Quadratic Equation and Quadratic Equation and Function of Second GradeFunction of Second Grade

OBJECTIVES:

• Know and apply mathematical concepts associated with the study of the quadratic function.

• Graph a quadratic function, determining vertex, axis of symmetry and concavity.

• Display graphic features of a parabola through discriminant analysis.

• Determine the intersection of the parabola with the Cartesian axes.

• Determine the roots of an equation of 2nd degree.

Content1. Quadratic function

2. 2nd degree equation

1.1 Y axis intercept1.2 Concavity1.3 Axis of symmetry and vertex

2.1 Roots of a quadratic equation2.2 Properties of the roots2.3 Discriminate

1.4 Discriminate

1. Quadratic Function It is of the form

f(x) = ax2 + bx + c

Examples:

And its graph is a parabola.

a) If f(x) = 2x2 + 3x + 1

b) If f(x) = 4x2 - 5x - 2

a = 2, b = 3 y c = 1

a = 4, b = -5 y c = -2

con a =0; a,b,c IR

1.1. Intersection with Y axisIn the quadratic function f (x) = ax2 + bx + c, the coefficient c indicates the ordinate of point Y where the parabola intersects the axis

x

y

x

y

c(0,C)

1.2. ConcavityIn the quadratic function f (x) = ax2 + bx + c, the coefficient a indicates whether the parabola is concave up or down.

If a> 0, is concave up

If a <0, is concave downward

Then, the parabola intersects the Y axis at the point (0, - 4) and is concave upward.

x

y

Example:In the function f (x) = x2 - 3x - 4, a = 1 and c = - 4.

(0,-4)

The value of "b" in the equation allows to know the movement horizontal parabola and the "a" concave.

Be the quadratic function f (x) = ax ² + bx + cThen

IF a>0 y b<0 The parabola opens upward and is oriented to                                right.

IF a>0 y b>0 The parabola opens upward and is oriented to                                left

IF a<0 y b>0 The parabola opens downward and is oriented to right

IF a<0 y b<0 The parabola opens downward and is oriented                                left

1.3 The importance of the value of "a" and "b"

Ej. f(x)=2x² - 3x +2

Ej. f(x)=x² + 3x - 2

Ej. f(x)=-3x² + 4x – 1

Ej. f(x)=-x² - 4x + 1

1.4. Axis of symmetry and vertex

The axis of symmetry is the line through the vertex of the parabola, and is parallel to the axis Y.

x

y Axis of symmetry

vertex

The vertex of a parabola is the highest or lowest point of the curve, as its concavity.

IF f(x) = ax2 + bx + c , Then:

b) Its vertex is:

a) Its axis of symmetry is:

2a 2aV = -b , f -b

4a -b , 4ac – b2

2aV =

-b2a

x =

Example:

2·1 -2x =

In the function f(x) = x2 + 2x - 8, a = 1, b = 2 y c = - 8, then:

V = ( -1, f(-1) )

a) Its axis of symmetry is:

x = -1

b) Its vertex is:

V = ( -1, -9 )

2a -bx =

-b , f -b2a 2a

V =

f(x)

V = ( -1, -9 )

x = -1Axis of symmetry:

vertex:

1.It means that the function is moved to the left or  right, h units and opens upward or downward.Ex. 1) y=2(x-3)² (↑→) 2) y=-3(x-4)² (↓→)

If y=ax² any quadratic function, then:

2. y =a(x+h)² means that the function is moved to the left or                                  right h units and opens up or down.Ex. 1) y= 4(x+2)² (↑←) 2) y=-(x+1)² (↓←)

1.5 1.5 Behavior of the function according to "a", "h" and "k"Behavior of the function according to "a", "h" and "k"

x

yx

y

xy

3. y=a(x-h)² ± k means that the function is moved to the right or left k units up or down.Ex. 1) y=5(x-1)² - 4 (↑→↑) 2) y=-3(x-7)² + 6 (↓→↓)

4. y=a(x + h)² ± k means that the function is moved to the right or left k units up or down.

Ex. 1) y=(x+6)² - 5 (↑←↑) 2) y=-5(x+3)² + 3 (↓←↓)

Obs. V(h,k) is the vertex of the parabola.

xy

f the parabola is opened upward, the vertex is a minimum and if the parabola is open downward, the vertex is a maximum.

The discriminant is defined as:

Δ = b2 - 4ac

a) If the discriminant is positive, then the parabola intersects two points on the axis X.

Δ > 0

1.6. Discriminate

If the discriminant is negative, then the      NO parabola intersects the axis X.

Δ < 0

c) If the discriminant is zero, then the      parabola intersects at a single point to the X axis is

tangent to it.

Δ = 0

x2x1

2. Quadratic EquationA quadratic or quadratic equation is of the form:

ax2 + bx + c = 0, con a ≠ 0

Every quadratic equation has two solutions or roots. If these are real, correspond to points of intersection of the parabola f (x) = ax2 + bx + c with the x-axis

x2 x

y

x1

Example:In the function f (x) = x2 - 3x - 4, the associated equation: x2 - 3x - 4 = 0, has roots -1 and 4.Then the parabola intersects the X axis at those points.

2.1. Roots of an equation of 2nd degreeFormula for determining the solutions (roots) of a quadratic equation:

- b ± b2 – 4ac

2ax =

Example:Determine the roots of the equation: x2 - 3x - 4 = 0

-(-3) ± (-3)2 – 4·1(- 4)

2x =

3 ± 9 + 162

x =

3 ± 252

x =

2x = 3 ± 5

2x = 8

2x = -2

x1 = 4 x2 = -1

You can also obtain the roots of the equation by factoring the product of binomials:

x2 - 3x - 4 = 0(x - 4)(x + 1) = 0

(x - 4)= 0 ó (x + 1)= 0x1 = 4 x2 = -1

2.2. Properties of the rootsIf x1 and x2 are the roots of a quadratic equation of the form ax2 + bx + c = 0, then:

-bax1 + x2 =

cax1 · x2 =

Δax1 - x2 = ±

1)

2)

3)

Given the roots or solutions of a quadratic equation, you can determine the equation associated with them.

                           a (x - x1) (x - x2) = 0

In a quadratic equation, the discriminant

Δ = b2 - 4ac

a) If the discriminant is positive, then the quadratic equation has two real solutions x1, x2 and distinct.

The parabola intersects at two points to the axis X.

Δ > 0

2.3. Discriminate

Provides information on the nature of the roots.

x1, x2 are real and          x1 ≠ x2

x2x1

b) If the discriminant is negative, then the quadratic equation has no real solution.

The parabola NO X axis intersects

Δ < 0

x1, x2 are complex conjugates          x1 = x2

c) If the discriminant is zero, then the quadratic equation has two real and equal roots.

The parabola intersects at a single point to the axis X.

Δ = 0

x1, x2 are real and          x1 = x2

x2x1=

1.-Questions2-Exercises:a)y = x²-2.b)y=4x²-3.c)y=x²-6+x.d)Y=x²-5x+6.e)Y=9-x².