Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(x^{2} y-x^{2}+4 y=0\)

Step-by-Step Solution

Answer

The graph of the given equation has both \(x\) - and \(y\) -intercepts at the origin, \(0\).

Step 1: Finding the x-intercepts

To find the \(x\)-intercepts, set \(y\) equal to zero and solve for \(x\). The equation becomes \(x^{2}*0 - x^{2} + 4*0 = 0\), which simplifies to \(-x^{2} = 0\). Therefore, \(x = 0\). Thus, the equation has an \(x\)-intercept of \(x =0\).

Step 2: Finding the y-intercepts

To find the \(y\)-intercepts, set \(x\) equal to zero and solve for \(y\). The equation becomes \(0^{2}*y - 0^{2} + 4*y = 0\),which simplifies to \(4y=0\). Therefore, \(y = 0\). Thus, the equation has a \(y\)-intercept of \(y = 0\).

Key Concepts

Intercepts of Polynomial Equations

The concept of finding intercepts is a foundational aspect of polynomial equations. When we discuss the intercepts of a polynomial equation, we are referring to the points where the graph of the polynomial crosses the x-axis and y-axis. The intersecting points on the axes are critical in understanding the shape and behavior of the graph.

To find the x-intercept, we set the y value to zero and solve the polynomial equation for x. This process effectively gives us the points where the graph touches or crosses the x-axis. Conversely, to locate the y-intercept, we set x to zero and solve the equation for y, yielding the points where the curve intersects the y-axis. Knowing these points is particularly helpful when graphing the polynomial by hand or analyzing its characteristics.

In essence, intercepts are identified by solving the equation of the polynomial first for one variable (setting the other to zero) and then for the other. It’s worth noting that while linear equations have a single x and y intercept, higher-degree polynomials might have multiple x-intercepts and potentially a single y-intercept, depending on the nature of the equation.

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics, where you manipulate an equation to find the values of the variables that satisfy it. The approach taken to solve an equation can significantly change depending on the type of equation and the complexity involved. For linear equations, the process is often straightforward, but for polynomial equations, it might involve factoring, using the quadratic formula, graphing, or employing numerical methods.

When finding intercepts, you are essentially solving the equation with one of the variables set to zero. Similar to intercept-finding, the broader practice of solving algebraic equations relies on a systematic approach: simplifying expressions, isolating the variable in question, and performing operations that 'undo' the equation's complexity until the solution is reached.

This process might include adding or subtracting terms, multiplying or dividing by coefficients, or applying more advanced techniques like completing the square. Solving algebraic equations is a critical skill that helps us not only with graphing but also with understanding algebraic relationships and functions.

Graphing Quadratic Equations

Graphing quadratic equations is an essential part of understanding the behavior of these second-degree polynomials. A quadratic equation is typically in the form of \( ax^{2} + bx + c = 0 \), and its graph is a parabola, either opening upwards or downwards depending on the leading coefficient. The vertex of the parabola reveals the maximum or minimum value of the function, and this is a key feature when sketching the graph.

While intercepts provide crucial points, the overall shape of the graph and how it relates to the x and y-axis is determined by additional factors such as the equation's coefficients and discriminant. Using intercepts to assist in sketching the graph, we find that if the equation has real and distinct x-intercepts, the parabola crosses the x-axis at these points. If there is one x-intercept, the vertex lies on the x-axis, and if there are no real x-intercepts, the parabola does not cross the x-axis at all.

Through the process of graphing, one can visualize the effects of the 'a', 'b', and 'c' coefficients on the direction, width, and location of the parabola, respectively. This visualization aids in understanding the quadratic's behavior and provides a powerful tool for solving a variety of applied problems in both pure and applied maths.