Forms of a Quad. eq. & , Domain and Range of Quadratic Functions

Forms of a Quad. eq. & , Domain and Range of Quadratic Functions

By Chi Onungwa

Open Ended

The graph of a quadratic function looks like a smile or frown depending on the a value being positive or negative or real world situation. The graph of a quadratic is called a Parabola.

Some example scenarios whose can be represented by a Parabola are:

the trajectory of a baseball pitcher throwing a baseball to the umpire

or the trajectory of swimmer swinging from a rope before diving in.

Can you think of scenario whose graph might look like a parabola?

Learning Target

​

• I can identify key characteristics of a quadratic equation in vertex form (​ vertex, axis of symmetry, domain, range)

Subject | Subject

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​a>0

Ex: (x-1)²​-4

​a<0

Ex: -(x-2)²​+3

​key characteristics of a Parabola

​Forms of Quadratic Equations

​Take Notes

Subject | Subject

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Vertex form of a parabola

f(x) = a(x-h)²+k ​​

​

if a>0 then ​ the graph looks like a smile/has a minimum value

if a<0 then ​ the graph looks like a frown/has a maximum value

(h,k) is the vertex ​

x=h is the axis of axis of symmetry​

Domian: (-∞​,∞) or all real numbers

Range: [K,∞​) if a smile or (-∞, K] if a frown(a<0)

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​

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Subject | Subject

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​Ex 1

y= (x+1)²​

this can be rewritten as ​ y=(x+1)² +0

​

The graph will look like a (frown or smile​ ) since leading coefficient is positive

Vertex: (-1,0)

axis of symmetry: x=-1

Domian: (-∞​,∞) or all real numbers

Range: [0,∞​) since a smile

​​​

​

Subject | Subject

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​Ex 1

​Here's the graph of parabola to the right

y= (x+1)²​

this can be rewritten as ​ y=(x+1)² +0

​

The graph will look like a (frown or smile​ ) since leading coefficient is positive

Vertex: (-1,0)

axis of symmetry: x=-1

Domian: (-∞​,∞) or all real numbers

Range: [0,∞​)

​​​

​

Subject | Subject

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​Ex 2

f(x)= ​-2(x-2)²+3

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​ looks like a frown

​vertex: (-2,3)

​axis of symmetry: x=-2

Domian: (-∞​,∞) or all real numbers

Range: (-∞,3]

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Subject | Subject

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​Ex 3

f(x)= ​2(x+2)²-3

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​ looks like a smile

​vertex: (-2,-3)

​axis of symmetry: x=-2

Domian: (-∞​,∞) or all real numbers

Range: [-3,∞)

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Subject | Subject

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​Try

​On a piece of paper

Tell whether the graph of the following looks like a smile or a frown and the vertex, axis of symmetry, domain and range for the following then sketch a rough graph.​

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Subject | Subject

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​1. -(x+1)​²+4

vertex:

Axis of symmetry: ​

Domain:

Range: ​

​

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​2. (x-2)​²-3

vertex:

Axis of symmetry: ​

Domain:

Range: ​

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​Domain

Domain of a quadratic will always be all real numbers except in REAL WORLD situations and specific situations like those shown in these graphs

Subject | Subject

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Multiple Choice

What is the vertex of the function   y = (x - 3)² + 5

• (-3, 5)
• (3, -5)
• (-3, -5)
• (3, 5)

Multiple Choice

Given: y = 2(x - 5) + 6, the vertex is

• (2, -5)
• (-5, 6)
• (2, 6)
• (5, 6)

Multiple Choice

Find the vertex.

• (-3,-4)
• (-2,-4)
• (2,-4)
• (2,4)

Multiple Choice

What is the vertex of this function

y= -(x+1)²

• (-1,1)

• (1.0)

• (1,-1)

• (-1,0)

Multiple Choice

Does the graph of
-2(x + 5) + 2
have a minimum or a maximum?

• Minimum
• Maximum
• It has neither.
• It has both.

Multiple Choice

Does the graph of

2(x + 5) + 3

have a minimum or a maximum?

• Minimum
• Maximum
• It has neither.
• It has both.

