QUADRATIC FUNCTIONS
Quadratic function: Is one of the form f(x) = ax2 + bx + c, where a 0.
First we are going to distinguish the values for a,b,c
Examples: a) f(x) = 4x2 + 6x + 1 a=4 b=6 c=1
b) f(x) = 5x2 – 2x a = 5 b = -2 c= 0
c) f(x) = 4x2 – 1 a=4 b=0 c=1
The graph of a quadratic function is a curve called a parabola
Concavity: If a 0, the function is concave up.
If a 0, the function is concave down.
Intersection with the ordinate axis or “y” axis: (0,c).
Intersection with the abscissa axis or “x” axis:
First we are going to analyze first the discriminant
= b2 – 4ac.
If 0, the graph of the function will have two x-intercepts.
If 0, the graph of the function will have no x-intercepts.
If = 0, The graph of the function will have an x-intercept.
Vertex: The vertex is an ordered pair, indicating the maximum or minimum point according to
the concavity. If it is concave up it will be a minimum point and if it is concave down, it will be a
−𝑏 −∆
maximum point : ( 2𝑎 , 4𝑎 )
The axis of symmetry always passes through the vertex of the parabola. The x -coordinate of the
−𝑏
vertex is the equation x = 2𝑎
Intervals of monotonicity:
Each Interval indicates where the function increases or decreases.
Range of a function:
−∆
If the function is concave up 𝑡ℎ𝑒 𝑟𝑎𝑛𝑔𝑒 𝑖𝑠 [ 4𝑎 , +∞[
−∆
If the function is concave down the range is]−∞, 4𝑎 ]
, Example 1: f(x) = x2 – 3x + 2
a= 1 b = -3 c=2
a) Concavity: a=1, it is concave up.
b) Intersection with the ordinate axis or “y” axis: (0,2)
c) Intersection with the abscissa axis or “x” axis:
= b2 – 4ac
= (-3)2 – 4(1· 2) = 1 the graph of the function will have two x-intercepts.
−𝑏+ √∆
X1 = 2𝑎
−−3+ √1
X1 = =2
2·1
−𝑏− √∆
X2 = 2𝑎
−−3− √1
X2 = =1
2·1
Intersection with the abscissa axis or “x” axis: (2,0) y (1,0)
−𝑏 −∆ −−3 −1 3 −1
d) Vertex: ( 2𝑎 , 4𝑎 ) = ( 2·1 , 4·1) = (2 , )
4
−𝑏 −−3 3
e) Axis of symmetry; x= 2𝑎 x= x=2
2·1
f) Intervals of monotonicity: it is concave up, then
−𝑏 3
Decreases in ]−∞, 2𝑎 [ = ]−∞, 2 [
−𝑏 3
Increases in ] , +∞ [ = ] , +∞ [
2𝑎 2
−∆
g) Range: [ 4𝑎 , +∞[
−1 −1
[4·1 , +∞[ = [ 4 , +∞[
Quadratic function: Is one of the form f(x) = ax2 + bx + c, where a 0.
First we are going to distinguish the values for a,b,c
Examples: a) f(x) = 4x2 + 6x + 1 a=4 b=6 c=1
b) f(x) = 5x2 – 2x a = 5 b = -2 c= 0
c) f(x) = 4x2 – 1 a=4 b=0 c=1
The graph of a quadratic function is a curve called a parabola
Concavity: If a 0, the function is concave up.
If a 0, the function is concave down.
Intersection with the ordinate axis or “y” axis: (0,c).
Intersection with the abscissa axis or “x” axis:
First we are going to analyze first the discriminant
= b2 – 4ac.
If 0, the graph of the function will have two x-intercepts.
If 0, the graph of the function will have no x-intercepts.
If = 0, The graph of the function will have an x-intercept.
Vertex: The vertex is an ordered pair, indicating the maximum or minimum point according to
the concavity. If it is concave up it will be a minimum point and if it is concave down, it will be a
−𝑏 −∆
maximum point : ( 2𝑎 , 4𝑎 )
The axis of symmetry always passes through the vertex of the parabola. The x -coordinate of the
−𝑏
vertex is the equation x = 2𝑎
Intervals of monotonicity:
Each Interval indicates where the function increases or decreases.
Range of a function:
−∆
If the function is concave up 𝑡ℎ𝑒 𝑟𝑎𝑛𝑔𝑒 𝑖𝑠 [ 4𝑎 , +∞[
−∆
If the function is concave down the range is]−∞, 4𝑎 ]
, Example 1: f(x) = x2 – 3x + 2
a= 1 b = -3 c=2
a) Concavity: a=1, it is concave up.
b) Intersection with the ordinate axis or “y” axis: (0,2)
c) Intersection with the abscissa axis or “x” axis:
= b2 – 4ac
= (-3)2 – 4(1· 2) = 1 the graph of the function will have two x-intercepts.
−𝑏+ √∆
X1 = 2𝑎
−−3+ √1
X1 = =2
2·1
−𝑏− √∆
X2 = 2𝑎
−−3− √1
X2 = =1
2·1
Intersection with the abscissa axis or “x” axis: (2,0) y (1,0)
−𝑏 −∆ −−3 −1 3 −1
d) Vertex: ( 2𝑎 , 4𝑎 ) = ( 2·1 , 4·1) = (2 , )
4
−𝑏 −−3 3
e) Axis of symmetry; x= 2𝑎 x= x=2
2·1
f) Intervals of monotonicity: it is concave up, then
−𝑏 3
Decreases in ]−∞, 2𝑎 [ = ]−∞, 2 [
−𝑏 3
Increases in ] , +∞ [ = ] , +∞ [
2𝑎 2
−∆
g) Range: [ 4𝑎 , +∞[
−1 −1
[4·1 , +∞[ = [ 4 , +∞[
