Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=x^{2}(x-1)(x-2) $$

Step-by-Step Solution

Answer

The graph has x-intercepts at (0,0), (1,0), (2,0) and a y-intercept at (0,0) with no symmetries.

Step 1: Identifying Symmetries

The given equation is \( y = x^2(x-1)(x-2) \). To check for symmetry, there are three primary tests: Even Function: \( f(-x) = f(x) \), Odd Function: \( f(-x) = -f(x) \), and symmetry with respect to the y-axis or origin. For this problem, substituting \( -x \) into the equation yields \( y = (-x)^2(-x-1)(-x-2) = x^2(-x-1)(-x-2) \), which simplifies to a different equation. Therefore, the function is neither even nor odd, and there is no symmetry.

Step 2: Finding x-intercepts

To find the x-intercepts, set \( y = 0 \) and solve the equation \( x^2(x-1)(x-2) = 0 \). The solutions are the values of \( x \) that make each factor zero: \( x^2 = 0 \) gives \( x = 0 \); \( x-1 = 0 \) gives \( x = 1 \); \( x-2 = 0 \) gives \( x = 2 \). Thus, the x-intercepts are at \( (0,0) \), \( (1,0) \), and \( (2,0) \).

Step 3: Finding y-intercepts

To find the y-intercept, set \( x = 0 \) and solve for \( y \). Substitute \( x = 0 \) into the equation \( y = x^2(x-1)(x-2) \), resulting in \( y = 0^2(0-1)(0-2) = 0 \). Thus, the y-intercept is at \( (0,0) \).

Step 4: Plotting the Graph

Plot the graph using the intercepts found: \( (0,0) \), \( (1,0) \), and \( (2,0) \). The presence of \( x^2 \) in the equation implies that the curve will touch the x-axis at \( x = 0 \) without crossing it, indicating a multiplicity of 2. It crosses the axis at \( x = 1 \) and \( x = 2 \). The function will be positive between \( x = 0 \) and \( x = 1 \) and negative between \( x = 1 \) and \( x = 2 \). Measure the behavior around these intercepts to sketch the curve effectively.

Key Concepts

Symmetry in Functions

Understanding symmetry in functions can help in predicting the shape of the graph even before plotting it. For a function to exhibit **symmetry**, it needs to have a specific behavior under alteration of its variables:

• Even functions satisfy the condition: \( f(-x) = f(x) \). They are symmetrical about the y-axis. A common example is \( f(x) = x^2 \).
• Odd functions satisfy: \( f(-x) = -f(x) \). They are symmetrical about the origin. An example is \( f(x) = x^3 \).

In the given polynomial, substitute \( -x \) to check symmetry :

• \( y = (-x)^2(-x-1)(-x-2) = x^2(-x-1)(-x-2) \), which does not match \( y = x^2(x-1)(x-2) \) or \( y = -(x^2(x-1)(x-2)) \).

Thus, the polynomial is neither even nor odd and possesses no symmetry with respect to the y-axis or the origin. This means we need to consider the entire range of x-values while plotting.

X-Intercepts

To determine where the function crosses the x-axis, calculate the **x-intercepts** by solving \( y = 0 \). For a polynomial \( y = x^2(x-1)(x-2) \):

• Set each factor to zero: \( x^2 = 0 \), \( x-1 = 0 \), and \( x-2 = 0 \).
• Solving these gives the roots: \( x = 0 \), \( x = 1 \), and \( x = 2 \).

These x-values indicate where the curve touches or crosses the x-axis:

• \((0,0)\): Here the curve only touches the axis, due to multiplicity (it will not cross).
• \((1,0)\): The graph will cross at this point.
• \((2,0)\): The graph will also cross at this point.

Recognizing these intercepts assures us of the specific locations where the graph interacts with the x-axis.

Y-Intercepts

The **y-intercept** is a vital point when analyzing polynomial functions as it indicates where the graph intersects the y-axis.
For this, substitute \( x = 0 \) into the polynomial equation:

• \( y = 0^2(0-1)(0-2) \)
• This simplifies to \( y = 0 \).

Therefore, the y-intercept is \((0,0)\).
In cases where the x-intercept and y-intercept coincide, the curve only touches the origin once. This helps in confirming the behavior and path of the graph, especially when there is no symmetry.

Multiplicity of Roots

**Multiplicity** tells us the behavior of the graph at its roots. For the polynomial \( y = x^2(x-1)(x-2) \), examine the exponents of each factor to understand their multiplicity:

• \(x^2\) has a multiplicity of 2, indicating that at \( x = 0 \), the curve just "kisses" the x-axis, but does not cross it.
• \(x-1\) has a multiplicity of 1. This means the function crosses the axis linearly at \( x = 1 \).
• Similarly, \(x-2\) has a multiplicity of 1, so the curve also crosses the x-axis clearly at \( x = 2 \).

Understanding multiplicity is crucial as it dictates whether the graph touches or crosses the axis at these roots. This helps in crafting a more accurate sketch of how the function behaves locally around each intercept.