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6.6 Analyzing Graphs of

Quadratic Functions


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Graphing the parabola y= f(x) = ax2

Consider the equation y= x2

0 1 41

(

1, 1) (0, 0) (1, 1) (2, 4)

y

x

4

(

2, 4)

Axis of symmetry: x =0( y=x2 is symmetr ic with

respect to the y-axis )

Vertex

(0, 0)

When a >0, the parabola

opens upwards and is called

concave up.

The vertex is called a

minimum point.

x 2 1 0 1 2

y


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For the function equation y= x2 , what is a?

a= 1 . What ifadoes not equal 1?

Consider the equation y=

4x2 .

0 44

(0, 0)

16y

16

(2,16) (1,4) (1,4) (4,16)x

When a


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Properties of the Parabola f(x) = ax2

The graph off(x) = ax2 is a parabola with thevertex at the origin and the yaxis as the line ofsymmetry.

Ifais positive, the parabola opens upward, ifaisnegative, the parabola opens downward.

Ifa is greater than 1 (a > 1), the parabola is

narrower then the parabola f(x) = x2.

Ifa is between 0 and 1 (0 < a < 1), the parabolais wider than the parabola f(x) = x2.


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Algebraic Approach: y=

4x2

3

x

2 1 0 1 2

-4x2 -16 -4 0 -4 -16

-4x2-3 -19 -7 -3 -7 -19

Numerical Approach:Graphical Approach:

Graphing the parabola

y= f(x) = ax2 + k

Consider the equation y=

4x2

3 . What is a?

a=4

x

Vertex

(0, -3)


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y=

4x2

3 .

x

Vertex(0, -3)

y=

4x2

In general the graph of y= ax2 + kis the graphofy= ax2shifted vertically kunits. Ifk> 0, the

graph is shifted up. Ifk< 0, the graph is shifted

down. (P. 267)

The graph y=4x2 is shifted

down 3 units.


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a=4. What effect does the 3 have on the function?

y

x

y=

4x2

y=

4(x3)2

Consider the equation y=4(x3)2 . What is a?

The axis of symmetry is x= 3.

x

2 1 0 1 24 x2 -16 -4 0 -4 -16

-4 (x-3)2 -100 -64 -36 -16 -4

Numerical Approach:

3-36

0

Axis of symmetry is shifted 3 units

to the right and becomes x= 3


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Vertex Form of a Quadratic Function

The quadratic function

f(x) = a(xh)2+ k, a 0

The graph offis a parabola .

Axis is the vertical line x = h.

Vertex is the point (h, k).

Ifa> 0, the parabola opens upward.

Ifa< 0, the parabola opens downward.


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6.6 Analyzing Graphs of Quadratic Functions

Afamily of graphsa group of graphs that displays one or

more similar characteristics. Parent graphy = x2

Notice that adding a constant to x2 moves the graph up.

Notice that subtracting a constant from x before squaring itmoves the graph to the right.

y = x2

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


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Vertex Form Each function we just looked at can be written in the form

(xh)2 + k, where (h , k) is the vertex of the parabola, andx = h is its axis of symmetry.

(xh)2 + kvertex form

Equation Vertex Axis of Symmetry

y = x2 ory = (x0)2 + 0

(0 , 0) x = 0

y = x2 + 2 ory = (x0)2 + 2

(0 , 2) x = 0

y = (x3)2 or

y = (x

3)2

+ 0

(3 , 0) x = 3


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Graph TransformationsAs the values of h and k change, the graph of

y = a(xh)2 + k is the graph of y = x2

translated.

| h | units leftifh is negative, or |h| units right

ifh is positive.

| k | units up ifkis positive, or | k | units down

ifkis negative.

G h Q d ti F ti i V t F


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Graph a Quadratic Function in Vertex Form Analyze y = (x + 2)2 + 1. Then draw its graph. This function can be rewritten as y = [x(-2)]2 + 1.

