"Getting to Know Parabolas" Quadratic Function PPT Courseware 4

Description

"Getting to Know Parabolas" Quadratic Function PPT Courseware 4

learning target

1. Explore the method and properties of the graph of the quadratic function y=x2, and gain experience in using images to study the properties of functions.

2. Be able to use the point drawing method to create an image of y=x2, and be able to recognize and understand the properties of the quadratic function y=x2 based on the image.

3. Be able to draw the image of the quadratic function y=-x2, and compare its similarities and differences with the image of y=x2, and initially establish the connection between the expression of the quadratic function and the image.

New course introduction

1. Definition of quadratic function

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

2. What are the main steps in drawing the graph of a function?

(1) List; (2) Draw points; (3) Connect lines.

Combination of numbers and shapes, intuitive feeling

In the quadratic function y=x2, what is the law of y changing as x changes?

Do you want to understand its properties intuitively?

Can you use the point tracing method to draw the graph of the quadratic function y=x2?

Observe the expression of y=x2, choose the appropriate x value, and calculate the corresponding y value to complete the following table:

Observe the image and answer the question string

(1) Can you describe the shape of the image? Communicate with your peers.

(2) Is the image an axially symmetrical figure? If so, what is its axis of symmetry? Please find several pairs of symmetry points and communicate with your peers.

(3) Does the image have an intersection with the x-axis? If so, what are the intersection coordinates?

Do it

(1) What is the shape of the graph of the quadratic function y=-x2?

(2) Think about it first, and then make an image of it.

(3) What is its relationship with the graph of the quadratic function y=x2?

Observe the image and answer the question string

(1) Can you describe the shape of the image? Communicate with your peers.

(2) Does the image have an intersection with the x-axis? If so, what are the intersection coordinates?

(3) When x<0, as the value of x increases, how does the value of y change? What about when x>0?

Properties of the quadratic function y=ax2

1. The vertex of the parabola y=ax2 is the origin, and the axis of symmetry is the y-axis.

2. When a>0, the parabola y=ax2 is above the x-axis (except for the vertex), its opening is upward, and it extends upward infinitely;

When a<0, the parabola y=ax2 is below the x-axis (except for the vertex), its opening is downward, and it extends downward infinitely.

3. When a>0, on the left side of the symmetry axis, y decreases as x increases; on the right side of the symmetry axis, y increases as x increases. When x=0, the function y has the smallest value.

When a<0, on the left side of the symmetry axis, y increases as x increases; on the right side of the symmetry axis, y decreases as x increases. When x=0, the function y The value is the largest.

induction

The image and properties of function y=ax2(a≠0)

Generally, the axis of symmetry of the parabola y=ax2 is the y-axis, and the vertex is the origin. When a>0, the opening of the parabola is upward, and the vertex is the lowest point of the parabola. The larger a, the smaller the opening of the parabola; when a<0, the opening of the parabola is toward ____, and the vertex is the most ____ point of the parabola, a The larger the opening of the parabola, the more _____.

The larger the |a|the more _____ the opening.

Appreciation of examples

1. It is known that the parabola y=ax2 passes through point A (-2, -8).

(1) Find the functional analytical formula of this parabola;

(2) Determine whether point B (-1, - 4) is on this parabola.

(3) Find the coordinates of the point on the parabola whose ordinate is -6.

Solution: (1) Substituting (-2, -8) into y=ax2, we get -8=a(-2)2,

The solution is a= -2, and the analytical formula of the function is y= -2x2.

(2) Because -4≠-2(-1)², point B(-1,-4) is not on this parabola.

(3) From -6=-2x2, we get x2=3, x=±√3, so there are two points with the ordinate of -6, they are (√3, -6) (-√3, -6)

2. Fill in the blanks: (1) The vertex coordinate of the parabola y=2x2 is _____, and the axis of symmetry is _____. On the _____ side, y increases with the increase of x; on the _____ side, y increases with the increase of x. It decreases as x increases. When x=_____, the value of function y is minimum, and the minimum value is _____. The parabola y=2x2 is in the _____ square of the x-axis (except for the vertex).

(2) The parabola _____ is on the _____ side of the x-axis (except for the vertex), on the left side of the symmetry axis, y follows the _____ of x; on the right side of the symmetry axis, y follows the _____ of x , when x=0, the value of function y is the largest, the maximum value is _____, when x_____0, y<0.

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For more information about the PPT courseware "Quadratic Functions and Parabolas", please click the Quadratic Functions and Parabolas ppt tag.

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Update Time: 2024-10-04

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