SWBAT use quadratic functions to model real-life problems.
(Lesson 2- Chapter 3-1)
Warm up
1)Determine the quadratic function whose vertex is (1, - 5) and whose
y-intercept is – 3.
SWBAT use quadratic functions to model real-life problems.
(Lesson 2- Chapter 3-1)
Many applications involve finding the maximum and minimum value of a quadratic function. You can find the maximum or minimum value of a quadratic function by locating the vertex of the graph of a function.
Vertex of a Parabola
a)If a > 0, has minimum at . b) If a < 0, has maximum at .
Example 1) A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 degrees with respect to the ground. The path of the baseball is given by the function , where f(x) is the height the baseball (in feet) and x is the horizontal distance from home plate (in feet).
What is the maximum height reached by the baseball?
Example 2) A small local soft-drink manufacturer has daily production costs of
C= 70,000 – 120x + 0.075x, where C is the total cost (in dollars) and x is the number of units produced.
How many units should be produced each day to yield a minimum cost?
Example 3) The path of a diver is given below where y is the height (in feet) and x is the horizontal distance from the end of the diving board (in feet).
What is the maximum height of the diver?
Example 4) The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by
where p is the price charged per unit (in dollars).
a)Find the revenue earned for each price per unit given below
$20
$25
$30
b)Find the unit price that will yield a maximum revenue.
What is the maximum revenue?
Example 5) The marketing department at Texas Instruments has found that, when certain calculators are sold at a price p dollars per unit, the number x of calculators sold is given by the demand equation.
x- = 21,000 – 150p
a)Find a model that expresses the revenue R as function of the price p.
b)What is the domain of R.
c)What unit price should be used to maximize revenue?
d)If this price is charged, what is the maximum revenue?
e)How many units are sold at this price?
f)Graph R.
Practice
1)Write the function in standard form. Sketch the graph of the quadratic function without a graphing calculator. Identify the vertex, axis of symmetry, and x- intercepts.
2)Write the function in Vertex Form. Sketch the graph of the quadratic function without a graphing calculator. Identify the vertex, axis of symmetry, and x- intercepts.
3)Write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point.
Vertex (- 2,5) point (0,9)