Linear Algebra and its Applications

David C. Lay, Steven R. Lay and Judi J. McDonald

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 483 pages

ISBN: 978-03219822384

Chapter 1: Q31E (page 7)

Find the polynomial of degree 2[a polynomial of the form $f (t) = a + bt + {ct}^{2}$ ] whose graph goes through the points localid="1659342678677" $(1, - 1), (2, 3) and (3, 13) .$ Sketch the graph of the polynomial.

Short Answer

Expert verified

The polynomial of degree 2 whose graph goes through the points $(1, - 1), (2, 3) and (3, 13) is f (t) = 1 - 5 t + 3 t^{2} .$ The graph is,

Step by step solution

Consider the points and substitute these in the standard equation

A polynomial of degree 2 is of the form $f (t) = a + bt + {ct}^{2}$ . Consider a polynomial of degree 2 and substitute given point in them as:

role="math" localid="1659342005093" $\begin{array}{l} f_{1} (t) = a + b t_{1} + c_{1}^{2} \\ - 1 = a + b (1) + a {(1)}^{2} \end{array}$

role="math" localid="1659342115449" $\begin{array}{l} f_{2} (t) = a + b t_{2} + c_{2}^{2} \\ 3 = a + b (2) + a {(2)}^{2} \end{array}$

$\begin{array}{l} f_{3} (t) = a + b t_{3} + c_{3}^{2} \\ 13 = a + b (3) + a {(3)}^{2} \end{array}$

Rearrange the terms of the above equations

Consider the simplified equations.

$- 1 = a + b + c . . . . . . . (1) 3 = a + 2 b + 4 c . . . . . . (2) 13 = a + 3 b + 9 c . . . . . . (3)$

Solve the above equations (1), (2) and (3)

Arrange the equations (1), (2) and (3) into matrix form:

$(\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{matrix}) (\begin{matrix} a \\ b \\ c \end{matrix}) = (\begin{matrix} - 1 \\ 3 \\ 13 \end{matrix})$

Upon solving the values of a,b and c are obtained as $a = 1, b = - 5, c = 3 .$ .

Substitute these values in the standard equation of the 2 degree polynomial. $\begin{array}{l} f (t) = (1) + (- 5) t + (3) t^{2} \\ f (t) = 1 - 5 t + 3 t^{2} \end{array}$

The graph of the polynomial $f (t) = 1 - 5 t + 3 t^{2}$ is,

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91影视

Find the polynomial of degree 2[a polynomial of the form $f (t) = a + bt + {ct}^{2}$ ] whose graph goes through the points localid="1659342678677" $(1, - 1), (2, 3) and (3, 13) .$ Sketch the graph of the polynomial.

Short Answer

Step by step solution

Consider the points and substitute these in the standard equation

Rearrange the terms of the above equations

Solve the above equations (1), (2) and (3)

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Most popular questions from this chapter

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91影视

Find the polynomial of degree 2[a polynomial of the form f(t)=a+bt+ct2] whose graph goes through the points localid="1659342678677" (1,-1),(2,3)and(3,13).Sketch the graph of the polynomial.

Short Answer

Step by step solution

Consider the points and substitute these in the standard equation

Rearrange the terms of the above equations

Solve the above equations (1), (2) and (3)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Find the polynomial of degree 2[a polynomial of the form $f (t) = a + bt + {ct}^{2}$ ] whose graph goes through the points localid="1659342678677" $(1, - 1), (2, 3) and (3, 13) .$ Sketch the graph of the polynomial.