THE GRAPH OF A QUADRATIC FUNCTION THE GRAPH OF A QUADRATIC FUNCTION

THE GRAPH OF A QUADRATIC THE GRAPH OF A QUADRATIC FUNCTIONFUNCTION

QUADRATIC QUADRATIC FUNCTIONSFUNCTIONSQUADRATIC QUADRATIC FUNCTIONSFUNCTIONS

A quadratic function is a function A quadratic function is a function of the formof the form

f(x) = ax f(x) = ax 2 2 + bx + c+ bx + c

where a, b & c are real numbers and a where a, b & c are real numbers and a 0 0

The domain of a quadratic The domain of a quadratic function is all real numbers.function is all real numbers.

GRAPHS OF GRAPHS OF QUADRATIC QUADRATIC FUNCTIONSFUNCTIONS


As we’ve already seen, f(x) = xAs we’ve already seen, f(x) = x2 2 graphs into a PARABOLA.graphs into a PARABOLA.

This is the simplest quadratic This is the simplest quadratic function we can think of. We will function we can think of. We will use this one as a model by which use this one as a model by which to compare all other quadratic to compare all other quadratic functions we will examine.functions we will examine.

VERTEX OF A VERTEX OF A PARABOLAPARABOLA

VERTEX OF A VERTEX OF A PARABOLAPARABOLA

All parabolas have a VERTEX, the All parabolas have a VERTEX, the lowest or highest point on the lowest or highest point on the graph (depending upon whether it graph (depending upon whether it opens up or down.)opens up or down.)

AXIS OF SYMMETRYAXIS OF SYMMETRYAXIS OF SYMMETRYAXIS OF SYMMETRY

All parabolas have an AXIS OF All parabolas have an AXIS OF SYMMETRY, an imaginary line SYMMETRY, an imaginary line which goes through the vertex which goes through the vertex and about which the parabola is and about which the parabola is symmetric.symmetric.

HOW PARABOLAS HOW PARABOLAS DIFFERDIFFER

HOW PARABOLAS HOW PARABOLAS DIFFERDIFFER

Some parabolas open up and Some parabolas open up and some open down.some open down.

Parabolas will all have a different Parabolas will all have a different vertex and a different axis of vertex and a different axis of symmetry.symmetry.

Some parabolas will be wide and Some parabolas will be wide and some will be narrow.some will be narrow.

y = (x - 3)y = (x - 3)2 2 - - 44

x y

6

5

4

3

2

1

0

5

0

-3

-4

-3

0

5

Y-intercept

Rootsor x intercepts

VertexA

xis of Sym

metry

Example



The general form of a quadratic The general form of a quadratic function is:function is:

f(x) = axf(x) = ax22 + bx + c + bx + c

The position, width, and The position, width, and orientation of a particular orientation of a particular parabola will depend upon the parabola will depend upon the values of a, b, and c.values of a, b, and c.



Compare f(x) = xCompare f(x) = x22 to the to the following:following:

f(x) = 2xf(x) = 2x22 f(x) = .5x f(x) = .5x2 2 f(x) = f(x) = -.5x-.5x22

If a > 0, then the parabola opens If a > 0, then the parabola opens upup

If a < 0, then the parabola opens If a < 0, then the parabola opens downdown



Now compare f(x) = xNow compare f(x) = x22 to the to the following:following:

f(x) = x f(x) = x 22 + 3+ 3 f(x) = x f(x) = x 22 - 2 - 2

Vertical shift upVertical shift up Vertical shift Vertical shift downdown



Now compare f(x) = xNow compare f(x) = x22 to the to the following:following:

f(x) = (x + 2)f(x) = (x + 2)22 f(x) = (x – 3)f(x) = (x – 3)22

Horizontal shift Horizontal shift to the leftto the left

Horizontal shift Horizontal shift to the rightto the right



When the general form of a quadratic When the general form of a quadratic function f(x) = axfunction f(x) = ax22 + bx + c is + bx + c is changed to the vertex form:changed to the vertex form:

f(x) = a(x - h) f(x) = a(x - h) 22 + k + k

We can tell by horizontal and vertical We can tell by horizontal and vertical shifting of the parabola where the shifting of the parabola where the vertex will be.vertex will be.

The parabola will be shifted h units The parabola will be shifted h units horizontally and k units vertically.horizontally and k units vertically.



Thus, a quadratic function written in Thus, a quadratic function written in the form the form

f(x) = a(x - h) f(x) = a(x - h) 22 + k + k

will have a vertex at the point (h,k).will have a vertex at the point (h,k).

