quadratic functions, how to compare the size of abc

As can be seen from the figure, a 0 (opening down), because -b 2a 0 (axis of symmetry greater than 0 less than 1), so b 0

Because the intersection of the image of this function with the y-axis is above the x-axis, c 0 (the intercept is c, and the intercept is greater than 0 from the graph), so abc<0 is correct.

When x=-1, y=a-b+c 0 (this is the assumption x=1, which is a common assumption in math problems.) Bringing the assumption of x=1 into the equation yields y=a-b+c, as can be seen from the graph, when x=-1, y<0).

So a+c b

So (a+c)*2>b*2 is wrong. (a+c)*2>b*2 is simplified i.e. a+c>b, which is inconsistent with a+c b)

When x=2, y=4a+2b+c 0 (this is the assumption x=2, which is a common assumption in math problems. Bringing the assumption of x=2 into the equation yields y=4a+2b+c, as can be seen from the graph, when x=2, y<0).

So 2a+b -c 2(-c 2 0, we have found c>0).

i.e. 2a+b 0

So 2a+b>0 is wrong.

When x=-2, y=4a-2b+c 0 (as can be seen from the figure).

So 4a-2b+c>0 is wrong.

So only 1} of the four conclusions is correct

The image opening is down, indicating a 0, the axis of symmetry of the image x=-b 2a is on the right side of the y axis, indicating -b 2a 0, so b 0

When x=0, y=ax*2+bx+c=c, it means that the intersection of the image of the function and the y-axis is the value of c, and the intersection is above the x-axis, so c 0

The quadratic function curve is symmetrical to x=-b 2a, and it can be seen from the figure that y>0 when x=1, and the symmetry axis is greater than 0, so y must be less than 0 when x=-1, that is, y=a-b+c 0

x=2 can be set on the right side of the figure, so y<0

The image opening is downward, a<0, the axis of symmetry of the quadratic function is -b 2a, because -b 2a > 0, so b>0, the intersection of the image and the y axis is above the x-axis, so c>0, and then there is no specific image, and it really can't be done....I'm sorry....

The opening size of the quadratic function is determined by the quadratic term coefficient a, and the smaller the absolute value of a, the larger the opening; The greater the absolute value of a, the smaller the opening. a, b, c are constants, a≠0, and a determines the opening direction of the function. a>0, the opening direction is upward; a<0, the opening direction is downward.

Generally, a function of the form y=ax +bx+c(a≠0) is called a quadratic function, where a is called the quadratic coefficient, b is the primary coefficient, and c is a constant term. x is the independent variable and y is the dependent variable. The highest number of independent variables to the right of the equal sign is 2.

The size of the opening of the secondary function of the second collapse is determined by the quadratic term coefficient a, and the smaller the absolute value of a, the larger the opening. The greater the absolute value of a, the smaller the opening. The basic representation of a quadratic function is y=ax +bx+c(a≠0).

The highest order of the quadratic function must be quadratic, and the image of the quadratic function is a parabola with the axis of symmetry parallel to the y-axis or coincident with the y-axis.

The quadratic term coefficient a determines the direction and size of the opening of the quadratic function image.

When a 0, the quadratic function image opens upward; When a 0, the parabola opens downwards.

a|The larger it is, the smaller the opening of the quadratic function image.

Factors that determine the position of the axis of symmetry.

4.The primary coefficient b and the quadratic coefficient a together determine the position of the axis of symmetry.

When a and b have the same sign (i.e., ab 0), the axis of symmetry is left on the y-axis;

Since the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -

b 2a When a is different from b (i.e., ab 0), the axis of symmetry is on the right side of the y axis. Since the axis of symmetry is on the right, the axis of symmetry should be greater than 0, i.e. -

b 2a > 0, so b 2a should be less than 0, so a and b should have different signs.

It can be simply remembered as the left is the same as the right, that is, when a and b have the same sign (i.e., ab 0), the axis of symmetry is on the left of the y axis; When A is under a different sign from B.

i.e. ab 0), the axis of symmetry is to the right of the y axis.

In fact, b has its own geometric meaning: the analytic expression of the tangent of the quadratic function image at the intersection of the quadratic function image with the y-axis (the primary function).

The value of the slope k. It can be obtained by finding a derivative of the quadratic function.

The factor that determines the intersection of the quadratic function image with the y-axis.

5.The constant term c determines the intersection of the quadratic function image with the y-axis.

The quadratic function image intersects (0,k) with the y-axis

The number of points where the quadratic function image intersects with the x-axis.

6.The number of points where the quadratic function image intersects with the x-axis.

a0 or a>0; When kk=0, the quadratic function image has 1 intersection point with the x-axis.

When a0 and k>0, there is no intersection between the quadratic function image and the x-axis.

When a>0, the function obtains the minimum value ymix=k at x=h, which is an increasing function in the range of xh (i.e., y decreases with the increase of x), and the opening of the quadratic function image is upward, and the value range of the function is y>k

When a>0, the function obtains a maximum value ymax=k at x=h, and increases the function in the range of x>h, and when x=0, the symmetry axis of the parabola is the y-axis, and the function is an even function.

y=ax^2+bx+c

then x=-1y=a-b+c

So it's a function value of x=-1.

y=a(x＋b/2a)²-b²/4a＋c

Look at the image opening downwards, a 0

Axis of symmetry - b 2a = 1, i.e. b =

2a，b＞0……①

There are two intersections with the x-axis: =b -4ac=b -2bc 0, i.e. b 2c ......②

The positive half axis of the y-axis: c-b 4a=c b 2b=c b 0 has , gets.

2c＜3b

The opening is up, a 0, and the opening is down, a 0

Axis of Symmetry - B 2A

Judge the size of B.

Let the intersection of x=0 and y-axis be c

Determine the size of c.

Solution: As can be seen from the graph, a 0, because -b 2a 0, so b 0

Because the intersection of the image of the function with the y-axis is above the x-axis, c 0 and abc <0 are correct.

When x=-1, y=a-b+c 0

So a+c b

So (a+c)*2>b*2 is wrong.

When x=2, y=4a+2b+c 0

So 2a+b -c 2(-c 2 0).

i.e. 2a+b 0

So 2a+b>0 is wrong.

When x=-2, y=4a-2b+c 0 (as can be seen from the figure), so 4a-2b+c>0 is wrong.

So only 1} of the four conclusions is correct