"The relationship between quadratic functions and quadratic equations" PPT download

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"The relationship between quadratic functions and quadratic equations" PPT download

Part One: Situation Import

We already know that the relationship between the height h(m) of a vertically thrown object and its movement time t(s) can be expressed by the formula h=-5t2+v0t+h0, where h0(m) is the height when it is thrown, v0( m/s) is the velocity when thrown.

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 40m/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?

(2) What do the vertical coordinates mean?

(3) How many seconds does it take for the ball to hit the ground?

How many solutions do you have? Communicate with your peers.

(1) What is the relationship between h and t?

Solution: It is a quadratic function h=-5t2+40t.

(2) How many seconds does it take for the ball to hit the ground? How many ways do you have to solve it? Communicate with your peers.

Solution: 8s. You can use images or solve the equation -5t2+40t=0

PPT on the relationship between quadratic functions and quadratic equations, part 2: thinking and exploration

Find the coordinates of the intersection points of the graphs of the quadratic functions y=x2+2x, y=x2-2x+1, y=x2-2x+2 and the x-axis respectively, and make a sketch.

Thoughts: The point of intersection with the x-axis is to find the solution to this equation when y=0, and then write it as the coordinates of the point.

Observe the graphs of the following quadratic functions y=x2+2x, y=x2-2x+1, y=x2-2x+2.

(1) How many intersection points does each image have with the x-axis?

(2) How many roots does the quadratic equation x2+2x=0, x2-2x+1=0 have?

Let’s verify, does the quadratic equation x2-2x+2=0 have roots?

(3) What is the relationship between the graph of the quadratic function y=ax2+bx+c and the coordinates of the intersection point of the x-axis and the root of the quadratic equation ax2+bx+c=0?

Summarize and organize:

There are three situations where the graph of the quadratic function y=ax2+bx+c intersects the x-axis:

1. There are two intersection points,

2. There is an intersection point,

3. There is no intersection.

When the graph of the quadratic function y=ax2+bx+c intersects with the x-axis, the abscissa of the intersection is the value of the independent variable x when y=0, that is, the quadratic equation ax2+bx+c=0 root.

PPT on the relationship between quadratic functions and quadratic equations of one variable, part three: in-depth understanding

1. A football is kicked upward from the ground. Its height h (m) from the ground can be expressed by the formula h=-4.9t2+19.6t. Among them, t(s) represents the time that has passed since the football was kicked.

(1) When t=1, what is the height of the football?

(2) What value of t makes h the largest?

(3) How long does it take for the ball to hit the ground?

(4) What is the actual significance of the roots of the equation -4.9t2+19.6t=0? Can you represent it on the diagram?

(5) What is the actual significance of the roots of equation 14.7=-4.9t2+19.6t? Can you represent it on the diagram?

Solution: (1) When t=1, h=14.7

(2) ∵h=-4.9(t-2) 2+19.6 ∴When t=2, h is the largest

(3) For h=-4.9t2+19.6t, the ball landing means h=0, that is, -4.9t2+19.6t=0. The solution is t1=0 (dropped), t2=4.

That is, after the football is kicked out, the ball hits the ground 4 seconds later.

(4) The actual meaning of the root of the equation -4.9t2+19.6t=0 is the time for the ball to leave the ground and fall to the ground. The figure is represented by the abscissa of the intersection of the parabola and the x-axis.

(5) Solving equation 14.7=-4.9t2+19.6t, we get t=1, t=3, which means that when the ball is kicked out for 1 second and 3 seconds, the height from the ground is 14.7 meters. The figure is represented by a parabola and a straight line h=14.7 The abscissa coordinate of the intersection point.

2. It is known that the graph of the quadratic function y=kx2-7x-7 intersects with the x-axis, and find the value range of k.

Correct solution: This function is a quadratic function,

∴k≠0, and has an intersection with the x-axis,

∴△=(-7)2-4×k×(-7)= 49＋28k≥0,

Obtain k≥-9/4, that is, k≥-9/4 and k≠0

Tips: ① Because it is a quadratic function, k≠0;

②There is an intersection point, so it should be △≥0.

3. Fill in the blanks

(1) The coordinates of the intersection point of the parabola y=-3(x-2)(x+5) and the x-axis are (2,0) (-5,0)

(2) The number of intersection points of the parabola y=x2-2x+3 and the x-axis is ____.

(3) The parabola y=2x2+8x+m has only one intersection point with the x-axis, then m=___

PPT on the relationship between quadratic functions and quadratic equations, Part 4: Class summary

There are three situations where the graph of the quadratic function y=ax2+bx+c intersects the x-axis:

1. There are two intersection points,

2. There is an intersection point,

3. There is no intersection.

When the graph of the quadratic function y=ax2+bx+c intersects with the x-axis, the abscissa of the intersection is the value of the independent variable x when y=0, that is, the quadratic equation ax2+bx+c=0 root.

Generally, when y takes a constant value, the quadratic function is a quadratic equation of one variable.

Keywords: Free download of Hebei Education Edition mathematics PPT courseware for the second volume of ninth grade, PPT download on the relationship between quadratic functions and quadratic equations, .PPT format;

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Update Time: 2024-09-05

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