Question

For a nonzero constant \(a\), find the intercepts of the graph of \(\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right)\). Then test for symmetry with respect to the \(x\) -axis, the \(y\) -axis, and the origin.

Short answer

The intercepts are \((0, 0), (a, 0), (-a, 0)\), and the graph is symmetric with respect to both axes and the origin.

Step-by-step solution

Identify intercepts with the x-axis

To find the intercepts with the x-axis, set \(y = 0\) in the given equation and solve for \(x\). The equation becomes \[ \left( x^2 + 0^2 \right)^2 = a^2 \left( x^2 - 0^2 \right) \Rightarrow x^4 = a^2 x^2 \].Solving this, \(x^4 = a^2 x^2\) implies \(x^2(x^2 - a^2) = 0\). Therefore, \(x = 0\) or \(x^2 = a^2\). Thus, \(x = \text{0, }a, -a}\).So, the x-intercepts are \((0, 0), (a, 0), (-a, 0)\).

Identify intercepts with the y-axis

To find the intercepts with the y-axis, set \(x = 0\) in the given equation and solve for \(y\). The equation becomes \[ \left( 0^2 + y^2 \right)^2 = a^2 \left( 0^2 - y^2 \right) \Rightarrow y^4 = -a^2 y^2 \]. Solving this, \(y = 0\) is the only solution. Thus, the y-intercept is \((0, 0)\).

Test symmetry with respect to the x-axis

For symmetry with respect to the x-axis, replace \(y\) with \(-y\) in the equation. The equation becomes \[(x^2 + (-y)^2)^2 = a^2(x^2 - (-y)^2) \Rightarrow (x^2 + y^2)^2 = a^2(x^2 - y^2)\], which is the same as the original equation. Therefore, the graph is symmetric with respect to the x-axis.

Test symmetry with respect to the y-axis

For symmetry with respect to the y-axis, replace \(x\) with \(-x\) in the equation. The equation becomes \[(-x^2 + y^2)^2 = a^2((-x)^2 - y^2) \Rightarrow (x^2 + y^2)^2 = a^2(x^2 - y^2)\], which is the same as the original equation. Therefore, the graph is symmetric with respect to the y-axis.

Test symmetry with respect to the origin

For symmetry with respect to the origin, replace \((x, y)\) with \((-x, -y)\). The equation becomes \[(-x^2 + (-y)^2)^2 = a^2((-x)^2 - (-y^2)) \Rightarrow (x^2 + y^2)^2 = a^2(x^2 - y^2)\], which is the same as the original equation. Therefore, the graph is symmetric with respect to the origin.

Key concepts

x-intercepts

The x-intercepts of a graph are the points where the graph intersects the x-axis. This happens when the value of y is zero. For the equation \(\left( x^2 + y^2 \right)^2 = a^2 \left( x^2 - y^2 \right)\), set \(y = 0\) and solve for \(x\).Substituting \(y = 0\), the equation simplifies to \(x^4 = a^2 x^2\). This can be factored as \(x^2 (x^2 - a^2) = 0\), leading to solutions \(x = 0, a, -a\).Thus, the x-intercepts are \( (0, 0), (a, 0), \text{and} (-a, 0)\).Understanding x-intercepts is crucial as they show where the graph crosses the x-axis. This can help in visualizing the graph and understanding its shape and behavior.

y-intercepts

The y-intercepts are the points where the graph intersects the y-axis, occurring when the value of x is zero. For the given equation, \(\left( x^2 + y^2 \right)^2 = a^2 \left( x^2 - y^2 \right)\), set \(x = 0\) and solve for \(y\).Substituting \(x = 0\), the equation simplifies to \(y^4 = -a^2 y^2\). The only real solution to this equation is \(y = 0\).Therefore, the y-intercept is at \( (0, 0)\).Finding y-intercepts helps us understand where the graph crosses the y-axis, giving more information about the graph’s overall layout.

graph symmetry

Graph symmetry means that a graph looks the same when reflected over a specific line (like the x-axis or y-axis) or point (like the origin).Symmetry with respect to the x-axis:
Replace \(y\) with \(-y\) in the equation. The simplified equation remains the same, indicating symmetry about the x-axis.Symmetry with respect to the y-axis:
Replace \(x\) with \(-x\). The simplified equation is the same, showing symmetry about the y-axis.Symmetry with respect to the origin:
Replace \((x, y)\) with \((-x, -y)\). The equation again simplifies to the original form, so the graph is symmetric about the origin.Recognizing symmetry helps in sketching graphs more efficiently and understanding their properties better.

algebraic equations

Algebraic equations represent mathematical relationships using variables, constants, and arithmetic operations. In graphing, they describe the set of points that form a curve or line on the coordinate plane.Consider the equation \(\left( x^2 + y^2 \right)^2 = a^2 \left( x^2 - y^2 \right)\).

• This equation involves squares and higher powers, indicating it forms a more complex curve.
• The constants (a) and variables (x,y) interact to determine the shape and position of the graph.

Understanding the structure of algebraic equations is essential for analyzing their graphs. It allows breaking down the equation to find intercepts, tests for symmetry, and ultimately graphing the relationship represented by the equation.