Discover Key Features of Quadratic Functions: Modeling Real-Life Scenarios

Function	Analysis	Domain
$f (x) = 3 x$	Multiplication by $3$ is defined for all real numbers.	All real numbers
$g (x) = x$	Square roots are not defined for negative numbers.	All non-negative numbers — that is, $x \geq 0$
$h (x) = \frac{1}{x}$	Division by zero is undefined.	All real numbers except $0$ — that is, $x \neq = 0$

Function

Analysis

Domain

f (x) = 3 x

Multiplication by

3

is defined for all real numbers.

All real numbers

g (x) = x

Square roots are not defined for negative numbers.

All non-negative numbers — that is,

x \geq 0

h (x) = \frac{1}{x}

Division by zero is undefined.

All real numbers except

0

— that is,

x \neq = 0

Function	Domain
$f (x) = 2 x$	All integers
$g (x) = x^{2}$	All real numbers
$h (x) = 4$	All real numbers

Function

Domain

f (x) = 2 x

All integers

g (x) = x^{2}

All real numbers

h (x) = 4

All real numbers

Function	Domain	Analysis	Range
$f (x) = 2 x$	All integers	The function takes any integer input and produces an output that is an even number, as each input is multiplied by $2 .$	All even numbers
$g (x) = x^{2}$	All real numbers	The function takes any real number input and produces an output that is a non-negative number, as each input is squared.	All non-negative numbers — that is, $y \geq 0$
$h (x) = 4$	All real numbers	The function takes any real number input and sends it to $4 .$	Only the number $4$ — that is, the range is ${4}$

Function

Domain

Analysis

Range

f (x) = 2 x

All integers

The function takes any integer input and produces an output that is an even number, as each input is multiplied by

2 .

All even numbers

g (x) = x^{2}

All real numbers

The function takes any real number input and produces an output that is a non-negative number, as each input is squared.

All non-negative numbers — that is,

y \geq 0

h (x) = 4

All real numbers

The function takes any real number input and sends it to

4 .

Only the number

4

— that is, the range is

{4}

$a$ -value	Range
$a > 0$	$[k, \infty)$
$a < 0$	$(- \infty, k]$

a

-value

Range

a > 0

[k, \infty)

a < 0

(- \infty, k]

a The quadratic function is given in standard form.

f (x) = x^{2} - 6 x + 5

To find its vertex the function will be rewritten in the vertex form by completing the square.

f (x) = x^{2} - 6 x + 5

▼

Write in vertex form

Rewrite

Rewrite $6 x$ as $2 \cdot 3 x$

f (x) = x^{2} - 2 \cdot 3 x + 5

AddDiffZero

$a = a + 3^{2} - 3^{2}$

f (x) = x^{2} - 2 \cdot 3 x + 3^{2} - 3^{2} + 5

FacNegPerfectSquare

$a^{2} - 2 a b + b^{2} = (a - b)^{2}$

f (x) = (x - 3)^{2} - 3^{2} + 5

CalcPow

Calculate power

f (x) = (x - 3)^{2} - 9 + 5

SubTerm

Subtract term

f (x) = (x - 3)^{2} - 4

With the quadratic function written in vertex form it is now possible to find its vertex.

f (x) = (x - 3)^{2} - 4

Therefore, the vertex is at

(3, - 4) .

Since the parabola opens upward this is a minimum.

b There are no given restrictions on the function, so it can be assumed that its domain consists of all real numbers.

Domain : all real numbers

From Part A it is known that the vertex of the parabola at

(3, - 4)

is a minimum. Therefore, all the values of the function are greater than or equal to

- 4 .

Range : [- 4, \infty)

c The

y -

intercept can be found by evaluating the function at

x = 0 .

f (x) = x^{2} - 6 x + 5

Substitute

$x = 0$

f (0) = 0^{2} - 6 (0) + 5

▼

Simplify right-hand side

CalcPow

Calculate power

f (0) = 0 - 6 (0) + 5

Multiply

f (0) = 0 - 0 + 5

AddSubTerms

Add and subtract terms

f (0) = 5

Therefore, the

y -

intercept is

5 .

To find the

x -

intercepts begin by setting the function equal to

0 .

x^{2} - 6 x + 5 = 0

The resulting expression is a quadratic equation. By looking at the graph it seems that

x = 1

and

x = 5

are solutions to the equation. This can be verified by solving the equation by factoring.

x^{2} - 6 x + 5 = 0

▼

Factor

Rewrite

Rewrite $- 6 x$ as $- 5 x - x$

x^{2} - 5 x - x + 5 = 0

FactorOut

Factor out $x$

x (x - 5) - x + 5 = 0

FactorOut

Factor out $- 1$

x (x - 5) - (x - 5) = 0

FactorOut

Factor out $(x - 5)$

(x - 1) (x - 5) = 0

ZeroProdProp

Use the Zero Product Property

x - 1 = 0 x - 5 = 0 (I) (II)

▼

(I), (II):

Solve for

x

AddEqn

$(I):$ $LHS + 1 = RHS + 1$

x = 1 x - 5 = 0

AddEqn

$(II):$ $LHS + 5 = RHS + 5$

x = 1 x = 5

The solutions to the quadratic equation are

x = 1

and

x = 5 .

Therefore, the

x -

intercepts are

1

and

5 .

The coordinate plane needs to be extended in order to see the intercepts.

d It was found in Part A that the vertex is at

(3, - 4)

and that it is a minimum. Since the domain of the function is all real numbers the function will be decreasing from negative infinity until it reaches the vertex. Then it will start increasing. Remember that the

vertex

is not included in decreasing or increasing intervals.

Knowing this, the decreasing and increasing intervals can be stated.

Decreasing Interval : (- \infty, 3) Increasing Interval : (3, \infty)

e The

x -

intercepts of the function were found in Part C. It can be noted that the graph is

below

the

x -

axis between these intercepts. Knowing that the points at which the function is equal to zero are not to be included, the negative interval can be stated.

Negative Interval : (1, 5)

The graph is

above

the

x -

axis for

x -

values less than

1

and greater than

5 .

Once again, these values are not included in the positive intervals.

Positive Intervals : (- \infty, 1) and (5, \infty)

The graph is divided into a negative interval and two positive intervals.

f It can be noted by taking a look at the graph that as

x

extends to the left or to the right infinitely the function increases in either case. Therefore, its end behavior is up and up.

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Catch-Up and Review

Exploring Parabolas

Extra

Domain of a Quadratic Function

Range of a Quadratic Function

Range

Vertex of a Parabola

Optimizing a Fence

Answer

Hint

Solution

Increasing and Decreasing Intervals of Quadratic Functions

Increasing and Decreasing Intervals

A Roller Coaster Described by Intervals

Hint

Solution

Intercepts of Quadratic Functions

Intercept

The Motion of a Ball Described by a Quadratic Function

Answer

Hint

Solution

Positive and Negative Intervals of Quadratic Functions

Positive and Negative Intervals

Filling a Birdbath

Answer

Hint

Solution

End Behavior of Quadratic Functions

End Behavior

Analyzing a Quadratic Function

Answer

Hint

Solution

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