Graphing a Quadratic Function: ( ) = + ..., Study notes of Algebra

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Graphing a Quadratic Function: 𝒇

Quadratic Functions are second degree polynomials (i.e. highest power of the domain variable is 2 ).

Quadratics can be written in several forms - General Form, Standard Form (also called Vertex Form ), and

Factored form*. The graph of a Quadratic Function is called a Parabola. It’s general shape is curved and looks

like a “U”. The “U” is right side up if “a” is positive (𝑎 > 0 ) , and it is upside down if “a” is negative (𝑎 < 0 ).

The Vertex (h, k) is either the lowest (right side up) or the highest (upside down) point on the parabola. The

Axis of Symmetry is a vertical line that visually cuts the parabola in half and is written as 𝑥 = ℎ.

General Form (𝑎, 𝑏, 𝑐 ∈ ℝ )

2

The y-intercept ( 0 , 𝑐) of the graph is easily

identifiable from General Form.

The x-intercept(s) (if any) can be found by factoring

and/or using the quadratic formula.

𝑥 =

−𝑏 ± √

𝑏

2

− 4 𝑎𝑐

2 𝑎

The Vertex (ℎ, 𝑘), Min/Max value (𝑘), and Axis of

Symmetry (𝑥 = ℎ) can be found by completing the

square or by using the vertex formula:

−𝑏

2 𝑎

Standard (Vertex) Form (𝑎, ℎ, 𝑘 ∈ ℝ )

2

 The Vertex (ℎ, 𝑘),

 The Min/Max value (𝑘) of the function, and

 The Axis of Symmetry (𝑥 = ℎ)

are all easily identifiable from Vertex Form.

The x-intercept(s) (if any) can be found by using the

square root property.

The y-intercept can be found by evaluating 𝑓( 0 ).

2

2

Parabolic Graph of a Quadratic Function

Axis of Symmetry

y-intercept

x- intercepts,

also called real

“zeros”

Vertex: (𝒉, 𝒌)

“k” is the Min or Max value of the function.

“h” is the domain value that results in the Min/Max.

Distance k (Up/Down)

Distance h (Rt/Lft)

(from Origin)

Origin

This Parabola is

“Face Up”

Practice Graphing Quadratic Functions  𝑓

2

2

Examples:

Graph the following Quadratic given in

General Form: 𝑓(𝑥) = − 3 𝑥

2

Identify the Vertex: (Calculate)

2

Find the x-intercept(s): (Factor or use the

Quadratic Formula)

2

2

x-intercepts: (− 4 , 0 ), ( 2 , 0 )

Find the y-intercept:

Find the axis of symm: 𝑥 = − 1

Extra Points: Use point plotting if needed.

Graph the following Quadratic given in

Standard (Vertex) Form: 𝑓(𝑥) = 3

2

Identify the Vertex : (from the formula)

Find the x-intercept(s): (Square root property)

2

2

2

4

3

2

4

3

2

√ 3

x-intercepts: (− 1 +

2 √

3

3

2 √

3

3

Find the y-intercept: 𝑓( 0 ) = 3 ( 1 )

2

Find the axis of symm: 𝑥 = − 1

Extra Points: Use point plotting if needed.

𝑎 < 0 ,

So facing DOWN

Vertex (Max)

y-intercept

x-intercepts

Axis of Symmetry

Range: (−∞, 𝑘]

Domain: (−∞, ∞)

𝑎 > 0 ,

So facing UP

x-intercepts

y-intercept

Vertex (Min)

Axis of Symmetry

Range: [𝑘, ∞)

Domain: (−∞, ∞)

Due to the “-“ sign in Vertex Form,” h” is the opposite of

the number you see.

Note: “a” is the same number

in both forms!

Graph the following Quadratic Functions given in Standard (Vertex) Form. Find the Vertex,

y-intercept, and x-intercept(s) if they exist. State the Domain and the Range. Also find and

show the Axis of Symmetry. State whether the parabola opens up or down.

2

1

4

2

Answer: Answer:

Vertex:

Vertex:

y-intercept: ( 0 , 5 ) y-intercept: ( 0 , − 6 )

x-intercept(s): (

4 + √

6

2

4 − √

6

2

, 0 ) x-intercept(s): 𝑁𝑜𝑛𝑒

Axis of Symmetry: 𝑥 = 2 Axis of Symmetry: 𝑥 = − 4

Domain: ℝ or (−∞, ∞) Domain: ℝ or (−∞, ∞)

Range:

[

− 3 , ∞) Range:

]

Opens: Up Opens: Down