10(.98282059854525)xy

4010(.5)x

y

10(.98)xy Put each in your calculator and check what y equals when x = 90.

Mystery Solved!!!

Chapter 2: Functions and Models

Lesson 6: Quadratic Models

Mrs. Parziale

Vocabulary

• Quadratic models: models based on quadratic functions

2( )0

f x ax bx cwhere a

Properties of Quadratics:

• Parabola opens upward for a>0 (positive) and downward for a<0 (negative).

• Vertex is the place where the axis of symmetry and the parabola intersect.

• Vertex is the minimum for a>0 and maximum for a<0

2( ) 0f x ax bx c where a

• Domain: all real numbers• Range: depends on the location of the

vertex. (range ≥ min or range ≤ max)• The x-intercepts are the solution to f(x) = 0

(done either by factoring or the quadratic formula).

• The y-intercept is (0, c).

2( ) 0f x ax bx c where a

Example 1:

Consider f(x) = 2x2 – 9x + 3(a) Find its x- and y-intercepts(b) Tell whether the parabola has a maximum or

minimum point, and find the coordinates of the vertex.

Example 2: Physics application – Famous quadratic model = Newton’s

Formula for Height of an Object Thrown Vertically

where g = accel. due to gravity (9.8 for m/s2 and 32 for ft/s2).

A projectile shot from a tower 10 feet high with an upward velocity of 100 feet/second.

(a) Approximate the relationship between height (h) in feet and time (t) in seconds after the projectile is shot. (formula)

(b) How long will the projectile be in the air?

20 0

12

h gt v t h

Example 3: A pizza is sliced by a number of straight cuts as shown below. The table shows the greatest number of pieces f(n) into which it can be sliced by n cuts.

a. Fit a quadratic model to these data.

b. Use your model to find the greatest number of pizza pieces produced by 5 straight cuts.

n 0 1 2 3 4f(n) 1 2 4 7 11

Closure

• Given the following quadratic function:

• What is the y-intercept?• Does this parabola have a min or max?• How do you find the zeros (x-intercepts)?• What is the domain? Range?

2( ) 2 5 3f x x x