Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=x^{2}(x-1)^{2} $$

Short answer

The graph intersects the x-axis at (0, 0) and (1, 0), touches but does not cross at these points, indicating local minimums at these x-values.

Step-by-step solution

Checking for Symmetries

To test for symmetry with respect to the y-axis, replace \( x \) with \( -x \) in the equation and simplify.\[ \begin{align*} y &= (-x)^{2}((-x) -1)^{2} \ &= x^2(x+1)^2 \ \end{align*}\]The equation is not equal to \( y = x^2(x-1)^2 \), so it is not symmetric with respect to the y-axis. To test for symmetry with respect to the x-axis, replace \( y \) with \( -y \) and see if the equation remains unchanged: since it's a polynomial in \( x \) with positive coefficients, such symmetry doesn't exist. Polar symmetry is irrelevant here as it's not a relevant transformation.

Finding the y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \).\[ y = 0^2(0-1)^2 = 0 \]Thus, the y-intercept is the point \((0, 0)\).

Finding the x-Intercepts

To find the x-intercepts, set \( y = 0 \) and solve for \( x \).\[ x^2(x-1)^2 = 0 \]The solutions are when \( x = 0 \) or \( (x-1) = 0 \). Solving these gives \( x = 0 \) or \( x = 1 \). Thus, the x-intercepts are at \((0,0)\) and \((1,0)\).

Analyzing the Behavior at Intercepts

For \( y = 0 \), the graph will intersect the x-axis at roots. Since at \( x = 0 \) and \( x = 1 \) both have even multiplicities, the graph touches and does not cross these points. Meaning, there will be a local minimum.

Sketching the Graph

Based on the intercepts and symmetry: - Start at \( (0,0) \) with a minimum. - Rise to \( (1,1) \) then back to \( (1,0) \). The direction and multiplicity suggest a parabola shape.

Key concepts

Symmetry in Graphs

Symmetry in graphs is a visual and mathematical concept that helps us understand how a polynomial function behaves. There are different types of symmetry you might encounter: symmetry concerning the y-axis, x-axis, or the origin. For polynomial functions, the most common type of symmetry is y-axis symmetry. We check this by replacing every instance of \( x \) with \( -x \) in the function. If the resulting equation is the same as the original, the graph is symmetric about the y-axis. Unfortunately, if it changes, as in our example, it means no y-axis symmetry is present.

- **Checking y-axis Symmetry:** Replace \( x \) with \( -x \). If \( y = x^2(x-1)^2 \) becomes \( y = x^2(x+1)^2 \), there's no symmetry.
- **Checking x-axis Symmetry:** Substitute \( y \) with \( -y \). In our case, symmetry doesn't apply since it’s a polynomial with positive coefficients.
Understanding symmetry aids in predicting how the graph might look and behave even before plotting it. Symmetry helps us quickly sketch one part of the graph and replicate it across the axis of symmetry.

Finding Intercepts

Identifying the x-intercepts and y-intercepts of a polynomial graph is crucial for sketching its basic structure. Intercepts are the points where the graph crosses the axes. Let's break it down further:

- **Finding y-intercepts:** Set \( x = 0 \) in the equation. For the equation \( y = x^{2}(x-1)^{2} \), set \( x = 0 \) to get the y-intercept. This results in \( y = 0 \), so the graph passes through the origin \( (0,0) \).
- **Finding x-intercepts:** Set \( y = 0 \), and solve \( y = x^2(x-1)^2 = 0 \). The solutions are \( x = 0 \) and \( x = 1 \), providing the intercepts at \( (0,0) \) and \( (1,0) \).

These calculations are straightforward and provide fundamental information about where the graph touches or crosses the axes, which aids significantly in graphing.

Even Multiplicity

Multiplicity refers to the number of times a root appears in the factorization of a polynomial. This indicates how the graph behaves at the x-intercepts and how the polynomial graph behaves near these points. Specifically, if a root has an **even multiplicity**, the graph touches the x-axis at the intercept but does not cross it. This means that at these points, the graph appears to "bounce" off the axis instead of slicing through it.

- **Identifying Even Multiplicity:** In the polynomial \( y = x^2(x-1)^2 \), both roots, \( x = 0 \) and \( x = 1 \), appear twice. Therefore, they have an even multiplicity.
- **Effect on Graphing:** Even multiplicity causes the graph to "touch and turn" at the intercepts without crossing. This creates a local minimum or maximum at those points, making the graph smoother compared to when there's odd multiplicity.

Understanding the concept of even multiplicity helps in predicting the behavior of the graph around its zeroes and enhances our graph sketching process.

Sketching Graphs

Sketching polynomial graphs involves using all the information derived from checking symmetries, finding intercepts, and identifying multiplicities to plot an accurate and visually coherent graph. The primary goal is to outline the general shape the polynomial graph will take.

To start the sketch, depict intercepts at crucial points like \((0,0)\) and \((1,0)\). Acknowledge that at both x-intercepts, the graph neither crosses nor pierces through the x-axis due to the even multiplicity. Here’s a simple guide to sketching with this data:

- **Starting Point:** Begin at \((0,0)\), which is a point of minimum for this graph.
- **Direction from Intercepts:** Move upwards towards \((1,1)\) before coming back down to \((1,0)\), illustrating the "bounce."
The graph will generally take a shape resembling a parabola due to its even-degree nature and positive leading coefficient. Always keep in mind that understanding intercepts and multiplicities can make sketching much more intuitive.