"Determining the Expression of a Quadratic Function" PPT Courseware 2

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"Determining the Expression of a Quadratic Function" PPT Courseware 2

Find the analytical formula of the quadratic function using the method of undetermined coefficients

1. General formula: y=ax²+bx+c (a, b, c are constants, a ≠0)

To find the analytical formula of the quadratic function y=ax²+bx+c, the key is to find the values ​​of the undetermined coefficients a, b, and c.

From the known conditions (such as the coordinates of three points on the quadratic function graph), list the system of equations about a, b, c, and find a, b, c, you can write the analytical expression of the quadratic function.

2. The vertex formula y=a(x-h)²+k (a, h, k are constants a≠0).

1. If the coordinates of the vertex of the parabola and the coordinates of another point on the parabola are known, the analytical expression of the function is set as the vertex expression y=a(x-h)²+k.

2. In particular, when the vertex of the parabola is the origin, h=0, k=0, the analytical formula of the function can be set as y=ax².

3. When the symmetry axis of the parabola is the y-axis, h=0, the analytical formula of the function can be set as y=ax²+k.

4. When the vertex of the parabola is on the x-axis, k=0, the analytical formula of the function can be set as y=a(x-h)².

3. The intersection formula y=a(x-x1)(x-x2). (a, x1, x2 are constants a≠0)

When the parabola has two intersection points with the x-axis (x1,0), (x2,0), the quadratic function y=ax2+bx+c can be transformed into the intersection formula y=a(x-x1)(x-x2 ). Therefore, when the parabola has two intersection points with the x-axis (x1,0) and (x2,0), the analytical formula of the function can be set as y=a(x-x1)(x-x2). By substituting the coordinates of a point into it, a can be solved and the analytical formula of the parabola can be obtained.

The intersection formula is y=a(x-x1)(x-x2). x1 and x2 are the abscissas of the two intersection points of the parabola and the x-axis respectively. These two intersection points are symmetrical about the symmetry axis of the parabola, then the straight line x=x1+ x2/2 is the axis of symmetry of the parabola.

1. General steps for finding the analytical expression of a quadratic function:

One assumption, two columns, three solutions, and four reductions.

2. Determination of several commonly used analytical expressions for quadratic functions

1. General type

Given the coordinates of three points on the parabola, the general formula is usually chosen.

2. Vertex type

The coordinates of the vertex on the parabola (the axis of symmetry or the maximum value) are known, and the vertex formula is usually selected.

3. Intersection type

Given the coordinates of the intersection point of the parabola and the x-axis, select the intersection formula.

4. Translational type

When the parabola is translated, only the vertex coordinates change in the analytical expression of the function. The original function can be converted into the vertex expression first, and then according to the rule of "add left and subtract right, add up and subtract", the analysis of the new function can be obtained Mode.

Learn and apply to deepen understanding

1. A certain parabola is obtained by translating the parabola y=ax2 to the right by one unit length and then upward by one unit length, and the parabola passes through the point (3,-3). Find the expression of the parabola.

Vertex coordinates (1, 1) let y=a(x-1)2+1

2. It is known that the symmetry axis of the quadratic function is the straight line x=1, the ordinate of the lowest point P on the image is -8, and the image also passes through the point (-2,10), find the expression of this function.

Vertex coordinates (1,-8) let y=a(x-1)2-8

3. It is known that the distance between the graph of the quadratic function and the two intersection points of the x-axis is 4, and when x=1, the function has a minimum value of -4. Find this expression.

Vertex coordinates (1,-4) let y=a(x-1)2-4

4. The abscissas of the intersection points of a parabola and the x-axis are 2 and 6, and the maximum value of the function is 2. Find the expression of the function.

Vertex coordinates (4, 2) let y=a(x-4)2+2

Do it

Choose the optimal solution and find the following analytical formula of the quadratic function:

1. It is known that the image of the parabola passes through the points (1,4), (-1,-1), and (2,-2). Let the analytical formula of the parabola be ________.

2. It is known that the vertex coordinates of the parabola are (-2,3) and passes through the point (1,4). Let the analytical formula of the parabola be ___________.

3. It is known that the quadratic function has a maximum value of 6 and passes through points (2, 3) and (-4,5). Let the analytical formula of the parabola be ________.

4. It is known that the axis of symmetry of the parabola is the straight line x=-2 and passes through points (1,3) and (5,6). Let the analytical formula of the parabola be _______.

5. It is known that the parabola intersects with the x-axis at points A(-1,0), B(1,0), and passes through point (2,-3). Let the analytical formula of the parabola be ______.

question group training

1. It is known that the maximum value of the quadratic function is 2, the vertex of the image is on the straight line y=x+1, and the image passes through the point (3, -6), find the analytical formula of the quadratic function.

2. It is known that the vertex coordinates of the parabola are at the point where it intersects with the axis. Find the analytical formula of this parabola.

3. It is known that the parabola passes through three points A(-2,0), B(1,0), and C(0,2). Find the analytical expression of this parabola.

4. According to the following conditions, find the analytical formula of the quadratic function.

(1), the image passes through three points (0, 0), (1, -2), (2, 3);

(2), the vertex (2, 3) of the image, and passes through the point (3, 1);

(3). The image passes through (-1, 0), (3, 0), (0, 3).

[Discuss one discussion]

Through solving the above problems, can you understand what is the general method used to find the expression of a quadratic function?

Can you summarize the general steps for solving the above problem?

1. If there is no coordinate system, an appropriate rectangular coordinate system should be established first;

2. Assume the expression of the parabola;

3. Write the coordinates of the relevant points;

4. List of equations (or systems of equations);

5. Solve equations or systems of equations and find undetermined coefficients;

6. Write the expression of the function;

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"Determining the Expression of a Quadratic Function" PPT courseware:

"Determining the Expression of a Quadratic Function" PPT courseware learning objectives 1. Be able to use the undetermined coefficient method to find the expression of a quadratic function; (Key points) 2. Be able to set up the corresponding expression of a quadratic function based on known conditions Form, it is easier to find the expression of quadratic function. ..

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Update Time: 2024-09-07

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