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]

Dominika and Emily are looking at a computer simulation of a rocket that will land on the moon. The following function describes the rocket's movement towards the moon's surface.

h (t) = \frac{t ^{2}}{9 0} - \frac{2 0 t}{3} + 1000

Here,

h

is the height in meters above the surface of the moon and

t

is the time in seconds after the rocket begins its landing. At what height above the moon's surface does the rocket begin its landing?

The given function models the rocket's height above the moon's surface t seconds after it begins its landing. This means that t=0 represents the initial time at which the rocket begins its landing. With this in mind, we will substitute t=0 into the given function rule and evaluate the result to find the height at which the rocket begins its landing. Let's do it!

Therefore, the rocket begins its landing when it is 1000 meters above the moon's surface. h(t)=t^2/90-20t/3+ 1000

Notice that the quadratic function is given in its standard form. f(x)=ax^2x+bx+ c In this form, c represents the constant term of the function. In other words, c is the value for f(0). f(0)= c Therefore, we can identify the value of h(0) by just looking the constant term of the given function which is 1000. l f(x)=ax^2x+bx+ c [0.5em] h(t)=t^2/90-20t/3+ 1000

A valley of the roller coaster that Dominika and Emily saw at the park can be modeled using the following function.

h (x) = 0.5 x^{2} + 5 x + 14.5

Here,

h

represents the height of the cart in meters above the ground. How high above the ground is the cart when it is at its lowest point?

We can find the height of the lowest point of this section of the roller coaster by finding the minimum of the function that describes it. To do so, we will rewrite the given function in vertex form. h(x)=a(x-h)^2+k Let's start by splitting the given function into factors and then factor out 0.5.

Now, we will rewrite the parenthesis by completing the square.

Great! With the quadratic function written in vertex form we can identify the coordinates of the vertex. Let's rearrange the signs of the terms as a final step! h(x)=0.5(x+5)^2+2 [0.5em] ⇓ [0.5em] h(x)=0.5(x-( -5))+ 2 The vertex of the function is at ( -5, 2). Since a>0 the parabola opens upwards, so 2 is the minimum of the function. Knowing this, we can conclude that the cart is 2 meters above the ground at its lowest point.

It is now Emily's turn at Shoot 'N' Hoop. She will shoot a basketball and try to win a teddy bear for this carnival game. Emily's shot is represented by the following equation and graph.

Vertical throw as function of time graph

Find the

x -

intercept of the function. Round the answer to two decimal places.

We are asked to find the x-intercepts of the quadratic equation, which is already written in standard form. We will begin by setting the given equation equal to 0 and identify the values of a, b, and c. ax^2+ bx+ c=0 ⇓ -16x^2+ 24x+ 8=0 In this case, a= -16, b= 24, and c= 8. We can now enter these values into the Quadratic Formula to solve this quadratic equation for x. x=- b±sqrt(b^2-4ac)/2a Let's substitute the values of a, b, and c into the formula!

Since negative values of time do not make sense in this scenario — the ball cannot be in the air for a negative amount of time — we will only consider the positive solution. The x-intercept represents the time the ball remains in the air, and we will round it down to two decimal places.

The ball is in the air for 1.78 seconds.

Key Features of Quadratic Functions

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

Key Features of Quadratic Functions

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