Chapter 10 Quadratic Equations and Functions. Section 5 Graphing Quadratic Functions Using Properties. Section 10.5 Objectives. 1 Graph Quadratic Functions of the Form f ( x ) = ax 2 + bx + c 2 Find the Maximum or Minimum Value of a Quadratic Function
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Quadratic Equations and Functions
Graphing Quadratic Functions Using Properties
1 Graph Quadratic Functions of the Formf (x) = ax2 + bx + c
2 Find the Maximum or Minimum Value of a Quadratic Function
3 Model and Solve Optimization Problems Involving Quadratic Functions
The Vertex of a Parabola
Any quadratic function f(x) = ax2 + bx + c, a 0, will have vertex
The x-intercepts, if there are any, are found by solving the quadratic equation
f(x) = ax2 + bx + c = 0.
(– 2, 12)
16Graphing Using Properties
f(x) = –2x2 – 8x + 4
The vertex is (–2, 12).
The axis of symmetry is x = – 2.
The y-intercept is f(0) = –2(0)2 – 8(0) + 4 = 4
The x-intercepts occur where f(x) = 0.
–2x2 – 8x + 4 = 0
Use the quadratic formula to determine the x-intercepts.
x 4.4 x 0.4
will be the maximum value of f.
The vertex will be the lowest point on the graph if a > 0 and
will be the minimum value of f.Maximum and Minimum Values
The graph of a quadratic function has a vertex at
a > 0
a < 0
aApplications Involving Maximization
The revenue received by a ski resort selling x daily ski lift passes is
given by the function R(x) = – 0.02x2 + 24x. How many passes must be sold to maximize the daily revenue?
Step 1: Identify We are trying to determine the number of passes that must be sold to maximize the daily revenue.
Step 2: Name We are told that x represents the number of daily lift passes.
Step 3: Translate We need to find the maximum of R(x) = – 0.02x2 + 24x.
Step 5: Check
R(x) = – 0.02x2 + 24x
R(600) = – 0.02(600)2 + 24(600)
= – 0.02(360000) + 144000
= – 7200 + 14400
Step 6: Answer
The ski resort needs to sell 600 daily lift tickets to earn a maximum revenue of $7200 per day.