Linear function called a function of the form y = kx + b, defined on the set of all real numbers. Here k– slope (real number), b – free term (real number), x– independent variable.
In the special case, if k = 0, we get constant function y = b, the graph of which is a straight line parallel to the Ox axis passing through the point with coordinates (0; b).
If b = 0, then we get the function y = kx, which is direct proportionality.
b – segment length, which is cut off by a straight line along the Oy axis, counting from the origin.
Geometric meaning of the coefficient k – tilt angle straight to the positive direction of the Ox axis, considered counterclockwise.
Properties of a linear function:
1) The domain of definition of a linear function is the entire real axis;
2) If k ≠ 0, then the range of values of the linear function is the entire real axis. If k = 0, then the range of values of the linear function consists of the number b;
3) Evenness and oddness of a linear function depend on the values of the coefficients k And b.
a) b ≠ 0, k = 0, hence, y = b – even;
b) b = 0, k ≠ 0, hence y = kx – odd;
c) b ≠ 0, k ≠ 0, hence y = kx + b – function of general form;
d) b = 0, k = 0, hence y = 0 – both even and odd functions.
4) A linear function does not have the property of periodicity;
5) Intersection points with coordinate axes:
Ox: y = kx + b = 0, x = -b/k, hence (-b/k; 0)– point of intersection with the abscissa axis.
Oy: y = 0k + b = b, hence (0; b)– point of intersection with the ordinate axis.
Note: If b = 0 And k = 0, then the function y = 0 goes to zero for any value of the variable X. If b ≠ 0 And k = 0, then the function y = b does not vanish for any value of the variable X.
6) The intervals of constancy of sign depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b– positive when x from (-b/k; +∞),
y = kx + b– negative when x from (-∞; -b/k).
b) k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b– positive when x from (-∞; -b/k),
y = kx + b– negative when x from (-b/k; +∞).
c) k = 0, b > 0; y = kx + b positive over the entire definition range,
k = 0, b< 0; y = kx + b negative throughout the entire range of definition.
7) The intervals of monotonicity of a linear function depend on the coefficient k.
k > 0, hence y = kx + b increases throughout the entire domain of definition,
k< 0 , hence y = kx + b decreases over the entire domain of definition.
8) The graph of a linear function is a straight line. To construct a straight line, it is enough to know two points. The position of the straight line on the coordinate plane depends on the values of the coefficients k And b. Below is a table that clearly illustrates this.
5. Monomial The product of numeric and alphabetic factors is called. Coefficient is called the numerical factor of a monomial.
6. To write a monomial in standard form, necessary: 1) Multiply the numerical factors and put their product in first place; 2) Multiply powers with the same bases and place the resulting product after the numerical factor.
7. A polynomial is called algebraic sum of several monomials.
8. To multiply a monomial by a polynomial, You need to multiply the monomial by each term of the polynomial and add the resulting products.
9. To multiply a polynomial by a polynomial, It is necessary to multiply each term of one polynomial by each term of another polynomial and add the resulting products.
10. Through any two points you can draw a straight line, and only one.
11. Two straight lines or have only one common point, or do not have common points.
12. Two geometric figures are called equal if they can be combined by overlapping.
13. The point of a segment that divides it in half, that is, into two equal segments, is called the midpoint of the segment.
14. A ray emanating from the vertex of an angle and dividing it into two equal angles, is called the angle bisector.
15. The rotated angle is 180°.
16. An angle is called right if it is equal to 90°.
17. An angle is called acute if it is less than 90°, that is, less than a right angle.
18. An angle is called obtuse if it is more than 90°, but less than 180°, that is, more than a right angle, but less than a straight angle.
19. Two angles in which one side is common, and the other two are continuations of one another, are called adjacent.
20. The sum of adjacent angles is 180°.
21. Two angles are called vertical if the sides of one angle are continuations of the sides of the other.
22. Vertical angles are equal.
23. Two intersecting lines are called perpendicular (or mutually
perpendicular) if they form four right angles.
24. Two lines perpendicular to a third do not intersect.
25. Factor the polynomial- means to represent it as a product of several monomials and polynomials.
26. Methods of factoring a polynomial:
a) putting the common factor out of brackets,
b) use of abbreviated multiplication formulas,
c) method of grouping.
27.To factor a polynomial by taking the common factor out of brackets, you need:
a) find this common factor,
b) take it out of brackets,
c) divide each term of the polynomial by this factor and add the resulting results.
Signs of equality of triangles
1) If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.
2) If a side and two adjacent angles of one triangle are respectively equal to the side and two adjacent angles of another triangle, then such triangles are congruent.
3) If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.
