"Quadratic Functions and Quadratic Equations of One Variable" Quadratic Function PPT Courseware 4

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"Quadratic Functions and Quadratic Equations of One Variable" Quadratic Function PPT Courseware 4

Think of it from the time it takes for the ball to hit the ground

The relationship between the height h(m) of a vertically thrown object and the movement time t(s) can be expressed by the formula h=-5t²+v0t+h0, where h0(m) is the height when thrown, v0(m/s) is the speed when thrown. A small ball is thrown vertically upward from the ground at a speed of 20m/s. The relationship between the height h (m) of the small ball and the movement time t (s) is as shown in the figure, then

(1).What is the relationship between h and t? h=-5t²+20t

(2).①Can the flying height of the ball reach 15m? If so, how much flight time is required?

②Can the flying height of the ball reach 20m? If so, how much flight time is required?

③Can the flying height of the ball reach 20.5m? If so, how much flight time is required?

④How many seconds does it take for the ball to hit the ground?

Explore 1. Find the coordinates of the intersection points A and B of the quadratic function graph y=x²-3x+2 and the x-axis.

Solution: ∵A and B are on the axis,

∴Their vertical coordinates are 0,

∴Let y=0, then x²-3x+2=0

Solution: x1=1, x2=2;

∴A(1,0),B(2,0)

Did you find any connection between the solution x1 and x2 of the equation x²-3x+2=0 and the coordinates of A and B?

Conclusion 1: The solution to the equation x2-3x+2=0 is the abscissa of the two intersections of the parabola y=x²-3x+2 and the x-axis. Therefore, parabola and quadratic equation are closely related.

That is: if the two roots of the quadratic equation ax²+bx+c=0 are x1 and x2, then the coordinates of the two intersection points of the parabola y=ax²+bx+c and the axis are A ( ) and B ( ) respectively.

Conclusion 2: The number of intersections between the parabola y=ax²+bx+c and the x-axis can be explained by the roots of the quadratic equation ax²+bx+c=0:

1. △＞0--- The quadratic equation ax²+bx+c=0 has two unequal real roots. The parabola y=ax²+bx+c has two intersection points with the x-axis - intersection.

2. △= 0--- The quadratic equation ax²+bx+c=0 has two equal real root parabolas y=ax²+bx+c and the x-axis have the only common point - tangent (vertex).

3. △<0---The quadratic equation ax²+bx+c=0 has no real root, and the parabola y=ax²+bx+c has no common point with the x-axis—they are separated.

Example 1. It is known that the parabola y=x²+2x+m+1.

(1) If the parabola has only one intersection with the x-axis, find the value of m.

(2) If there is only one intersection point between the parabola and the straight line y=x+2m, find the value of m.

Example 2. Known quadratic function y=x²+kx+k-2

(1) Determine the intersection point of the above parabola and the X-axis

(2) Assume that the distance between the intersection of the parabola and the X-axis is 2√5, and find the value of k

Example 3 Assume that the image of the quadratic function y=-x²+(m-2)x+3(m+1) intersects the X-axis at two points A and B, intersects the y-axis at point C, and the line segments OA and OB The product of the length is equal to 6 (O is the origin of the coordinates)

Find: the value of m

Example 4 Use the graph of the function to estimate the roots of the quadratic equation x2+2x-10=0

(1). Use the point drawing method to draw the image of the quadratic function y=x²+2x-10;

(2). Observe and estimate the abscissa of the intersection point between the graph of the quadratic function y=x²+2x-10 and the x-axis;

It can be seen from the image that there are two intersection points between the image and the x-axis. One of its abscissas is between -5 and -4, and the other is between 2 and 3, which are approximately -4.3 and 2.3 respectively.

(3). Determine the solution to the equation x²+2x-10=0;

It can be seen that the approximate roots of the equation x²+2x-10=0 are: x1≈-4.3, x2≈2.3.

basic exercises

1. It is known that the vertex of the parabola y=x²-6x+a is on the x-axis, then a=_____; if the parabola has two intersections with the x-axis, then the range of a is _____;

2. It is known that the parabola y=x²-3x+a+1 has at most one intersection point with the x-axis, then the range of a is _____.

3. It is known that the two intersection points of the parabola y=x²+px+q and the x-axis are (-2, 0), (3, 0), then p=_____, q=_____.

4. Determine whether the following parabolas intersect with the x-axis. If they intersect, find the coordinates of the intersection points.

(1) y=6x²-2x+1 (2) y=-15x²+14x+8 (3) y=x²-4x+4

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Update Time: 2024-10-02

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