Problem 27 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ x^{4}-5 x^{2} y+4 y^{2}=0 $$

Step-by-Step Solution

Answer

Factor the equation, solve for y, and sketch the parabolas y = x^2 and y = x^2/4.

Step 1: Identify Symmetry

Notice that the equation involves terms with even powers of both variables, which suggests symmetry with respect to both axes and the origin.

Step 2: Factor the Equation

Rewrite the given equation. We recognize that both the left side and the right side can be factored: x^{4} - 5x^{2}y + 4y^{2} = (x^{2} - y)(x^{2} - 4y) = 0.

Step 3: Set the Factors to Zero

Set each factor from the factored equation equal to zero: 1. \( x^{2} - y = 0 \) 2. \( x^{2} - 4y = 0 \).

Step 4: Solve for y

Solve each equation for y: 1. \( y = x^{2} \) 2. \( y = \frac{x^{2}}{4} \)

Step 5: Sketch the Graphs

Plot the equations \( y = x^{2} \) and \( y = \frac{x^{2}}{4} \). Observe that these are parabolas opening upwards, with different widths. The first parabola (\( y = x^{2} \)) is narrower and the second one (\( y = \frac{x^{2}}{4} \)) is wider.

Step 6: Combine the Plots

Combine the graphs of both equations to obtain the final sketch. The final graph consists of both parabolas intersecting at the origin and symmetrically placed.

Key Concepts

symmetry in graphs

Symmetry plays a huge role in understanding graphs of equations. When you have an equation like \(x^4 - 5x^2y + 4y^2 = 0\), notice how the powers of both \(x\) and \(y\) are even. This indicates symmetry. In this case, the graph is symmetric with respect to the x-axis, y-axis, and the origin. Symmetry helps in predicting the shape and positioning of the graph without plotting every point. If you see even powers, you will often find that reflecting the graph over the x-axis or y-axis will result in the same curve.

factoring equations

Factoring is a key step in solving many algebraic equations. For the equation \(x^4 - 5x^2y + 4y^2 = 0\), we first rewrite it by grouping terms: \((x^2)^2 - 5(x^2)y + 4y^2 = 0\). Notice the pattern? This equation can be factored into: \((x^2 - y)(x^2 - 4y) = 0\). Factoring helps simplify complex polynomials into products of simpler binomials or polynomials, making it easier to solve.

solving for variables

Once we factor the equation, we move to solving for variables. Setting each factor equal to zero, we get: 1. \(x^2 - y = 0\) 2. \(x^2 - 4y = 0\). Solving these, we isolate \(y\): 1. \(y = x^2\) 2. \(y = \frac{x^2}{4}\). Solving for variables simplifies the problem, making it easier to understand and graph.

plotting parabolas

Plotting the results, we graph the individual equations \(y = x^2\) and \(y = \frac{x^2}{4}\). These are both parabolas opening upwards but with different widths. \(y = x^2\) shows a standard parabola, while \(y = \frac{x^2}{4}\) is wider. When combining these plots, you see the overall graph intersects at the origin and exhibits symmetry in placement, confirming earlier predictions on symmetry. Plotting this accurately provides a clear visual understanding of the equation.