Multiple Choice

What is the domain of the function?
*Hint: Domain is the set of all x-values included in the function.

• x≥3
• x≤3
• All real numbers
• -1≤x≤5

Multiple Choice

What is the domain of the function?

• D:{ x is greater than or equal to 3}
• D:{x is less than or equal to 3}
• D:{all real numbers}
• D:{ x is all real numbers between and including -1 and 5}

Multiple Choice

What is the domain of the function?

• D:{ x is greater than or equal to 3}
• D:{x is less than or equal to 3}
• D:{all real numbers}
• D:{ x is all real numbers between and including -1 and 5}

Multiple Choice

What is the domain of the function?

• 1<x<9

• 1 $$<$$  x $$\le$$  9

• 4 $$<$$  x $$\le$$  9

• 0 $$<$$  x $$\le$$  9

Multiple Choice

What is the domain of the function?

• 1<x $$\le$$  9

• 1 $$<$$  x < 9

• 4 $$<$$  x $$\le$$  9

• 0 $$<$$  x $$\le$$  9

Multiple Choice

What is the domain of the function?

• All real numbers from 0 to 10

• All real numbers

• The set of all integers greater than 0

• The set of all real numbers greater than 0

Multiple Choice

What is the range of the function?

• 1<x $$\le$$  9

• 0 $$<$$  x < 9

• 4 $$<$$  x $$\le$$  9

• 0 $$<$$  x $$\le$$  9

Multiple Choice

What is the range of the function?

• -4< y $$\le$$  5

• -1 $$<$$ y< 3

• -4 $$<$$ y $$\le$$  9

• -4 $$\le$$  y $$\le$$   $$\infty$$  

Multiple Choice

What is the range of the function?

• - $$\infty$$  < y $$\le$$  -4

• $$-\infty<y\le4$$

• -4 $$<$$ y $$\le$$  9

• -4 $$\le$$  y $$\le$$   $$\infty$$  

Multiple Choice

what is the range of the following function?

f(x)= 2(x+3)-4

• $$\left(-4,\infty\right)$$  

• $$\left(3,\infty\right)$$  

• $$\left(-\infty,4\right]$$  

• $$\left[-4,\infty\right]$$  

Multiple Choice

what is the range of the following function?

f(x)= 2(x+3)-2

• $$\left[-2,\infty\right)$$  

• $$\left(3,\infty\right)$$  

• $$\left(-\infty,-2\right)$$  

• $$\left[-4,\infty\right)$$  

Multiple Choice

Amanda and George are playing soccer on a field. Amanda kicks the ball in the air to George's feet, who is 16 feet away. The ball reaches a maximum height of 4 feet.

Which of the following parabolas represents the path that the ball took, where x is the horizontal distance from Amanda, and y is the vertical distance from the ground?

​Rewrite vertex form into Standard form

Subject | Subject

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​Finding the vertex standard form

Multiple Choice

$$y=4(x+2)^2-8$$ Convert the equation from vertex form to standard form. 

• $$y=4x^2+16x+8$$  

• $$y=4x^2+16x+16$$  

• $$y=16x^2+64x-8$$  

• $$y=16x^2+256x+8$$  

Multiple Choice

$$y=-5(x+2)^2-10$$   Convert the equation from vertex form to standard form.

• y = -5x² - 20x - 30

• y = -5x² - 4x - 10

• y = 25x² + 100x + 90

• y = 25x² - 10x - 30

Multiple Choice

Find the vertex of 
f(x) = -x² - 4x + 12.

• (-2, 16)
• (2, 0)
• (2, 4)
• (-2, 4)

Multiple Choice

Find the vertex of f(x)= -2x²-16x+3.

• (4, -93)
• (-4, 35)
• (-4,99)
• (8,3)

Multiple Choice

What is the vertex of the quadratic function:
y = x² - 6x + 11?

• (3,2)
• (-3,-2)
• (-3,2)
• (3,-2)

Multiple Choice

Determine the vertex of the parabola:
y = 5x² - 20x + 13

• (5, -7)
• (-2, -7)
• (2, -7)
• (-7, -2)