Then, h = -2 and k = 1

The vertex is at (h , k) = (-2 , 1), the axis of symmetry is x = -2. Thegraph is the same shape as the graph of y = x2, but is translate 2 unitsleft and 1 unit up.

Now use the information to draw the graph.

Step 1 Plot the vertex (-2 , 1)

Step 2 Draw the axis ofsymmetry, x = -2.

Step 3 Find and plot two

points on one sideof the axis symmetry,such as (-1, 2) and (0 , 5).

Step 4 Use symmetry to complete the graph.

G h T f ti


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Graph Transformations How does the value ofa in the general form

y = a(xh)2 + k affect a parabola? Compare the graphs

of the following functions to the parent function, y = x

2

.a.

b.

c.

d.

22y x

21

2y x

22y x

21

2y x

y = x2

a b

dc


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Example: Write a quadratic

function for a parabola with a vertex of

(-2,1) that passes through the point (1,-1). Since you know the vertex, use vertex form!

y=a(x-h)2+k

Plug the vertex in for (h,k) and the other pointin for (x,y). Then, solve for a.

-1=a(1-(-2))2+1

-1=a(3)2

+1-2=9a

a

9

21)2(

9

2 2

xy

Now plug in a, h, & k!


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Standard 9 Write a quadratic function in vertex form

Vertex form- Is a way of writing a quadratic equationthat facilitates finding the vertex.

y k= a(xh)2

The hand the krepresent the coordinates ofthe vertex in the form V(h, k).

The aif it is positive it will mean that our

parabola opens upward and if negative it will

open downward.

A small value for awill mean that our

parabola is wider and vice versa.


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Standard 9 Write a quadratic function in vertex form

Writey =x210x + 22 in vertex form.

Then identify the vertex.

y =x210x + 22 Write original function.

y+ ?= (x210x+ ?) + 22 Prepare to complete the square.

y+ 25= (x2

10x+ 25) + 22 Add 1022

( ) = (5)2

= 25 to each side.

y + 25 = (x5)2 + 22 Writex210x + 25 as abinomial squared.

y + 3 = (x5)2 Write in vertex form.

The vertex form of the function isy + 3= (x5)2.

The vertex is (5,3).

ANSWER


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EXAMPLE 1 Write a quadratic function in vertex form

Write a quadratic function for the parabola shown.

SOLUTION

Use vertex form because the

vertex is given.

y k= a(xh)2 Vertex form

y = a(x1)22 Substitute 1 forhand 2 fork.Use the other given point, (3, 2), to find a.2= a(3

1)2

2 Substitute 3 forxand 2 fory.

2 = 4a2 Simplify coefficient ofa.

1 = a Solve fora.


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EXAMPLE 1 Write a quadratic function in vertex form

A quadratic function for the parabola isy = (x1)22.

ANSWER


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EXAMPLE 1 Graph a quadratic function in vertex form

Graphy 5= (x + 2)2.1

4SOLUTION

STEP 1 Identify the constants

a = , h = 2, and k= 5.

Because a < 0, the parabola

opens down.

1

4

STEP 2 Plot the vertex

(h, k) = ( 2, 5) and draw the

axis of symmetryx =

2.


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EXAMPLE 1 Graph a quadratic function in vertex form

STEP 3 Evaluate the function for two values ofx.

x = 0:y = (0 + 2)2 + 5 = 414

x = 2:y = (2 + 2)2 + 5 = 11

4

Plot the points (0, 4) and

(2, 1) and their reflections in the

axis of symmetry.

STEP 4 Draw a parabola through the

plotted points.


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GUIDED PRACTICE for Examples 1 and 2

Graph the function. Label the vertex and axis of symmetry.

1. y = (x + 2)23 2. y =(x + 1)2 + 5


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GUIDED PRACTICE for Examples 1 and 2

3. f(x)= (x3)2412

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