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.