Quadratic function model
The conventional quadratic function is y=ax? +bx+c where a≠0 is similar to the letter n or u.
Properties of quadratic function
(1) A > 0, the function image opens upward; A < 0, the function image opens downward.
(2) axis of symmetry x= -b/(2a) (note: this is a straight line)
If A and B have the same sign (both positive and negative), the symmetry axis is on the left side of the Y axis (or the negative half axis of X).
If the signs of A and B are different (one is positive and the other is negative), the symmetry axis is on the right side of the Y axis (or the positive half axis of X);
Special: If b=0, the symmetry axis is the Y axis.
(3) Maximum. According to the image, we can find that the maximum of the quadratic function is at the point (-b/(2a), (4ac-b? )/(4a))
That is, the intersection of symmetry axis and quadratic function image.
When a > 0, the function is in x= -b/(2a) (4ac-b? )/(4a)
When a < 0, the function is in x= -b/(2a) (4ac-b? )/(4a)
(4) Intersection point with Y axis
Quadratic function and y axis intersect at point (0, c), so
When c > 0, the image of the quadratic function passes through the positive semi-axis of the Y axis.
When c=0, the image of the quadratic function passes through the origin (0,0).
When c > 0, the image of the quadratic function passes through the negative semi-axis of the Y axis.
(5) Intersection point with X axis
Make an axe? +bx+c=0, the intersection with the x axis is calculated.
From the existence of quadratic function solution, we can know that
1 when b? -4ac < 0, the equation has no root, so it does not intersect with the X axis.
2 when b? -4ac=0, then the equation has two identical real roots, so it is tangent to the x axis.
So this intersection point is the maximum of the original quadratic function.
Judging from the position of the symmetry axis, we can know the position of the intersection point.
3 when b? -4ac > 0, then the equation has two different real roots, so it has two intersections with the x axis.
In this case, according to Vieta's theorem, x1+x2 =-b/ax1* x2 = c/a.
Therefore, when a > 0, b > 0, and c > 0, both roots are negative, so the function is like the negative semi-axis passing through the X axis and has two intersections.
When a > 0, b > 0 and c < 0, the two roots are positive and negative, then the function image passes through the positive and negative half axes of the X axis respectively.
When a > 0, b < 0, c > 0, and both roots are positive roots, the function image intersects with the positive semi-axis of the X axis, and there are two intersections.
When a > 0, b < 0 and c < 0, the two roots are positive and negative, then the function image passes through the positive and negative half axes of the X axis respectively.
When a < 0, b > 0 and c > 0, the two roots are positive and negative, then the function image passes through the positive and negative half axes of the X axis respectively.
When a < 0, b > 0, and c < 0, both roots are positive roots, then the function image intersects with the positive semi-axis of the X axis, and there are two intersections.
When a < 0, b < 0, c > 0, and the two roots are positive and negative, the function image passes through the positive and negative half axes of the X axis respectively.
When a < 0, b < 0, and c < 0, both roots are negative, then the function is like the negative semi-axis passing through the X axis, and there are two intersections.
If c=0, one intersection is the origin, and the other is easy to judge.
If b=0, it must be a positive root and a negative root, then the function image passes through the positive and negative half axes of the X axis respectively (note: the premise is that there are two different roots).
(6) Determination of functional images:
There are many ways to do this, so I'll talk about it briefly.
Look at the opening direction first to determine whether it is U-shaped or N-shaped.
Then because the quadratic function image can be determined by three points.
Therefore, according to the intersection point with the Y axis, the position of the symmetry axis, the positive and negative of the maximum value or both, the specific relative position with the X and Y axes can be determined.
Finally, you can draw a sketch.
If LZ has anything unclear, you can ask.