Chapter 9.1, Problem 13CYU

(a)

To determine

Whether the function has a maximum or a minimum value.

Answer to Problem 13CYU

The given function has a maximum value.

Explanation of Solution

Given Information:

Given equation is :

  $y=-x^2+4x-3$

Concept Used :

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

  $f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

• If a > 0 ,
• The graph of $a{x}^2+bx+c$ opens upward.
• The lowest point on the graph is the minimum.
• If a < 0 ,
• The graph of $a{x}^2+bx+c$ opens downward.
• The highest point on the graph is the maximum.
• The maximum or the minimum is the vertex.

Calculation:

• In the equation $y=-x^2+4x-3$ ; comparing it with standard form of quadratic equation , we have a = -1 , b = 4 , c = -3 .
• Since the value of a is -1 , which is less than 0. So, the graph opens downward and hence the function has maximum value.

(b)

To determine

To Find:

The maximum or minimum value of the function.

Answer to Problem 13CYU

The maximum value is 1.

Explanation of Solution

Given Information:

Given equation is :

  $y=-x^2+4x-3$

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

Concept Used :

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

  $f(x)=a{x}^2+bx+c;a\ne 0$

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.

  $x=-\frac{b}{2a}$

Calculation:

• In the equation $y=-x^2+4x-3$ ; comparing it with standard form of quadratic equation , we have a = -1 , b = 4 , c = -3 .
• To find the maximum value:
• The maximum value is the y-coordinate of the vertex .
• The x-coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{4}{2(-1)}=2$
• Substitute x = 2 in the given function to find the y-coordinate of the vertex :

  $\begin{array}{l}y=-x^2+4x-3\\ $ y=-{(2)}^2+4(2)-\mathrm{3.........................}[x=2]$y=-4+8-3\\ y=1\end{array}$

So, the vertex is (2,1)

So, the maximum value of the function is 1.

(c)

To determine

To Find:

The domain and range of the function.

Answer to Problem 13CYU

Domain is all real numbers.

Range = $\{y\in R:y\le 1\}$

Where R= set of all real numbers.

Explanation of Solution

Given Information:

Given equation is :

  $y=-x^2+4x-3$

From part (b) , the vertex of the given function is (2,1).

Concept Used :

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

  $f(x)=a{x}^2+bx+c;a\ne 0$

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 $y=-x^2+4x-3$ ; comparing it with standard form of quadratic equation , we have a = -1 , b = 4 , c = -3 .
• The vertex is (2,1).
• The domain of the function is all real numbers .
• The range of a quadratic function is all real numbers less than or equal to the maximum.So, the range is $\{y\in R:y\le 1\}$ where R= set of all real numbers.