Quadratic Functions: Notes for Skills
A quadratic function is a function of the form y = ax2 + bx + c where a, b, c are real numbers and a
0
The graph of a quadratic function is called a parabola
y
y
y = x2
y = − x2
x
x
If a > 0 then the parabola opens_____________________
If a < 0 then the parabola opens_____________________
Vertex: the lowest point on a parabola that opens upward
the highest point on a parabola that opens downward
Concavity:
a parabola that opens upward is concave up
a parabola that opens downward is concave down
General Form of a Quadratic Functions: y = ax2 + bx + c
Standard Form of a Quadratic Functions: y = a(x −h)2 + k sometimes called Vertex Form
The point (h, k) is the vertex of the parabola
Axis of symmetry: the vertical line y = k through the vertex
A parabola is symmetric about the vertical line y = k that passes through the vertex.
Example: Find the vertiex and axis of symmetry for each parabola and determine whether the
parabola is concave up or concave down.
y = −2 (x−3)2 + 5
y = 3 (x+7)2 − 2
Completing the square to find the vertex of a parabola: y = 2x2 − 12x + 19
Completing the square to find the vertex of a parabola
y = ax2 + bx + c
b
y = a ( x2 + x ) + c
a
2
2
b
1 b
1 b
y + a    = a ( x2 + x +    ) + c
a
2 a
2 a
b 2
b 2
 = a ( x +
) + c
2a
2a



y + a 
2
 b 
b 2
) + c − a  
2a
 2a 
2
y = a ( x −h) + k
y =a(x+
Therefore the x coordinate of the vertex is h = 
b
2a

b
 ;
2a


The y coordinate of the vertex is f  
substitute h = 
b
back into the equation for the quadratic function to find the y coordinate of the vertex.
2a
Example: y = 2x2 − 12x + 19
x coordinate of vertex:
y coordinate of vertex:
Finding the minimum or maximum value of a quadratic function.
The minimum value of a quadratic function that opens upward is the y value at the vertex.
A quadratic function that opens upward does not have a maximum value.
The maximum value of a quadratic function that opens downward is the y value at the vertex.
A quadratic function that opens downward does not have a minimum value.
A question may ask you to
 find the minimum (or maximum) which is a y value.
 find the x value at which the minimum (or maximum) value occurs.
 find the coordinates (x,y) of the point that is the minimum (or maximum)
Read the question carefully so that you understand what it is asking for in the context of the
situation in the particular problem and answer it properly.
Applications finding the minimum or maximum value of a quadratic function:
Finding the maximum height of a projectile thrown into the air
Finding the maximum area of a fenced or partitioned region
Maximizing profit
Common applications of quadratic functions
Geometric problems involving area
Objects thrown straight up into the air
Objects dropped from a height
Projectile motion (height as function of horizontal distance)
Economic applications