Graphing Quadratic Equations + bx + c · 2018. 5. 14. · 11-1 Graphing Quadratics Equations.notebook March 29, 2016 11-1 Graphing Quadratic Equations Ex 1: Sketch the graph of the

11-1 Graphing Quadratics Equations.notebook March 30, 2016

y = -ax2 + bx + c Equations with x2 form Parabolas.

11-1 Graphing Quadratic Equations - Parabolas 1

y = ax2 + bx + cf(x) = ax2 + bx + c

y = mx + by = 2x + 2

Straight Line

Graphing Linear Equations

Graphing Quadratic Equations

y is the same as f(x)

y = ax2 + bx + c

When "a" is negative.When "a" is positive.

Equations with just x form Straight Lines.


11-1 Graphing Quadratic Equations

Ex 1:Sketch the graph of the following function.

Step1: Use make a T-Table.

x f(x) Same as "y"

01-12-2

Pick at least 5 numbers. Pick same number of positives & negatives.

Step2: Plug each number for x

f(x) Same as "y"

0 31 4-1 02 3-2

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

If x = 0, then y= -(0)2 + 2(0) +3 y = 3If x = , then y= -( 2 + 2( ) +3 y = 4If x = -1, then y= -(-1 2 + 2(-1) +3 y = 0If x = , then y= -( 2 + 2( ) +3 y = 3If x = -2, then y= -(-2 2 + 2(-2) +3 y = -5

Step3: Plot each point.

f(x)0

1-12

-2 -5

ANSWER

Parabola is upside downbecause equation is negative.



Checks for Example 1Sketch the graph of the following function.

f(x) = x + 2x + 1

01-12

-2

ANSWER

Parabola is upward becauseequation is positive.



Domain and Range

Domain: The x values

Range: The y values

(x, y)

Domain Range

x f(x) Same as "y"

Domain Range

01-12

-2

Where do the x's end or stretch out to?What is the limit of all the x's

What is the domain?

Arrows on parabola mean that they

Ex 2:

ANSWER

Domain = All Real Numbers

or

What is the range?

At what point do the y's end or stop?

ANSWER

Range = y 1

stop at +1

Ex 3:


11-1 Graphing Quadratic EquationsEx 4:

ANSWER

Domain = All Real Numbers

or

Range = y -2

y's stop at -2

ANSWER

State the domain and range of the following function.



Range

Ex 5:

Domain = x -4

x's stop at

ANSWER ANSWER

Range = All Real Numbers

or



Domain = -4 x 2

711-1 Graphing Quadratic EquationsEx 6:


DomainWhere do begin and end?

x starts at -4 and ends at +2.

ANSWER

RangeWhere do begin and end?

Range = -2 y 4

ANSWER

y starts at -2 and ends at +4.



Checks for Examples 2 through 6

-5 3

1)

DomainWhere do x's begin and end?

Domain = -5 x 3

ANSWER

Range

Where do y's begin and end?

-2

Range = -2 y 1

ANSWER



Domain = x 4

911-1 Graphing Quadratic EquationsChecks for Examples 2 through 6

2)


x's stop at

ANSWER ANSWER

Range = All Real Numbers

or

Domain Range(x's) (y's)



Ex 7:

y-intercept: where the graph crosses the y-axis.

Crosses at: (0, -2)

x-intercept: where the graph crosses the x-axis.

Crosses in two places at: (-1, 0) & (3, 0)

Step 1: Find the y-intercept ; set x = 0

x-intercepts: (1, 0) & (-3, 0)

y = x + 2x - 3f(x) same as y

y = ( ) + 3( ) - 3

y-intercept = (0, -3)

y = - 3y = - 3

Note: you must place x = 0

Step 2: Find the x-intercept ; set y = 0 & solve for x

0 = (x )(x + 3)Factors of Add the factors of

-1, +3 +2

x - 1 = 0 x + 3 = 0+ 1

x = 1 x = -3

+1

Note: you must place y = 0

y = x + 2x - 3= x + 2x - 3

ANSWERx-intercepts: (1, 0) & (-3, 0)y-intercept = (0, -3)

Find the x & y-intercepts of the graph of the following function.

admin

Text Box

2



Ex 8:Find the x & y-intercepts of the graph of the following function.

Step 1: Find the y-intercept ; set x = 0

x-intercepts: (-2, 0) & (-6, 0)

y = -x - 8x - 12f(x) same as y

y = -( ) - 8( ) - 12


y = -0 - 0 - 12y = - 12

Note: you must place x = 0

Step 2: Find the x-intercept ; set y = 0

0 = (x + 2)(x + 6)Factors of +12 Add the factors of

+2, +6 +8 x + 2 = 0 x + 6 = 0

x = -2 x = -6

Note: you must place y = 0

y = -x - 8x - 12= -x - 8x - 12

Step 3: Eliminate negative by multiplying everything by -1

(0 = -x - 8x - 12) (-1)

= x +8x +12Step 4: Solve for x.

How do we know it is anupside down parabola?

ANSWERx-intercepts: (-2, 0) & (-6, 0)y-intercept = (0, -12)


11-1 Graphing Quadratic EquationsChecks for Examples 7 & 8Find the x & y-intercepts of the graph of each function.



tip or maximum or minimum point of parabola.

Ex 9:Identify the vertex of each graph of the following functions.

y = -x - 2x - 6Step 1: Find the x-coordinate

To find x, use the following formula:

Step 2: Find the y-coordinate

The vertex is a point on a graph.

