert all the y-values of the points on the graph, while the x-values remain the same. The 'peaks' and 'valleys' of the graph simply flip over the x-axis. This transformation often changes the nature of the function's graph significantly, as it alters the direction of the function's increase and decrease. Students find it helpful to mark specific points on the original graph before reflecting to clearly see the effect of the reflection.