Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=x^{4}(x-1)^{4}(x+1)^{4} $$

Short answer

The graph is symmetric about the y-axis with intercepts at (-1, 0, 1) and both ends rising.

Step-by-step solution

Check for Symmetry

The given function is \( y = x^{4}(x-1)^{4}(x+1)^{4} \). The function is even since the power of \( x \) in each term is even. Therefore, the graph is symmetric with respect to the y-axis.

Find x-intercepts

To find the x-intercepts, set \( y = 0 \). Thus, we solve the equation \( x^{4}(x-1)^{4}(x+1)^{4} = 0 \). The solutions are \( x = 0 \), \( x = 1 \), and \( x = -1 \). These are the x-intercepts.

Find y-intercept

To find the y-intercept, set \( x = 0 \) in the equation. Calculate \( y = 0^{4}(0-1)^{4}(0+1)^{4} = 0 \). Thus, the y-intercept is 0.

Determine End Behavior

As \( x \to \pm \infty \), the highest degree term \( x^{12} \) will dominate, meaning \( y \to \infty \). Thus, the graph rises to positive infinity at both ends.

Sketch the Graph

Using the symmetry about the y-axis, the x-intercepts (-1, 0, and 1), the y-intercept (0), and the end behavior (both ends rising), sketch the graph. The function will have local minima at \( x = -1, 0, 1 \), with the curve touching the x-axis and rising between these intercepts.

Key concepts

Symmetry in Functions

Symmetry in polynomial functions can greatly simplify the process of graphing. For the given function, \( y = x^4(x-1)^4(x+1)^4 \), symmetry plays a crucial role as it's an even function. An even function means that for every point \((x, y) \), there is an equivalent point \((-x, y) \) on the graph. This is because when plugging in \(-x\) for \(x\) in the function, you end up with the same value of \(y\), due to all exponents being even. This leads to a graph that is symmetrical about the y-axis. Recognizing symmetry can help reduce the workload because visualizing one side allows you to replicate it on the other side seamlessly.

Finding Intercepts

Finding the intercepts of a polynomial function provides vital points through which the graph passes.

• **X-Intercepts:** To find where the graph crosses the x-axis, you set \( y = 0 \). For the function \( y = x^4(x-1)^4(x+1)^4 \), this results in the equation \( x^4(x-1)^4(x+1)^4 = 0 \), leading to solutions of \( x = 0, 1, \) and \(-1 \). These are the x-intercepts where the graph touches or crosses the x-axis.
• **Y-Intercept:** The y-intercept occurs where the graph crosses the y-axis, so set \( x = 0 \). Substituting this in gives \( y = (0)^4((0)-1)^4((0)+1)^4 = 0 \), showing that the y-intercept is 0. Understanding intercepts helps to plot key points accurately, serving as anchor points for drawing the graph.

End Behavior of Polynomials

Understanding a polynomial's end behavior is crucial for predicting how the graph behaves as \( x \) approaches very large positive or negative values.The given function’s highest degree term \( x^{12} \) dictates its end behavior. As \( x \to \pm \infty \), the dominant term \( x^{12} \) becomes hugely positive. This results in \( y \to \infty \) in both positive and negative directions of \( x \).Observing end behavior helps anticipate how the graph behaves outside the range of intercepts. Both ends of the graph rising tells us that, no matter what happens in between, the graph ascends as \( x \) moves outward.

Sketching Graphs of Polynomial Functions

Sketching the graph of a polynomial comprises using information collected from symmetry, intercepts, and end behavior. Here's how it can be efficiently executed:

• **Symmetry:** Because the function is even and symmetric about the y-axis, sketching one half will help with the other.
• **Interceptions:** Plot the x-intercepts at \(-1, 0, \) and \(1\), and the y-intercept at \(0\).
• **End Behavior:** Visualize both ends rising due to the dominant term \(x^{12}\).

With these elements, sketch the curve starting from the left, touching the x-axis at \(x = -1\), rising back up, dipping again at \(x = 0\), and repeating the pattern at \(x = 1\). The symmetry ensures an accurate reflection on either side of the y-axis, and the visual should emerge clearly.