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Chapter 2: Problem 3

A ___ function is a second-degree polynomial function, and its graph is called a ___.

Short Answer

Expert verified
A second-degree polynomial function is called a quadratic function and its graph is called a parabola.

Step by step solution

01

Identify the type of function

A function defined by an equation of the form \( ax^{2} + bx + c \), where \(a, b,\) and \(c\) are constants and \(a \neq 0\), is a second-degree polynomial function. This type of function is called a quadratic function.
02

Identify the name of the graph

The graph of a quadratic function is a curve. This curve is called a parabola.

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Key Concepts

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

Second-degree polynomial
A second-degree polynomial, also known as a quadratic polynomial, is a type of mathematical function that plays a vital role in algebra and calculus. It follows the standard form:\[ ax^2 + bx + c \]where:
  • \(a\), \(b\), and \(c\) are constants
  • \(a eq 0\)
The presence of \(x^2\) as the highest power ensures that this is considered a second-degree polynomial. These functions are extensively used because they model a wide range of natural phenomena such as projectiles, areas, and various other scientific applications. Understanding the elements of the polynomial, like the coefficient \(a\) which determines the "width" and direction of the parabola, is crucial for analyzing real-world scenarios.
Parabola
In mathematics, the term 'parabola' specifically refers to the U-shaped curve that represents the graphical appearance of a quadratic function. A key feature of a parabola is its symmetry about a vertical line called the axis of symmetry. The standard equation for a parabola derived from a quadratic function is:\[ y = ax^2 + bx + c \]
  • The direction of the parabola (whether it opens upward or downward) is determined by the sign of \(a\).
  • If \(a > 0\), the parabola opens upward.
  • If \(a < 0\), it opens downward.
The highest or lowest point on the parabola is known as the vertex, which plays an essential role in defining the shape and position of the curve. Recognizing and interpreting the vertex can help solve practical problems like maximizing profit or minimizing cost.
Graph of quadratic function
Visualizing a quadratic function involves plotting its graph, which results in a parabola. This graph is crucial for comprehending the behavior and properties of quadratic functions. Important features visible from this graph include:
  • The vertex, the highest or lowest point.
  • The y-intercept, where the graph crosses the y-axis, is at \(y = c\).
  • The x-intercepts, if they exist, are where the graph crosses the x-axis and can be found using the quadratic formula or by factoring.
The axis of symmetry, a vertical line running through the vertex, is another significant attribute and can be calculated using the formula:\[ x = -\frac{b}{2a} \]By examining the graph, students can understand how changes in the quadratic equation's coefficients affect the parabola's position and shape, aiding in the prediction and understanding of different mathematical and real-world situations.

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Most popular questions from this chapter

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$

The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$2 x^{2}+b x+5=0$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=3 x^{3}+2 x^{2}+x+3$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)
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