The eighth grade mathematics function learning method is as follows
First, understand the connotation and essence of quadratic function.
Quadratic function y=ax2 +bx+c(a? 0, a, b and c are constants) contains two variables, X and Y. As long as we determine one of the variables first, we can find the other variable by analytical formula, that is, we can get a set of solutions. And a set of solutions is the coordinates of a point, in fact, the image of quadratic function is a graph composed of countless such points.
Familiar with the images and properties of several special quadratic functions.
1. Observe the shapes and positions of y=ax2, y=ax2+k, y=a(x+h)2 images by tracing points, and get familiar with the basic features of their respective images. On the contrary, according to the characteristics of parabola, we can quickly determine which analytical formula it is.
2. Understand the translation formula of images? Add and subtract, add left and subtract right? .
y=ax2？ y=a(x+h)2+k？ Add and subtract? It's about k, left plus right minus? It's about H.
In a word, if the coefficients of quadratic terms of two quadratic functions are the same, their parabolas have the same shape, but the translation of parabolas is essentially the translation of vertices because of their different coordinates and positions. If parabolas are in general form, they should be converted into vertices and then translated.
3. Through drawing and image translation, we understand and make it clear that the characteristics of analytical expressions are completely corresponding to the characteristics of images. When solving problems, we should have a picture in mind and see the function to reflect the basic characteristics of its image in our hearts.
4. On the basis of being familiar with the function image, through observing and analyzing the characteristics of parabola, we can understand the properties of quadratic function, such as increase and decrease, extreme value and so on. Distinguish the coefficients A, B, C, △ of quadratic function and the symbols of algebraic expressions composed of coefficients by images.
Third, we should make full use of parabola? Vertex? The role of.
1, can be accurate and flexible? Vertex? . As y=a(x+h)2+K? Vertex (-h, k), for other forms of quadratic function, we can turn it into a vertex and find the vertex.
2. Understand the relationship between vertex, symmetry axis and maximum value of function. If the vertex is (-h, k) and the symmetry axis is x=-h, the maximum (minimum) value of y = k；; On the other hand, if the symmetry axis is x=m and the maximum value of y is n, then the vertex is (m, n); Understanding the relationship between them can achieve the effect of analyzing and solving problems.
3. Draw a sketch with vertices. In most cases, we only need to draw a sketch to help us analyze and solve problems. At this time, we can draw the approximate image of parabola according to the vertex and opening direction of parabola.
Understand and master the solution of the intersection of parabola and coordinate axis.
Generally speaking, the coordinates of a point consist of abscissa and ordinate. When we find the intersection of parabola and coordinate axis, we can give priority to one of the coordinates, and then find the other coordinate by analytical formula. If the equation has no real root, it means that the parabola and the X axis do not intersect.
From the process of finding the intersection point above, we can see that the essence of finding the intersection point is to solve the equation, which is related to the discriminant of the root of the equation, and the number of times the parabola intersects the X axis is determined by the discriminant of the root.
Quadratic functions are all parabolic functions (its function trajectory is like the trajectory of a ball, of course this is not important), so we can grasp the quadratic function by grasping its function image.
Pay attention to several points in the function image (standard formula y = ax 2+bx+c, a is not equal to 0):
1, the opening direction is related to the quadratic coefficient a, indicating that the opening is upward, and vice versa.
2. There must be an extreme point, which is also the maximum point. If the opening is upward, it is easy to imagine that this extreme point should be the minimum point, and vice versa. The abscissa of the extreme point is -b/2a. Extreme points are prone to application problems.
3. it doesn't necessarily intersect with the x axis. When judging the root of the formula. =b^2-4ac<； 0, there is no intersection point, that is, the equation ax 2+bx+c = 0? No real number solution? (can't say there is no solution! You'll know in high school) What if? =0, then there is just an intersection point, that is, we say that the X axis is tangent to the function image. The corresponding equation has a unique real number solution. ? & gt0, there are two intersections, and the corresponding equation has two real number solutions.
4. Inequality. Clear the above three points, we can certainly solve the inequality of reference function image.
Learning formula of mathematical function in grade two of junior high school
Discrimination of positive proportional function
To judge the proportional function, the test is divided into two steps.
One quantity means another quantity, right or wrong.
If there is, it depends on the value. All real numbers must be there.
The authenticity of the proportional function needs to be identified in two steps.
One quantity means another quantity, right?
Look at the numerical values, all you need is real numbers.
Distinguishing the proportional function, the measurement can be divided into two steps.
Images and properties of proportional function
Line, passage and origin of proportional function graph.
K is positive one, three, negative two and four, and the changing trend is in the heart.
K is low on the left and high on the right, climbing in the same direction.
K is negative, the left is high and the right is low, and the mountain is one big and one small.
linear function
Linear function diagram straight line, passing point.
K is low on the left and high on the right, and the higher you climb, the higher you climb.
K is negative, higher left and lower right, and lower and lower.
K is called the slope b intercept, and the zero intercept becomes a positive function.
inverse proportion function
Inverse hyperbola, crossing point.
K is plus one, three, minus two, four, and the two axes are its asymptotes.
K is high on the left and low on the right, and one or three quadrants slide down the mountain.
K is negative, low left and high right, and the second and fourth quadrants are like climbing mountains.
quadratic function
When the quadratic equation changes from zero to y, a quadratic function appears.
All real numbers define the domain, and images are called parabolas.
A parabola has an axis of symmetry, and both sides are monotonous and opposite.
A determines the opening and size, and the intersection of spools is called the vertex.
Vertex is either high or low. Up and down are conspicuous.
If you want to draw a parabola, you can also translate the pursuit point.
Extract formulas and set vertices, and then select them in two ways.
After drawing the list, connect the lines and keep the translation rules in mind.
Add parentheses to the left and right, and add and subtract redundant numbers.
For y, you get a quadratic function.
The image is called parabola, which defines all the real numbers in the domain.
A set the opening and size, with the opening facing upward.
The absolute value is large and the opening is small, and the opening is negative.
Parabola has an axis of symmetry, and the increase and decrease characteristics can be seen in the figure.
The intersection of axes is called the vertex, and the ordinate of the vertex is the most valuable.
If you want to draw a parabola, trace the point and translate two roads.
Select the fixed vertex of the formula and draw all translation points.
After the list is drawn, connect the lines, and three points roughly define the whole picture.
If you want to translate, it is not difficult to draw a basic parabola first
The vertex moves to the new position, and the size of the opening follows the foundation.