(x, y)

x-coordinate y-coordinate

a b

y = -x2 - 2x - 6a = -1b = -

?

-2

2

-2

2

2

-2Simplify

x-coordinate =

To find y, plug x-coordinate into original equation.

y = -x - 2x - 6

-(-1) - 2(-1) - 6- 1 +2 - 6

1 - 6

y-coordinate =

ANSWER vertex =






16

Shortcut:3 things you need in order to graph f(x) = ax2+bx+c

Two ways to graph f(x) = ax2+bx+c :

Step 1: Find vertex. Use

Ex10: Graph the function using vertex and x & y-intercepts.Also state the domain and range

Step 2: Find the y-intercept, set x = 0

Step 3: Find the x-intercepts, set y = 0 & solve for x.

Step 4: Plot Vertex y-intercept & x-intercepts on graph.Also state Domain & Range of graph.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

Vertex = (-2, -1)y-intercept =x-intercepts =Domain =Range =

Vertex = (-2, -1)y-intercept = (0, 3)x-intercepts =Domain =Range =

y = (0 2 +4(0) + 3y = x2 + 4x + 3

y-intercept = (0, 3)

box filled to showwork.

= x2 + 4x + 3

= (x )(x

Don't forget that y = 0

x-intercepts are: (-1, 0) & (-3, 0)

x= -1 x= -3

Vertex = (-2, -1)y-intercept = (0, 3)x-intercepts = (-1, 0) & (-3, 0)Domain =Range =

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

1) Find vertex2) Find y-intercept3) Find x-intercepts

1) Using T-Table2) Shortcut

y = x2 + 4x + 3f(x) = x2 + 4x + 3

Vertex = (-2, -1)y-intercept = (0, 3)x-intercepts = (-1, 0) & (-3, 0)Domain = All Real numbersRange = y -1

x y

ANSWER

Note: You must have this box filledto show your work.

= x2 + 4x + 3Reverse FOIL.

11-1 Graphing Quadratic Equations - Parabolas


17

Checks for Example 10

1)y = x + 2x - 3

Vertex =y-intercept =x-intercepts =Domain =Range =

Graph the function using vertex and x & y-intercepts.Also state the domain and range

x-coordinate:

s:

Vertex (-1, -4)y-intercept = (0, -3)x-intercepts =Domain =Range =

Vertex (-1, -4)y-intercept =x-intercepts =Domain =Range =

Vertex (-1, -4)y-intercept = (0, -3)x-intercepts = (1, 0) & (-3, 0)Domain =Range =

Vertex (-1, -4)y-intercept = (0, -3)x-intercepts = (1, 0) & (-3, 0)Domain = All Real NumbersRange = y -4

ANSWER




18

Checks for Example 10

2)y = -x - 4x

Vertex = (-2, 4)y-intercept =x-intercepts =DomainRange

Vertex = (-2, 4)y-intercept = (0, 0)x-intercepts =Domain =Range =

s:

Vertex = (-2, 4)y-intercept = (0, 0)x-intercepts = (0, 0) & (-4, 0)Domain =Range =

ANSWER Vertex = (-2, 4)y-intercept = (0, 0)x-intercepts = (0, 0) & (-4, 0)Domain = All Real NumbersRange = y 4

Graph the function using vertex and x & y-intercepts.Also state the domain and range

Vertex =y-intercept =x-intercepts =Domain =Range =



19

Axis of symmetry: vertical line that divides a parabola into two halves.

Where does the parabola split in two? At x = 3



20

Ex11:Graph the function using vertex and y-intercept &

Step 1: Find the vertex

a b

y = -x2 - 4x - 6a = -1b = -4

-4

2

-4

2

4

-2Simplify

x-coordinate =

To find y, plug x-coordinate into original equation.

y= -x - 4x - 6

-(-2) - 4(-2) - 6-2 -2

- 4 +8 - 6

4 - 6

y-coordinate =

ANSWER vertex =

To find x-coordinate:

y = -x - 4x - 6

(-2, ?

(-2, -2)

Vertex =y-intercept =axis of symmetry =

Vertex = (-2, -2)y-intercept =axis of symmetry =



21

Vertex = (-2, -2) y-intercept = (0, -6) axis of symmetry =

ex 11 -continued

Step 2: Find the y-intercept, set x = 0


y = -(0)2 - 4(0) - 6y = -x2 - 4x - 6

y = 0 - 0 - 6

Step 3: To find axis of symmetry, plot vertex & y-intercept.y = -x2 - 4x - 6

Vertex = (-2, -2) y-intercept = (0, -6) axis of symmetry = x= -2

Step 4: Finish graph using axis of symmetry.

Vertex = (-2, -2)y-intercept = (0, -6)axis of symmetry = x= -2

ANSWER




22

Graph the function using vertex and y-intercept &Checks for Example 11:

Vertex = (2, -2) y-intercept = (0, 6) axis of symmetry = x= 2

y =

1)



23

Graph the function using vertex and y-intercept &

Vertex = (-2, 1) y-intercept = (0, -7) axis of symmetry = x= -2


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Graphing Quadratic Equations + bx + c · 2018. 5. 14. · 11-1 Graphing Quadratics Equations.notebook March 29, 2016 11-1 Graphing Quadratic Equations Ex 1: Sketch the graph of the