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Chapter 3: Problem 41

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$g(t)=\frac{1}{2} t^{4}-\frac{1}{2}$$

Short Answer

Expert verified
The real zeros of the function are \( t = 1 \) and \( t = -1 \).

Step by step solution

01

Set the Function Equal to Zero

Start by setting the function equal to zero to find the values of \( t \) that make the function zero. So the equation to solve is \( \frac{1}{2} t^{4} - \frac{1}{2} = 0 \).
02

Solve the Equation

Next, solve the equation for \( t \). Multiply every term by 2 to get rid of the fraction: \( t^{4} - 1 = 0 \). Add 1 to both sides for \( t^{4} = 1 \). The solutions are \( t = 卤\sqrt[4]{1} \) which result in \( t = 卤1 \).
03

Confirm Results using a Graphing Utility

Plot the function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \) using a graphing utility. The function should intersect the x-axis at \( t = 1 \) and \( t = -1 \), confirming that these are indeed the zeros of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Solutions
To find the real zeros of a polynomial function algebraically, one must solve for the variable that makes the function equal to zero. In the given exercise, the function is set to zero: \( \frac{1}{2} t^{4} - \frac{1}{2} = 0 \). Here are some key steps and tips to keep in mind:
  • Eliminate Fractions: It's easier to handle equations without fractions. Multiply both sides by the denominator, which in this case, clears the fraction and gives us \( t^{4} - 1 = 0 \).
  • Rearrange and Simplify: Simple algebraic manipulation like adding or subtracting terms from both sides can help isolate terms. For example, adding 1 to both sides yields \( t^{4} = 1 \).
  • Solve for the Variable: You can now solve the equation by taking the fourth root of both sides. Remember, \( t^{4} = 1 \) gives us solutions \( t = \pm 1 \).
By utilizing these strategies, you can confidently solve polynomial equations algebraically.
Graphical Confirmation
Graphical confirmation is a stellar way to verify the solutions you got algebraically. By plotting the polynomial function on a graph, you can visually identify the zeros where the curve crosses the x-axis.Let's break it down:
  • Use a Graphing Utility: There are various tools, such as graphing calculators or computer software, to help with plotting functions. For this function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \), inputting it correctly should show the parabola.
  • Analyze the Graph: Look for points where the graph intersects the x-axis. These are your zeros. Here, it should show intersections at \( t = 1 \) and \( t = -1 \).
Graphical methods provide a visual check and help in understanding the behavior of the polynomial function across its domain.
Polynomial Functions
Polynomial functions are a fundamental part of algebra. They consist of terms with variables raised to different powers and their coefficients.In the case of our function \( g(t) = \frac{1}{2} t^{4} - \frac{1}{2} \), it is a quartic polynomial. Here's why it matters:
  • Degree of the Polynomial: The highest power of the variable, here it's 4, determines the general shape and number of solutions. A quartic polynomial can have up to four real roots. However, in this case, due to the nature of the equation, only \( t = 1 \) and \( t = -1 \) are the real solutions.
  • End Behavior: As a polynomial of even degree, the ends of the graph behave similarly, both turning up or down. For \( \frac{1}{2} t^{4} \), both ends rise up as \( t \) becomes very large or very small.
Understanding polynomial functions helps in predicting their graphical behavior and finding their algebraic solutions.

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Most popular questions from this chapter

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=4 x^{2}-x^{3}$$

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=x^{2}-25$$

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=x^{2}-4 x+1$$

Use the graph of \(y=x^{3}\) to sketch the graph of the function. $$f(x)=(x+3)^{3}$$

Regression Problem Let \(x\) be the number of units (in tens of thousands) that a computer company produces and let \(p(x)\) be the profit (in hundreds of thousands of dollars). The table shows the profits for different levels of production. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Units, } x & 2 & 4 & 6 & 8 & 10 \\ \hline \text { Profit, } p(x) & 270.5 & 307.8 & 320.1 & 329.2 & 325.0 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Units, } x & 12 & 14 & 16 & 18 & 20 \\ \hline \text { Profit, } p(x) & 311.2 & 287.8 & 254.8 & 212.2 & 160.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for \(p(x)\). (c) Use a graphing utility to graph your model for \(p(x)\) with the scatter plot of the data. (d) Find the vertex of the graph of the model from part (c). Interpret its meaning in the context of the problem. (e) With these data and this model, the profit begins to decrease. Discuss how it is possible for production to increase and profit to decrease.
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