Problem 12 - Contemporary Precalculus

Question

In Exercises \(7-12,\) find the graph of the equation in the standard window. $$y=2 x^{4}-2 x^{3}-2 x^{2}-2 x+5$$

Step-by-Step Solution

Answer

Answer: The important points for the function are the y-intercept (0, 5) and two x-intercepts, approximately (-0.43, 0) and (1.06, 0). The graph is a curve with one or more turning points, passing through these points.

Step 1: Find the y-intercept

The y-intercept is the point where the function intersects the y-axis. To find the y-intercept, we simply set x = 0 and solve for y: $$y = 2(0)^4 - 2(0)^3 - 2(0)^2 - 2(0) + 5$$ $$y = 5$$ So the y-intercept is (0, 5).

Step 2: Find the x-intercepts (if possible)

The x-intercepts are the points where the function intersects the x-axis. To find the x-intercepts, we set y = 0 and solve for x: $$0 = 2x^4 - 2x^3 - 2x^2 - 2x + 5$$ Unfortunately, this polynomial equation is not easily solvable by factoring or other basic methods due to its high degree. Instead, we will rely on technology (such as a graphing calculator or software) to approximate the x-intercepts.

Step 3: Use a graphing calculator or software to plot the function and find the x-intercepts

Using a graphing calculator or software, enter the following function: $$y = 2x^4 - 2x^3 - 2x^2 - 2x + 5$$ After graphing the function, we can see that it has two x-intercepts: approximately (-0.43, 0) and (1.06, 0).

Step 4: Plot the points and draw the graph

Now that we have the y-intercept and the x-intercepts, we can plot these points on a graph in the standard window: - Y-intercept: (0, 5) - X-intercepts: (-0.43, 0) and (1.06, 0) Draw a smooth curve through these points to complete the graph of the function. Remember to label the graph with the equation \(y = 2x^4 - 2x^3 - 2x^2 - 2x + 5\) and the important points found above (y-intercept and x-intercepts).

Key Concepts

y-intercept

Understanding the y-intercept of a polynomial function is an essential component in graphing. The y-intercept is the point where the curve crosses the y-axis. To find this point, you simply evaluate the function when the independent variable (usually denoted as x) is zero. For instance, in our exercise, we calculate the y-intercept of the polynomial function y=2 x^{4}-2 x^{3}-2 x^{2}-2 x+5 by substituting x with 0, resulting in the value y=5. Therefore, the y-intercept is the point (0, 5).

The y-intercept gives you a starting reference point for graphing the entire polynomial function on a coordinate plane. It is crucial because it also represents the function's output when all the independent variables are set to zero, showing where the graph will start its journey across the coordinate system.

x-intercepts

Similarly, the x-intercepts are points at which the graph of the polynomial function intersects the x-axis. Mathematically, x-intercepts occur where the function output, or y-value, is zero. If you need to find the x-intercepts for a polynomial equation, such as y=2 x^{4}-2 x^{3}-2 x^{2}-2 x+5, you will set y to zero and solve for x.

Finding the x-intercepts by hand can sometimes be complicated, especially for higher-degree polynomials. In such cases, we use technology like graphing calculators to help us approximate these values. In the given polynomial, technological tools reveal that we have two x-intercepts at approximately (-0.43, 0) and (1.06, 0). Sharing these points, along with the y-intercept, provides a clearer picture of the function's graph.

polynomial equation

A polynomial equation is an algebraic expression that involves a sum of powers of a variable multiplied by coefficients. The general form of a polynomial equation in one variable x is a_{n} x^{n} + a_{n-1} x^{n-1} + ... + a_{1} x + a_{0}, where a represents the coefficients and n indicates the degree of the polynomial.

Graphing a polynomial equation involves plotting several points, which are determined by choosing input values for x and computing corresponding outputs y. This process results in a graph that could represent different shapes depending on the degree of the polynomial. Our exercise shows a fourth-degree polynomial, which may have up to four real x-intercepts and may fluctuate between increasing and decreasing intervals, creating peaks and valleys as observed in its graph.

graphing calculator

A graphing calculator is a powerful educational tool that assists in visualizing and solving complex mathematical problems. For graphing polynomial functions, a graphing calculator allows us to input the equation and observe its graphical representation. When using a graphing calculator to plot y=2 x^{4}-2 x^{3}-2 x^{2}-2 x+5, you would typically enter the expression as it is, adjust the viewing window to an appropriate scale, and then graph the function.

The calculator does not only provide a visual but also helps in finding approximate values of x-intercepts, maximums and minimums, and other critical points that may be difficult to determine manually. Especially with polynomials of higher degrees, a graphing calculator becomes an invaluable tool in bridging the gap between a raw equation and its tangible, visual graph, thus enhancing comprehension and learning.