1. 2.4: Quadratic Functions

2. is called a quadratic function.
Let a, b, and c be real numbers a  0. The function f (x) = ax2 + bx + c
is called a quadratic function.
The graph of a quadratic function is a parabola.
Every parabola is symmetrical about a line called the axis (of symmetry). x y
The intersection point of the parabola and the axis is called the vertex of the parabola.
f (x) = ax2 + bx + c
vertex
axis
Quadratic function

3. The leading coefficient of ax2 + bx + c is a.y
a > 0 opens upward
When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum.
f(x) = ax2 + bx + c
vertex minimum x y
vertex maximum
When the leading
coefficient is negative, the parabola opens downward and the vertex is a maximum.
f(x) = ax2 + bx + c
a < 0 opens downward
Leading Coefficient

4. Case1: a>0 y a > 0 opens upward Minimum value: k Range: x
vertex minimum
Minimum value: k
Range:
Increasing:
Decreasing:

5. Case2: a<0 y x Maximum value: k vertex maximum Range:
a < 0 opens downward
vertex maximum
Maximum value: k
Range:
Increasing:
Decreasing:

6. Def:
The standard form for the equation of a quadratic function is: f (x) = a(x – h)2 + k (a  0)
where
h=-b/2a
k=f(h)=f(-b/2a)
2) Vertex : (h, k)
3) Axis of symmetry : x=h

7. Quadratic Function in Standard Form
The standard form for the equation of a quadratic function is: f (x) = a(x – h)2 + k (a  0)
The graph is a parabola opening upward if a  0 and opening downward if a  0. The axis is x = h, and the vertex is (h, k).
Example: By completing the square method write the parabola
f (x) = 2x2 + 4x – 1in standard form and find the axis and vertex. x y
x = –1
f (x) = 2x2 + 4x – 1
f (x) = 2x2 + 4x – original equation
f (x) = 2( x2 + 2x) – factor out 2
f (x) = 2( x2 + 2x + 1) – 1 – 2 complete the square
f (x) = 2( x + 1)2 – standard form
a > 0  parabola opens upward .
(–1, –3)
h = –1, k = –3  axis x = –1, vertex (–1, –3).
Quadratic Function in Standard Form

8. Ex1: For the following functions
Write the function in the standard form
Find the vertex
Find the axis of symmetry
Find , if any, the maximum value of the function
Find , if any, the minimum value of the function
Find the range of the function
Find the interval(s) of increasing and decreasing
Sketch the graph of the function and show on the graph the intercept(s), the vertex, and the axis of symmetry
a) f(x)=2x2+4x+3
b) f(x)=-x2+2x+3

9. f (x) = a(x – h)2 + k standard form
Example: Find an equation for the parabola with vertex (2, –1) passing through the point (0, 1). y x
y = f(x)
(0, 1)
(2, –1)
f (x) = a(x – h)2 + k standard form
f (x) = a(x – 2)2 + (–1) vertex (2, –1) = (h, k)
Since (0, 1) is a point on the parabola: f (0) = a(0 – 2)2 – 1
1 = 4a –1 and
Example: Parabola

10. The maximum height of the ball is 15 feet.
Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:
The path is a parabola opening downward. The maximum height occurs at the vertex.
At the vertex,
So, the vertex is (9, 15).
The maximum height of the ball is 15 feet.
Example: Basketball

11. Let x represent the width of the corral and 120 – 2x the length.
Example: A fence is to be built to form a rectangular corral along the side of a barn 65 feet long. If 120 feet of fencing are available, what are the dimensions of the corral of maximum area?
barn
corral x
120 – 2x
Let x represent the width of the corral and 120 – 2x the length.
Area = A(x) = (120 – 2x) x = –2x x
The graph is a parabola and opens downward.
The maximum occurs at the vertex where
a = –2 and b = 120
120 – 2x = 120 – 2(30) = 60
The maximum area occurs when the width is 30 feet and the length is 60 feet.
Example: Maximum Area

12. Q85/227 Find two numbers whose sum is 8 and whose product is a maximum.
Ex: If x is a real number, then find the maximum area of
a rectangle of length 3+2x and width 1-2x.
Ex: If x=3 is the axis of symmetry of the parabola
y=-2x2+cx+2, then find c
The End
Example: Basketball