"Relationships Described by Quadratic Functions" Quadratic Function PPT Courseware 2

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"Relationships Described by Quadratic Functions" Quadratic Function PPT Courseware 2

Dare to explore

What circular patterns have you seen in your life? What does the size of their area depend on?

Let the radius of the circle be x and the area be y.

(1) Is y a function of x?

yes. Because when a value of x is given, a value of y is determined accordingly, so y is a function of x.

(2) If yes, please write the functional relationship between y and x;

The functional relationship between y and x is: y =лx 2

A rectangular lawn has a length and width of 100m and 80m respectively. Now we want to make two mutually perpendicular paths with a width of xm in the lawn. The remaining lawn area is ym². Is there a functional relationship between y and x? If so, please write down the functional relationship between y and x;

yes. Because when a value of x is given, a value of y is determined accordingly, so y is a function of x.

The functional relationship between y and x is: y=(100-x)(80-x)

That is y=x²-180x+8000.

Observation and induction

y = л x 2，y =- x²+30 x，

y=x²-180x+8000, y=100x²+200x+100.

1. From the functional formula obtained above, is y a proportional function of x? A one-time function?

Inverse function?

2. What characteristics do they have in common? Can you derive the definition of a quadratic function by analogy?

Definition: Generally, a function of the form y=ax²+bx+c (a, b, c are constants, a≠ 0) is called the quadratic function of x.

Which of the following functions are quadratic functions?

(1)y=3(x-1)²+1 (2)s=-2t²

(3)y=(x+3)²-x² (4)y=1/x²-x

(5)y=-1/2+3x² (6)y=2²+2x,

(7)s=1+t+5t²,

Several different representations of the quadratic function y=ax²+bx+c (a, b, c are constants, a≠0):

(1)y=ax²(a≠0). (2)y=ax²+c(a≠0).

(3)y=ax²+bx (a≠0) (4)y=ax²+bx+c (a≠0)

Apply what you learn

The side length of a square is 10cm. If a small square with side length x (x<10)cm is dug out from it, the area of ​​the remaining part is y.

(1) Write the relationship expression between y(cm2) and x(cm), and indicate what function it is;

Solution: y=100-x², which is a quadratic function

(2) When x is 2cm, 4cm, or 6cm, what are the corresponding y values?

Solution: When x=2, y=100-4=96;

When x=4, y=100-16=84;

When x=6, y=100-36=64;

∴When x is 2cm, 4cm, and 6cm, the corresponding y values ​​are 96 cm2, 84 cm2, and 64cm2 respectively.

There are 100 orange trees in an orchard, and each tree bears an average of 600 oranges. Now we are planning to plant more orange trees to increase production, but if we plant more trees, the distance between the trees and the sunlight received by each tree will It will decrease. According to empirical estimates, every time a tree is planted, each tree will bear 5 fewer oranges on average.

(1) Suppose x orange trees are planted in the orchard. How many orange trees are there in the orchard? How many oranges does each tree bear on average at this time?

There are (100+x) trees in the orchard, and each tree bears (600-5x) oranges on average.

(2) If the total output of oranges in the orchard is y, then please write down the functional relationship between y and x.

y=(100+x)(600-5x)=-5x²+100x+60000.

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Update Time: 2024-07-29

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