Question

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

Short answer

The graph of the function \(f(x) = x^4 - 4\) is the same shape as that of \(y = x^4\), but it is shifted downwards by 4 units.

Step-by-step solution

Understanding the function transformation

The given function \(f(x)=x^{4}-4\) is a transformation of the base function \(y=x^{4}\). The transformation involves a vertical shift of 4 units down. This is seen by the '-4' which is subtracted from \(x^{4}\) in the function \(f(x)=x^{4}-4\).

Plotting the original graph

Start by drawing the original graph \(y=x^{4}\). The graph of \(y=x^{4}\) is symmetrical about the y-axis and will rise steeply for both negative and positive x-values. It also touches the x-axis at the origin (0,0).

Plotting the transformed graph

Having plotted \(y=x^{4}\), to draw the given function \(f(x)=x^{4}-4\), visualize shifting the graph of \(y=x^{4}\) 4 units down. This shifted graph represents the function \(f(x)=x^{4}-4\). The steepness of the curve remains unchanged, but now the point where the graph touches the x-axis, which was at the origin (0,0), will now be at (0,-4).

Key concepts

Vertical Shift

When dealing with function transformations, a vertical shift happens when we add or subtract a constant from the function's output. For example, in the function \( y = x^4 - 4 \), the part "- 4" indicates a vertical shift. This means that we adjust the entire graph of \( y = x^4 \) downward by 4 units.
This transformation is simple: each point on the original graph moves down to a new position. Similarly, if a constant were added (like \( y = x^4 + 4 \)), the graph would shift up by 4 units, demonstrating a positive vertical shift.
Vertical shifts do not change the shape of the graph. The only effect they have is on the position, moving the graph either up or down depending on the signed constant.

Graph of Polynomial

Polynomials are functions that include various powers of \(x\). The simplest form is a monomial like \( y = x^n \). In our example, we specifically deal with a polynomial \( y = x^4 \), which is known as a quartic function.
The graph of \( y = x^4 \) has unique properties:

• It is steep and rises sharply as you move away from the origin, both to the left (negative \(x\)) and to the right (positive \(x\)).
• At the origin (0,0), the graph 'touches' the x-axis but doesn't cross it. This is typical for even-power polynomials like \( x^4 \), where the graph forms a "U"-shape.

Understanding these features helps you predict how similar polynomial graphs behave, especially when transformed, like in the transformation from \( y = x^4 \) to \( f(x) = x^4 - 4 \).

Symmetrical Graphs

Symmetry in graphs simplifies the analysis and drawing of functions. A graph is said to be symmetrical if one part mirrors another. For the function \( y = x^4 \), the graph is symmetrical about the y-axis.
This means that for every point \((x, y)\), there is a corresponding point \((-x, y)\) on the graph. This symmetry is a hallmark of even-degree polynomials like \( y = x^4 \). It ensures that as you draw one side, the other is a mirror image, reducing complexity.
Even after applying a vertical shift, this symmetry remains intact because the entire graph shifts uniformly. Thus, for \( f(x) = x^4 - 4 \), the graph is still symmetrical about the y-axis, just repositioned down by 4 units.