Problem 4 - Calculus with Concepts in Calculus

Question

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ x^{2}=y^{15}-y^{9} $$

Step-by-Step Solution

Answer

The intercepts are (0, 0), (0, 1), (0, -1); symmetric wrt y-axis only.

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). This gives: \[0 = y^{15} - y^{9}.\]Factor out the common term \(y^9\):\[y^9 (y^6 - 1) = 0.\]This equation has solutions \(y = 0\) and \(y^6 = 1\), which implies \(y = \, ext{either of}\sqrt[6]{1} = 1\) or \(y = -1\). Therefore, the y-intercepts are \((0, 0), (0, 1), (0, -1)\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substituting, we have:\[x^2 = 0^{15} - 0^{9} = 0.\]So, \(x = 0\). Therefore, the x-intercept is \((0, 0)\).

Step 3: Check for symmetry with respect to the x-axis

To check symmetry with respect to the x-axis, replace \(y\) with \(-y\) in the original equation and see if it remains unchanged. We have:\[x^2 = (-y)^{15} - (-y)^9.\]Since both exponents are odd, this simplifies to:\[x^2 = -y^{15} + y^9,\]which is not the same as the original equation. Thus, the graph is not symmetric with respect to the x-axis.

Step 4: Check for symmetry with respect to the y-axis

To check symmetry with respect to the y-axis, replace \(x\) with \(-x\) and see if the equation remains unchanged. The modified equation becomes:\[(-x)^2 = y^{15} - y^{9}.\]This simplifies to:\[x^2 = y^{15} - y^{9},\]which is identical to the original equation. So the graph is symmetric with respect to the y-axis.

Step 5: Check for symmetry with respect to the origin

To check symmetry with respect to the origin, replace both \(x\) with \(-x\) and \(y\) with \(-y\). The equation becomes:\[(-x)^2 = (-y)^{15} - (-y)^9.\]This simplifies to:\[x^2 = -y^{15} + y^9,\]which is not the same as the original equation. Thus, the graph is not symmetric with respect to the origin.

Key Concepts

Intercepts

Intercepts are points where the graph crosses the axes. These are crucial for understanding the behavior of the function and are indeed the first step in sketching a graph. The two types of intercepts often explored are:

• Y-intercept: This is where the graph meets the y-axis. To find it, set \( x = 0 \) in the equation and solve for \( y \). For the equation \( x^2 = y^{15} - y^{9} \), upon setting \( x = 0 \), it simplifies to \( 0 = y^9(y^6 - 1) \). Solving this gives \( y = 0, 1, \) and \(-1\). Hence, the y-intercepts are \((0, 0), (0, 1),\) and \((0, -1)\).
• X-intercept: This is where the graph meets the x-axis. Set \( y = 0 \) and solve for \( x \). Here, substituting \( y = 0 \) results in \( x^2 = 0 \), leading to \( x = 0 \). Thus, the x-intercept is \((0, 0)\).

Finding intercepts provides foundational points that guide the rest of the graphing process.

Symmetry with respect to axes

Graph symmetry can significantly simplify the graphing process. It tells us whether the graph reflects evenly across a particular axis or point.

• Symmetry with respect to the x-axis: A graph is symmetric with respect to the x-axis if replacing \( y \) with \( -y \) in the equation does not change it. For our equation, \( x^2 = (-y)^{15} - (-y)^9 \) becomes \( x^2 = -y^{15} + y^9 \). This result differs from the original, indicating no x-axis symmetry.
• Symmetry with respect to the y-axis: Replace \( x \) with \( -x \), and if the equation remains unchanged, it is symmetric about the y-axis. Substituting gives \( (-x)^2 = y^{15} - y^9 \), which simplifies back to \( x^2 = y^{15} - y^9 \), showing y-axis symmetry.
• Symmetry with respect to the origin: Replace both \( x \) with \( -x \) and \( y \) with \( -y \). If the equation remains unchanged, it has origin symmetry. Here, \( (-x)^2 = (-y)^{15} - (-y)^9 \) results in \( x^2 = -y^{15} + y^9 \), differing from the original equation, implying no origin symmetry.

Determining symmetry helps in understanding the basic structure of the graph and can make sketching more efficient.

Equation solving

Solving equations involves finding values that satisfy the equation, which is crucial for interpreting graph behaviors. In our scenario:

• Setting \( x = 0 \) or \( y = 0 \): This strategy isolates other variables, simplifying the equation considerably and making it easier to solve for the intercepts. It's a method that quickly reveals where the graph intersects the axes.
• Simplifying and factoring: Factoring enables us to break down complex expressions into products of simpler ones. For example, \( y^{15} - y^{9} \) becomes \( y^9(y^6 - 1) \). This step simplifies solving, as the factored form offers straightforward solutions like \( y = 0 \) or \( y^6 = 1 \), which further simplifies to \( y = 1 \) and \( y = -1 \).

Mastering equation solving techniques is indispensable for tackling any mathematical problem effectively and is especially beneficial when working with complex equations.

Polynomial functions

Polynomial functions are powerful tools in mathematics and play a pivotal role in understanding a wide array of scenarios.

• Structure and order: Polynomial functions like \( y^{15} - y^{9} \) consist of terms with variables raised to whole number powers. They're characterized by their degree, which in our case is a 15th degree in \( y \), indicating a complex graph structure with multiple turning points.
• Behavior and graph characteristics: The degree determines the graph's general shape, the number of roots, and how it behaves at extreme values. High-degree polynomials, like ours, can fluctuate significantly, leading to varied intercepts and complex symmetries.

Understanding polynomials allows us to predict and analyze the overall behavior of their graphs. Their properties are foundational in many applications across science and engineering.