Download Pre-Calculus Section 2

Pre-Calculus Section 2.5: Quadratic Functions; Maxima and Minima
Objective: TSWBAT find the maximum and minimum values of quadratic and other
functions.
Homework: pg 200 (1-7 odd, 15, 17, 19, 23, 27, 29-37 odd, 45-63 odd)
Daily Warm Up: Substitute into the difference quotient.
1. y  2 x 2  3x  2
Graphing Quadratic Functions Using the Standard Form
A quadratic function f ( x)  ax 2  bx  c can be expressed in the standard form
f ( x)  a( x  h)2  k
by completing the square. The graph of f is a parabola with vertex (h, k); the
parabola opens upward if a>0 or downward if a<0.
Example 1: Standard Form of a Quadratic Functions
Let f ( x)  2 x 2  12 x  23 .
(a) Express f in standard form.
(b) Sketch the graph of f.
Maximum or Minimum Values of Quadratic Functions
Let f be a quadratic function with standard form f ( x)  a( x  h)2  k . The
maximum or minimum value of f occurs at x = h.
If a > 0, then the minimum value of f is f(h) = k
If a < 0, then the maximum value of f is f(h) = k.
Example 2: Minimum Value of a Quadratic Function
Consider the quadratic function f ( x)  5 x 2  30 x  49 .
(a) Express f in standard form.
(b) Sketch the graph of f.
(c) Find the minimum value of f.
Example 3: Maximum Value of a Quadratic Function
Consider the quadratic function f ( x)   x 2  x  2 .
(a) Express f in standard form.
(b) Sketch the graph of f.
(c) Find the maximum value of f.
Maximum or Minimum Value of a Quadratic Function
The maximum or minimum value of a quadratic function
b
.
f ( x)  ax 2  bx  c occurs at x  
2a
 b 
If a>0, then the minimum value is f    .
 2a 
 b 
If a<0, then the maximum value is f    .
 2a 
Example 4: Finding the Maximum and Minimum Values of Quadratic Functions
Find the maximum or minimum value of each quadratic function.
(a) f ( x)  x 2  4 x
(b) f ( x)  2 x 2  4 x  5
Example 5: Maximum Gas Mileage for a Car
Most cars get their best gas mileage when traveling at a relatively modest speed. The
gas mileage M for a certain new car is modeled by the function
1
M ( s )   s 2  3s  31,
15  s  70
28
where s is the speed in mi/h and M is measured in mi/gal. What is the car’s best gas
mileage, and at what speed is it attained?
Using Graphing Devices to Find Extreme Values
If there is a viewing rectangle such that the point (a, f(a)) is the highest point on
the graph of f within the viewing rectangle (not on the edge), then the number f(a) is
called a local maximum value of f.
If there is a viewing rectangle such that the point (b, f(b)) is the lowest point on
the graph of f within the viewing rectangle, then the number f(b) is called a local
minimum value of f. In this case, f(b)  f(x) for all numbers x that are close to b.
Example 6: Finding Local Maxima and Minima from a Graph
Find the local maximum and minimum values of the function f ( x)  x3  8x  1 ,
correct to three decimals.
Example 7: A Model for the Food Price Index
A model for the food price index (the price of a representative “basket” of foods)
between 1990 and 2000 is given by the function
I (t )  0.0113t 3  0.0681t 2  0.198t  99.1
where t is measured in years since midyear 1990, 0  t  10 , and I(t) is scaled so that
I(3) = 100. Estimate the time when food was most expensive during the period 19902000.