The value of “a” will determine The value of “a” will determine whether the parabola opens up or whether the parabola opens up or down (positive or negative) and down (positive or negative) and whether the parabola is narrow or whether the parabola is narrow or wide.wide.

GRAPHS OF QUADRATIC GRAPHS OF QUADRATIC FUNCTIONSFUNCTIONS


f(x) = a(x - h) f(x) = a(x - h) 22 + k + k

Vertex (highest or lowest point): Vertex (highest or lowest point): (h,k)(h,k)

If a > 0, then the parabola opens If a > 0, then the parabola opens upup

If a < 0, then the parabola opens If a < 0, then the parabola opens downdown



Axis of SymmetryAxis of Symmetry

The vertical line about which the The vertical line about which the graph of a quadratic function is graph of a quadratic function is symmetric.symmetric.

x = hx = h

where h is the x-coordinate of the where h is the x-coordinate of the vertex.vertex.



So, if we want to examine the So, if we want to examine the characteristics of the graph of a characteristics of the graph of a quadratic function, our job is to quadratic function, our job is to transform the general form:transform the general form:

f(x) = axf(x) = ax22 + bx + c + bx + c

into the vertex form:into the vertex form:

f(x) = a(x – h)f(x) = a(x – h)22 + k + k



This will require to process of This will require to process of completing the square which is a completing the square which is a little differentlittle different than completing than completing the square to solve a quadratic the square to solve a quadratic

equation.equation.

Factor x2 + 6x + 9 (x + 3)(x + 3) or (x + 3)2

Perfect Square Trinomial

The factors are in the form(x + a)2 or (x - a)2.

Note the relationship between the middle term and the last term.The last term is one-half the middle term squared.

Find the value of the last term that will make the followingperfect square trinomials.

x2 + 14x + _______

x2 + 7x + _______

x2 - 3x + _______

(x + 7)249

Remember about Perfect Square Trinomials

1

26

2

= 32 = 9

49

49

4

x 7

2

2

x 3

2

2

Changing from Standard Form to Vertex Form

Write y = x2 + 10x + 23 in the form y = a(x - h)2 + k. Sketch the graph.

1. Bracket the first two terms.y = x2 + 10x + 23

y = (x2 + 10x + ____ - ____) + 23

( )

2. Add a value within the brackets to make a perfect square trinomial. Whatever you add must be subtracted to keep the value of the function the same.

25 25

3. Group the perfect square trinomial.

4. Factor the trinomial and simplify.

y = (x2 + 10x + 25) - 25 + 23

y = (x + 5)2 - 2

(-5, -2)

Changing from Standard Form to Vertex FormWrite y = 2x2 - 12x -11 in the form y = a(x - h)2 + k. Sketch the graph.

1. Bracket the first two terms.y = 2x2 - 12x - 11

y = 2(x2 - 6x + ____ - ____) - 11

( )

3. Add a value within the brackets to make a perfect square trinomial. Whatever you add must be subtracted to keep the value of the function the same.

9 9

4. Group the perfect square trinomial. When grouping the trinomial, remember to distribute the coefficient.

5. Factor the trinomial and simplify.

(3, -29)

2. Factor out the coefficient of the x2- term.

y = 2(x2 - 6x) - 11

y = 2(x2 - 6x + 9) - 18 - 11

y = 2(x - 3)2 - 29Multiply, when you remove this term from the brackets.

1892

y = -3x2 + 5x - 1

y = -3x2 + 5x - 1( )

y = -3(x2 - x) - 1

y = -3(x2 - x + ______ - ______ ) - 125

36

25

36

y = -3(x2 - x + ) - 175

36

y = -3(x - )2 + - 125

12

y = -3(x - )2 +25

12

12

12

y = -3(x - )2 +13

12

Vertex is5

6,13

12

Completing the Square

5

35

35

3

25

36

5

6

5

65

6

y = ax2 + bx + c

y = (ax2 + bx ) + c

ya(x2 b

ax) c

ya(x2 b

ax

b2

4a2 b2

4a2 ) c

ya(x2 b

ax

b2

4a2 ) a(b2

4a2 ) c

a

bac

a

bxay

4

4)

2(

22

The vertex is

( b

2a,4ac b2

4a).

