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Chapter 1: Problem 2

(a) determine the polynomial function whose graph passes through the given points, and (b) sketch the graph of the polynomial function, showing the given points. $$(2,4),(3,4),(4,4)$$

Short Answer

Expert verified
The polynomial function is \(y = 4\). It's graph is a straight horizontal line intercepting the y-axis at 4 and passing through the points (2,4), (3,4), and (4,4).

Step by step solution

01

Determine the polynomial function

Since the points all have the same y-value (4) and different x-values, a horizontal line that passes through all the points is the polynomial function. So the polynomial function is \(y = 4\)
02

Sketch the graph

Draw a horizontal line on a graph paper or a plotting software parallel to the x-axis at y = 4. It intersects the y-axis at 4 and passes through the points (2,4), (3,4), and (4,4).

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Key Concepts

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

Horizontal Line
A horizontal line in the context of polynomial functions is one where the function has a constant value for all x-values. This means that no matter what x-coordinate you plug into the function, the y-coordinate remains the same. In a graph, this results in a line that runs parallel to the x-axis. The general equation for a horizontal line is simply \(y = c\), where \(c\) is a constant.
  • In our particular example, the horizontal line passes through the points \((2,4)\), \((3,4)\), and \((4,4)\).
  • Here, the line's equation is \(y = 4\), meaning that regardless of the x-value, the y-value is always 4.
A horizontal line is a special kind of polynomial function called a constant function. Constant functions are the simplest form of polynomial functions, as they do not change with different x-values. They're also perfect examples to start understanding more complex polynomials.
Polynomial Graph Sketching
When sketching the graph of polynomial functions, it is crucial to understand the relationship between the algebraic expression and its graphical representation. For a polynomial function such as a horizontal line, the process is straightforward because the function's graph is simple.Here's how you can accurately sketch a graph for a horizontal polynomial function:
  • Start by drawing the x-axis and y-axis on your graph paper or plotting software.
  • Locate the constant y-value of the function, which is where the horizontal line will be drawn.
  • In this example, plot the line at \(y = 4\). This means you draw a straight line parallel to the x-axis that intersects the y-axis at the point \((0,4)\).
  • It's important to pass the horizontal line through all given points, \((2,4), (3,4), \) and \((4,4)\).
Sketching polynomial functions becomes more complex with higher-degree polynomials, but starting with simple horizontal lines is a good way to build foundational skills in graph sketching.
Interpolation
Interpolation is a method used in mathematics to estimate values between two known values. In the context of polynomial functions, particularly when dealing with a horizontal line, interpolation helps in understanding how the graph behaves between known data points.In our exercise, interpolation is fairly straightforward:
  • Since we have a horizontal line defined by the polynomial function \(y = 4\), any value of \(x\) between the known points \((2,4)\), \((3,4)\), and \((4,4)\) will also have a corresponding y-value of 4.
  • This means the function's value is consistent everywhere along the line between the known data points.
Interpolation becomes more beneficial as polynomial functions grow more complex, especially when dealing with curves or higher-degree polynomials. This basic example illustrates the principle that applies to more sophisticated polynomial interpolations.

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Most popular questions from this chapter

Find all values of \(\lambda\) (the Greek letter lambda) such that the homogeneous system of linear equations will have nontrivial solutions. $$\begin{array}{r} (\lambda-2) x+\quad y=0 \\ x+(\lambda-2) y=0 \end{array}$$

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Prove that if a polynomial function \(p(x)=\) \(a_{0}+a_{1} x+a_{2} x^{2}\) is zero for \(x=-1, x=0,\) and \(x=1,\) then \(a_{0}=a_{1}=a_{2}=0\) Getting Started: Write a system of linear equations and solve the system for \(a_{0}, a_{1},\) and \(a_{2}\) (i) Substitute \(x=-1,0,\) and 1 into \(p(x)\) (ii) Set the result equal to 0 (iii) Solve the resulting system of linear equations in the variables \(a_{0}, a_{1},\) and \(a_{2}\)

Consider the matrix \(A=\left[\begin{array}{rrr}1 & k & 2 \\ -3 & 4 & 1\end{array}\right]\) (a) If \(A\) is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. (b) If \(A\) is the augmented matrix of a system of linear equations, find the value(s) of \(k\) such that the system is consistent. (c) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. (d) If \(A\) is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of \(k\) such that the system is consistent.

In the 2007 Fiesta Bowl Championship Series on January 8 \(2007,\) the University of Florida Gators defeated the Ohio State University Buckeyes by a score of 41 to \(14 .\) The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth \(6,1,\) and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. The number of touchdowns was one more than three times the number of field goals. Write a system of equations to represent this event. Then determine the number of each type of scoring play. (Source: www.fiestabowl.org)
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