Problem 29 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = 2x^3-4x^2 \)

Step-by-Step Solution

Answer

The \( x \)-intercepts are \( x_1 = 0 \) and \( x_2 = 2 \) while the \( y \)-intercept is (0,0).

Step 1: Finding the \( x \)-intercepts

To find the \( x \)-intercepts we need to set \( y = 0 \) and solve for \( x \). So: \[0 = 2x^3-4x^2\]First, we factor out the common factor \( x^2 \):\[0 = x^2 (2x - 4)\]Next we set each factor to zero:\[x^2 = 0 \text{ and } 2x-4 = 0\]Solving these we get \( x_1 = 0, x_2 = 2 \) which are our \( x \)-intercepts.

Step 2: Finding the \( y \)-intercepts

To find the \( y \)-intercepts we need to set \( x = 0 \) and solve for \( y \). So: \[y = 2 * 0^3 - 4 * 0^2 = 0\]So, the \( y \)-intercept is (0,0).

Key Concepts

Finding x-intercepts

When delving into the world of algebraic functions, understanding how to locate the points where the graph of the function crosses the x-axis, known as the x-intercepts, is crucial. Imagine you're on a treasure hunt, and the x-intercept marks the spot on the map. To uncover this treasure, we search for when the output value, represented by the variable y, equals zero. An easy way to visualize this is by picturing a rollercoaster that briefly touches ground level—that point where it kisses the ground, is the intercept in question.

For quadratic equations and higher, such as the given cubic function, this often involves factoring. The process is similar to pulling out toys from a treasure chest; if possible, we look for common factors and pull them out of the equation. In our exercise, we plucked out an x², simplifying our task to finding the zeros of what remains. We're essentially solving a simpler puzzle within the larger riddle. Like splitting a lock's combination into smaller parts, we deal with each factor separately to find the x-intercepts, which are, in this case, x = 0 and x = 2.

Factoring Polynomials

The art of factoring polynomials is akin to dismantling a complex piece of machinery into its component parts. It's about breaking down expressions into the simplest building blocks—these are the 'factors' that, when multiplied together, give back the original polynomial. Think of it as reverse-engineering, where we're looking for the original blueprints of the algebraic structure.

For our exercise, the polynomial in question is cubic, giving it a bit more complexity. We first identify any common factors among the terms, just like looking for repeating patterns or pieces that clearly fit together. In the given function y = 2x³ - 4x², we can see that both terms share an x². Extracting this common piece streamlines the equation, leaving us with a clearer path to solve the puzzle. Factoring not only simplifies the equation but also paves the way to finding both x-intercepts and y-intercepts.

Solving Cubic Equations

Once we enter the realm of solving cubic equations, we're stepping up the game—it's like moving from checkers to three-dimensional chess. Unlike the easier to manage linear or quadratic equations, cubic equations can behave in unexpected ways, often leading to multiple turning points and possibly more than one x-intercept. To solve such a conundrum, one effective strategy is to use factoring, a method we've touched upon earlier, which can simplify our quest significantly.

Our goal here is to express the cubic equation 2x³ - 4x² = 0 as a product of factors that can each be set to zero. By pulling out the greatest common factor—x² in this case, and then examining the quadratic factor (2x-4) that remains, we create a clearer path to identify potential solutions. These solutions paint a picture of where the function's graph touches or crosses the x-axis and are the keys to solving the holistic cubic puzzle.