Problem 30 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = x^4-25 \)

Step-by-Step Solution

Answer

The x-intercepts are \( x = \sqrt[4]{25} \) and \( x = - \sqrt[4]{25} \), and the y-intercept is \( y = -25 \)

Step 1: Finding the x-intercepts

We set \( y = 0 \) in the equation and solve for x: \n\n\[ 0 = x^{4} - 25 \]\n\nThis can be rewritten as \n\n\[ x^{4} = 25 \]\n\nTaking the fourth root of both sides, we have \n\n\[ x = \pm(\sqrt[4]{25}) \]\n\nTherefore, the x-intercepts are \( x = \sqrt[4]{25} \) and \( x = - \sqrt[4]{25} \).

Step 2: Finding the y-intercept

We set \( x = 0 \) in the equation and solve for y: \n\n\[ y = 0^{4} - 25 \]\n\nTherefore, the y-intercept is \( y = - 25 \).

Key Concepts

Graph of an Equation

When you come across the 'graph of an equation,' you're dealing with a visual representation of all the solutions to that equation on a coordinate plane. To graph a polynomial equation like \( y = x^{4} - 25 \), you plot the points where the curve crosses the axes. These points are known as the intercepts.

The graph is essentially the set of all points \( (x, y) \) that make the equation true. In the case of \( y = x^{4} - 25 \), the graph will be a curve with certain symmetry because it's a function of an even-powered polynomial. It's crucial to find the intercepts first as they give you a starting point for drawing the curve on the graph. The x-intercepts occur where the graph crosses the x-axis (when \( y=0 \)), and the y-intercept is where the graph crosses the y-axis (when \( x=0 \)).

Solving Polynomial Equations

To solve polynomial equations, such as \( y = x^{4} - 25 \), the goal is to find the values of x that make the equation hold true. These solutions are also referred to as the 'roots' or 'zeros' of the equation.

Step 1 is to make one side of the equation zero, as seen in the textbook's example. This is because the Zero Product Property tells us that if the product of factors equal zero, then at least one of those factors must itself be zero. For quartic functions like this, you might need to factor or, as with the example, take the fourth root of both sides to solve for x.

Function Intercepts

Function intercepts are the points at which the graph of the function crosses the axes of the coordinate plane.

x-Intercepts (Roots)

For the x-intercepts, you're finding where the function's output, or y-value, is zero. This means setting the equation of the function equal to zero and solving for the x-values. In our example, they were found by solving \( x^{4} - 25 = 0 \), which gave the x-intercepts as positive and negative fourth roots of 25.

y-Intercept

The y-intercept is located by setting the x-value to zero and solving for y. This gives you the single point where the graph touches the y-axis. In our quartic equation, the y-intercept is straightforward to find: just plug in \( x = 0 \) to get \( y = -25 \).

Quartic Functions

Quartic functions are polynomials with a degree of four; that is, the highest power of x is four. The general form is \( ax^{4} + bx^{3} + cx^{2} + dx + e \), where a, b, c, d, and e are constants and \( a \) is not zero.

These functions can have up to four real roots (as in the x-intercepts). They display certain symmetry—since the highest degree term is even, the ends of the graph either both go up or down as x increases in magnitude. Also, quartic functions may have either one or two bends. The challenge usually lies in finding these x-intercepts, which involve setting y to zero and solving for x as with our equation \( x^{4} - 25 = 0 \), leading to two real x-intercepts for this particular function.