(a) Answer The given function has a minimum value. Explanation Given Information: Given equation is : y = 3 x 2 + 18 x − 21 Concept Used : Quadratic Functions: Quadratic functions are non-linear and are of the form: f ( x ) = a x 2 + b x + c ; a ≠ 0 This is the standard form of Quadratic function. If a > 0 , The graph of a x 2 + b x + c opens upward. The lowest point on the graph is the minimum. If a < 0 , The graph of a x 2 + b x + c opens downward. The highest point on the graph is the maximum. The maximum or the minimum is the vertex. Calculation: In the equation y = 3 x 2 + 18 x − 21 ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 . Since the value of a is 3 , which is greater than 0. So, the graph opens upward and hence the function has minimum value. (b) To determine To Find: The maximum or minimum value of the function. Answer The minimum value is -48. Explanation Given Information: Given equation is : y = 3 x 2 + 18 x − 21 And from part (a) , we have that the function has minimum value. Concept Used : Quadratic Functions: Quadratic functions are non-linear and are of the form: f ( x ) = a x 2 + b x + c ; a ≠ 0 This is the standard form of Quadratic function. Shape of the graph of a Quadratic function is called Parabola. Parabolas are symmetric about a central line called Axis Of Symmetry. The axis of symmetry intersects a parabola only at one point , called the Vertex. Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves. x = − b 2 a Calculation: In the equation y = 3 x 2 + 18 x − 21 ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 . To find the minimum value: The minimum value is the y-coordinate of the vertex . The x-coordinate of the vertex is x = − b 2 a = − 18 2 ( 3 ) = − 3 Substitute x = -3 in the given function to find the y-coordinate of the vertex : y = 3 x 2 + 18 x − 21 y = 3 ( − 3 ) 2 + 18 ( − 3 ) − 21......................... [ x = − 3 ] y = 27 − 54 − 21 y = − 48 So, the vertex is (-3,-48) So, the minimum value of the function is -48. (c) To determine To Find: The domain and range of the function. Answer Domain is all real numbers. Range = { y ∈ R : y ≥ − 48 } Where R= set of all real numbers. Explanation Given Information: Given equation is : y = 3 x 2 + 18 x − 21 From part (b) , the vertex of the given function is (-3,-48) and minimum of the function is at -48. Concept Used : Quadratic Functions: Quadratic functions are non-linear and are of the form: f ( x ) = a x 2 + b x + c ; a ≠ 0 This is the standard form of Quadratic function. The range of a quadratic function is all real numbers greater than or equal to the minimum , or all real numbers less than or equal to the maximum. Calculation: In the equation y = 3 x 2 + 18 x − 21 ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 . The vertex is (-3,-48). The domain of the function is all real numbers . The range of a quadratic function is all real numbers greater than or equal to the minimum.So, the range is { y ∈ R : y ≥ − 48 } where R= set of all real numbers.

1st Edition

ISBN: 9780078884801

Author: McGraw-Hill/Glencoe

Publisher: Glencoe/McGraw-Hill School Pub Co

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Question

Chapter 9.1, Problem 45PPS

(a)

To determine

Whether the function has a maximum or a minimum value.

(a)

Expert Solution

Answer to Problem 45PPS

The given function has a minimum value.

Explanation of Solution

Given Information:

Given equation is :

$y=3x2+18x−21$

Concept Used :

Quadratic Functions:
Quadratic functions are non-linear and are of the form:

$f(x)=ax2+bx+c;a≠0$

This is the standard form of Quadratic function.

If a > 0 ,
The graph of $ax2+bx+c$ opens upward.
The lowest point on the graph is the minimum.
If a < 0 ,
The graph of $ax2+bx+c$ opens downward.
The highest point on the graph is the maximum.
The maximum or the minimum is the vertex.

Calculation:

In the equation $y=3x2+18x−21$ ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 .
Since the value of a is 3 , which is greater than 0. So, the graph opens upward and hence the function has minimum value.

(b)

To determine

To Find:

The maximum or minimum value of the function.

(b)

Expert Solution

Answer to Problem 45PPS

The minimum value is -48.

Explanation of Solution

Given Information:

Given equation is :

$y=3x2+18x−21$

And from part (a) , we have that the function has minimum value.

Concept Used :

Quadratic Functions:
Quadratic functions are non-linear and are of the form:

$f(x)=ax2+bx+c;a≠0$

This is the standard form of Quadratic function.

Shape of the graph of a Quadratic function is called Parabola.
Parabolas are symmetric about a central line called Axis Of Symmetry.
The axis of symmetry intersects a parabola only at one point , called the Vertex.
Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

$x=−b2a$

Calculation:

In the equation $y=3x2+18x−21$ ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 .
To find the minimum value:
The minimum value is the y-coordinate of the vertex .

The x-coordinate of the vertex is $x=−b2a=−182(3)=−3$

Substitute x = -3 in the given function to find the y-coordinate of the vertex :

$y=3x2+18x−21 y=3 (−3) 2 +18(−3)−21.........................[x=−3]y=27−54−21y=−48$

So, the vertex is (-3,-48)

So, the minimum value of the function is -48.

(c)

To determine

To Find:

The domain and range of the function.

(c)

Expert Solution

Answer to Problem 45PPS

Domain is all real numbers.

Range = ${y∈R:y≥−48}$

Where R= set of all real numbers.

Explanation of Solution

Given Information:

Given equation is :

$y=3x2+18x−21$

From part (b) , the vertex of the given function is (-3,-48) and minimum of the function is at -48.

Concept Used :

Quadratic Functions:
Quadratic functions are non-linear and are of the form:

$f(x)=ax2+bx+c;a≠0$

This is the standard form of Quadratic function.

The range of a quadratic function is all real numbers greater than or equal to the minimum , or all real numbers less than or equal to the maximum.

Calculation:

In the equation $y=3x2+18x−21$ ; comparing it with standard form of quadratic equation , we have a = 3 , b = 18 , c = -21 .
The vertex is (-3,-48).
The domain of the function is all real numbers .
The range of a quadratic function is all real numbers greater than or equal to the minimum.So, the range is ${y∈R:y≥−48}$ where R= set of all real numbers.

Chapter 9.1, Problem 44PPS

Chapter 9.1, Problem 46PPS

Chapter 9 Solutions

Algebra 1

Ch. 9.1 - Prob. 67PPS Ch. 9.1 - Prob. 68HP Ch. 9.1 - Prob. 69HP Ch. 9.1 - Prob. 70HP Ch. 9.1 - Prob. 71HP Ch. 9.1 - Prob. 72HP Ch. 9.1 - Prob. 73HP Ch. 9.1 - Prob. 74HP Ch. 9.1 - Prob. 75STP Ch. 9.1 - Prob. 76STP

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