Using the general form, y = ax2 + bx + c, complete the square:

Completing the Square - The General Case

This IS the vertex BUT it is easier just to remember that the x-value is and then plug that in to the equation to get the y-value for the vertex.

a

b

2

a

bc

a

bxay

4)

2(

22

Find the vertex and the maximum or minimum value of

f(x) = -4x2 - 12x + 5

using the axis of symmetry, the vertex is

Find the x-value of the vertex:

x b

2a

x ( 12)

2( 4)

x 3

2

Find the y-value of the vertex:

Using the Vertex Formula

The vertex is 3

2,14

.

Therefore there is a maximum of

y = 14, when x = 3

2.

)2

(,2 a

bf

a

b

)2

(a

bfy

52

312

2

34

2

32

fy

Direction of the ParabolaDirection of the Parabola

If the coefficient If the coefficient of xof x22 is is positivepositive the the parabola will parabola will open up.open up.

If the coefficient If the coefficient of xof x22 is is negativenegative the the parabola will parabola will open downopen down..

CHARACTERISTICS OF THE CHARACTERISTICS OF THE GRAPH OF A QUADRATIC GRAPH OF A QUADRATIC

FUNCTIONFUNCTION

CHARACTERISTICS OF THE CHARACTERISTICS OF THE GRAPH OF A QUADRATIC GRAPH OF A QUADRATIC

FUNCTIONFUNCTIONf(x) = axf(x) = ax22 + bx + c + bx + c

2ab-

x :SYMMETRY OF AXIS 2ab-

f 2ab-

VERTEX

,

Parabola opens up and has Parabola opens up and has a minimum value if a > 0.a minimum value if a > 0.

Parabola opens down and Parabola opens down and has a maximum value if a < has a maximum value if a < 0.0.

EXAMPLEEXAMPLEEXAMPLEEXAMPLE

Determine without graphing Determine without graphing whether the given quadratic whether the given quadratic function has a maximum or function has a maximum or minimum value and then find the minimum value and then find the value. Verify by graphing.value. Verify by graphing.

f(x) = 4xf(x) = 4x22 - 8x + 3 - 8x + 3 g(x) = -2xg(x) = -2x22 + + 8x + 38x + 3

THE X AND Y INTERCEPTS THE X AND Y INTERCEPTS OF A QUADRATIC FUNCTIONOF A QUADRATIC FUNCTIONTHE X AND Y INTERCEPTS THE X AND Y INTERCEPTS

OF A QUADRATIC FUNCTIONOF A QUADRATIC FUNCTION1.1. Find the x-intercepts by setting the quadratic Find the x-intercepts by setting the quadratic

function equal to zero and solve by whatever function equal to zero and solve by whatever method is easiest.method is easiest.

2.2. If the discriminant bIf the discriminant b22 – 4ac > 0, the graph of f(x) – 4ac > 0, the graph of f(x) = ax= ax22+ bx + c has two distinct x-intercepts and + bx + c has two distinct x-intercepts and will cross the x-axis twice.will cross the x-axis twice.

3. If the discriminant b3. If the discriminant b22 – 4ac = 0, the graph of f(x) – 4ac = 0, the graph of f(x) = ax= ax22 + bx + c has one x-intercept and touches + bx + c has one x-intercept and touches the x-axis at its vertex.the x-axis at its vertex.

4. If the discriminant b4. If the discriminant b22 – 4ac < 0, the graph of f(x) – 4ac < 0, the graph of f(x) = ax= ax2 2 + bx + c has no x-intercept and will not + bx + c has no x-intercept and will not cross or touch the x-axis.cross or touch the x-axis.

5. Find the y-intercept by substituting x=0 into 5. Find the y-intercept by substituting x=0 into function. function.

GRAPHING QUADRATIC GRAPHING QUADRATIC FUNCTIONSFUNCTIONS

GRAPHING QUADRATIC GRAPHING QUADRATIC FUNCTIONSFUNCTIONS

Graph the functions below by Graph the functions below by hand by determining whether its hand by determining whether its graph opens up or down and by graph opens up or down and by finding its vertex, axis of finding its vertex, axis of symmetry, y-intercept, and x-symmetry, y-intercept, and x-intercepts, if any. Verify your intercepts, if any. Verify your results using a graphing results using a graphing calculator.calculator.

f(x) = 2xf(x) = 2x22 - 3 - 3 g(x) = xg(x) = x22 - 6x - 6x - 1- 1

h(x) = 3xh(x) = 3x22 + 6x + 6x k(x) = -2xk(x) = -2x22 + + 6x + 26x + 2

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THE GRAPH OF A QUADRATIC FUNCTION THE GRAPH OF A QUADRATIC FUNCTION