Warm-Up Quadratic Functions: Standard Form

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Warm-Up Quadratic Functions: Standard Form

Lesson

Question

Lesson Goals

?

Calculate the

of

the quadratic

function.

Graph the

quadratic function.

Calculate the

quadratic

function’s

.


Warm-Up

WK2 Words to Know

Fill in this table as you work through the lesson. You may also use the glossary

to help you.

in a function, the 𝑦-coordinate of the highest point on

the graph of the function or the largest value in the

range

in a parabola, the point at which the function goes from

increasing to decreasing, or vice versa

in a function, the 𝑦-coordinate of the lowest point on

the graph of the function or the smallest value in the

range

the point on a graph at which the graph crosses the

𝑦-axis

graphically, a point on a graph at which the graph

crosses or touches the 𝑥-axis; algebraically, an input of

a function that results in an output of 0

in a parabola, the line that passes through the vertex

and is the line of reflection

Quadratic Functions: Standard Form


Warm-Up

Quadratic Functions

The function 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is a quadratic function in

form.

𝑓 𝑥 = 3𝑥2− 4𝑥 − 10

Leading Coefficient (𝒙𝟐-Term)


𝒙-Term Coefficient Constant Term

𝑎 = 𝑏 = −4 𝑐 =


Identifying the Vertex of a Quadratic Function

Slide

2

When a local high school football

kicker punts the ball, the ball’s

trajectory follows a path that can

be modeled by the function

𝑓 𝑥 = −4.9𝑥2 + 20𝑥.

After 2.04 seconds, it reaches a

height of

20.4 meters.

Draw the axis of symmetry.

Instruction Quadratic Functions: Standard Form

The graph is symmetrical on either side of the axis of symmetry.

When the distance is the same from the axis of symmetry, the output will be

the same.

Heig

ht

(me

ters

)

Time (seconds)


Slide

2 The or maximum value of a quadratic function,

𝑓(𝑥) = 𝑎𝑥2+ 𝑏𝑥 + 𝑐, is called the vertex.


𝒈 𝒙 = −𝒙𝟐 + 𝟐𝒙 + 𝟑𝒇 𝒙 = 𝒙𝟐− 𝟐𝒙− 𝟖

𝑎 = 1, 𝑎 > 0

opens

vertex: min.

𝑎 = −1, 𝑎 < 0

opens

vertex: max.


Slide

4


Calculating a Vertex and an Axis of Symmetry

The vertex of a quadratic function in standard form, 𝑓(𝑥) = 𝑎𝑥2+ 𝑏𝑥 + 𝑐, is the

point −𝑏

2𝑎, 𝑓

−𝑏

2𝑎. The axis of is the vertical line 𝑥 =

−𝑏

2𝑎.

Find the axis of symmetry of the function 𝑓(𝑥) = 2𝑥2– 6𝑥 + 3.

𝑎 = 2 𝑏 =

𝑥 =−𝑏

2𝑎=

−( )

2( )=

6

4=3

2

Substitute the 𝑥-value back into the function to find 𝑓−𝑏

2𝑎.

𝑓3

2=

23

2

2

− 63

2+ 3

vertex: 3

2, −

3

2

axis of symmetry (a.o.s.):

= −3

2

−( )

2( )

23

2

2

− 63

2+ 3


Slide

4


The is −𝑏

2𝑎, 𝑓

−𝑏

2𝑎and

the axis of symmetry is the vertical

line 𝑥 =−𝑏

2𝑎.

𝑓 𝑥 = 2𝑥2– 6𝑥 + 3

vertex: 3

2, –

3

2

axis of symmetry: 𝑥 =3

2

6 Analyzing a Parabola

Vertex: (0, )

Domain: {𝑥|𝑥 is a real number}

Range: {𝑦│𝑦 ≥ −9}

Intervals increasing: ( , ∞)

Intervals decreasing: (−∞, )

(–4, 7)

(–3, 0)

(–2, –5)

(–1, –8)

(0, –9)

(1, –8)

(2, –5)

(3, 0)

(4, –7)


𝑦-intercept: (0, )

Slide

10


Calculating Intercepts

An is the point

where the graph crosses the 𝑥-axis and

the function’s value, or output, is 0.

𝑓(𝑥) = −𝑥2− 2𝑥 + 8

0 = −𝑥2− 2𝑥 + 8

0 = −(𝑥2+ 2𝑥 − 8)

0 = −1(𝑥− )(𝑥 + 4)

0 = 𝑥 − 2 0 = 𝑥 + 4

𝑥 =

𝑥-intercepts: (0, 2) and (0,−4)

Circle the 𝑥-intercepts.

An is the point

where the graph crosses the 𝑦-axis

and the function’s input is 0.

𝑓(𝑥) = –𝑥2 – 2𝑥 + 8

Circle the 𝑦-intercept.

𝑓(0) = – (0)2 – 2( ) + 8

= 0 − 0 + 8

𝑓(0) =

𝑥 = −4

−1(𝑥− )(𝑥 + 4)

– (0)2–2( ) + 8


Slide

12


Graphing a Quadratic Function

Graph the function 𝑓 𝑥 = 𝑥2 − 2𝑥 − 3.

Vertex:

𝑥 =−𝑏

2𝑎=− −2

2 1=

𝑓 1 = 12 − 2 1 − 3 =

( , )

Axis of symmetry:

𝑥 =

𝑥-intercepts:

= 𝑥2 − 2𝑥 − 3

0 = (𝑥 − 3)(𝑥 + 1)

𝑥 = ,

𝑦-intercept:

02 − 2 − 3 = −3

Find other points.

𝑓 4 = 42 − 2 4 − 3

𝑓 4 = 5

(4, 5)

Because of symmetry: (−2, 5)

Draw the parabola.

02 − 2 − 3


Slide

14


Analyzing Parabolas by 𝒙-Intercepts

Quadratic functions can have , 1, or 2 𝑥-intercepts, but they will always

have 𝑦-intercept.

𝑥-intercepts 𝑥-intercept

(𝑥-intercept = vertex)

𝑥-intercepts

The range does not

include , the

parabola does not

cross the 𝑥-axis.

The parabola touches

the 𝑥-axis at exactly

the vertex.

All other cases have 2

𝑥-intercepts.


Summary

Answer

What do the coefficients of a quadratic function in standard form

reveal about its graph?

Lesson

Question?

Slide

2 Review: Key Concepts

Key aspects of quadratic function graphs:

𝑓(𝑥) = 𝑎𝑥2+ 𝑏𝑥 + 𝑐

• : −𝑏

2𝑎, 𝑓

−𝑏

2𝑎

• Minimum (if 𝑎 > 0)

• Maximum (if 𝑎 < 0)

• Axis of symmetry: 𝑥 =

• 𝑥-intercepts when 𝑓(𝑥) = 0

• 𝑦-intercept when 𝑥 = 0



Summary

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Warm-Up Quadratic Functions: Standard Form