Educational minimum
1. Factorization using abbreviated multiplication formulas:
a 2 – b 2 = (a – b) (a + b)
a 3 – b 3 = (a – b) (a 2 + ab + b 2)
a 3 + b 3 = (a + b) (a 2 – ab + b 2)
2. Abbreviated multiplication formulas:
(a + b) 2 =a 2 + 2ab + b 2
(a – b) 2 = a 2 – 2ab + b 2
(a + b) 3 =a 3 + 3a 2 b + 3ab 2 + b 3
(a – b) 3 = a 3 – 3a 2 b + 3ab 2 – b 3
3. The segment connecting the vertex of a triangle with the midpoint of the opposite side is called median triangle.
4. The perpendicular drawn from the vertex of a triangle to the line containing the opposite side is called height triangle.
5. IN isosceles triangle the angles at the base are equal.
6. In an isosceles triangle, the bisector drawn to the base is the median and altitude.
7. Circumference called geometric figure, consisting of all points of the plane located at a given distance from a given point.
8. A segment connecting the center with any point on the circle is called radius circle .
9. A segment connecting two points on a circle is called its chord.
A chord passing through the center of a circle is called diameter
10. Direct proportionality y = kx , Where X – independent variable, To – a non-zero number ( To – proportionality coefficient).
11. Direct proportionality graph is a straight line passing through the origin of coordinates.
12. Linear function is a function that can be given by the formula y = kx + b , Where X – independent variable, To And b - some numbers.
13. Graph of a linear function- this is a straight line.
14 X – function argument (independent variable)
at – function value (dependent variable)
15. At b=0 the function takes the form y=kx, its graph passes through the origin.
At k=0 the function takes the form y=b, its graph is a horizontal line passing through the point ( 0;b).
Correspondence between the graphs of a linear function and the signs of the coefficients k and b
1. Two straight lines in a plane are called parallel, if they don't intersect.
A linear function is a function of the form y = kx + b, defined on the set of all real numbers. Here k is the slope (real number), b is the intercept (real number), x is the independent variable.
In the particular case, if k = 0, we obtain a constant function y = b, the graph of which is a straight line parallel to the Ox axis passing through the point with coordinates (0; b).
If b = 0, then we get the function y = kx, which is direct proportionality.
The geometric meaning of the coefficient b is the length of the segment that the straight line cuts off along the Oy axis, counting from the origin.
The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, calculated counterclockwise.
Properties of a linear function:
1) The domain of definition of a linear function is the entire real axis;
2) If k ≠ 0, then the range of values of the linear function is the entire real axis. If k = 0, then the range of values of the linear function consists of the number b;
3) Evenness and oddness of a linear function depend on the values of the coefficients k and b.
a) b ≠ 0, k = 0, therefore, y = b - even;
b) b = 0, k ≠ 0, therefore y = kx - odd;
c) b ≠ 0, k ≠ 0, therefore y = kx + b is a function of general form;
d) b = 0, k = 0, therefore y = 0 is both an even and an odd function.
4) A linear function does not have the property of periodicity;
Ox: y = kx + b = 0, x = -b/k, therefore (-b/k; 0) is the point of intersection with the abscissa axis.
Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the ordinate.
Note: If b = 0 and k = 0, then the function y = 0 vanishes for any value of the variable x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any value of the variable x.
6) The intervals of constant sign depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b - positive at x of (-b/k; +∞),
y = kx + b - negative for x of (-∞; -b/k).
b)k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b - positive at x from (-∞; -b/k),
y = kx + b - negative for x of (-b/k; +∞).
c) k = 0, b > 0; y = kx + b is positive throughout the entire domain of definition,
k = 0, b< 0; y = kx + b отрицательна на всей области определения.
7) The monotonicity intervals of a linear function depend on the coefficient k.
k > 0, therefore y = kx + b increases throughout the entire domain of definition,
k< 0, следовательно y = kx + b убывает на всей области определения.
8) The graph of a linear function is a straight line. To construct a straight line, it is enough to know two points. The position of the straight line on the coordinate plane depends on the values of the coefficients k and b. Below is a table that clearly illustrates this, Figure 1. (Fig. 1)
Example: Consider the following linear function: y = 5x - 3.
3) General function;
4) Non-periodic;
5) Points of intersection with coordinate axes:
Ox: 5x - 3 = 0, x = 3/5, therefore (3/5; 0) is the point of intersection with the x-axis.
Oy: y = -3, therefore (0; -3) is the point of intersection with the ordinate;
6) y = 5x - 3 - positive for x from (3/5; +∞),
y = 5x - 3 - negative at x of (-∞; 3/5);
7) y = 5x - 3 increases throughout the entire domain of definition;
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“Drawings for slides” - Optional course “World of Multimedia Technologies”. Drawings on slides. C) you can transfer the drawing by grabbing the middle with the mouse. Inserting pictures onto a slide. Municipal educational institution average comprehensive school No. 5. 95% of information is perceived by a person through the organs of vision...
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Properties and graphs tasks quadratic function cause, as practice shows, serious difficulties. This is quite strange, because they study the quadratic function in the 8th grade, and then throughout the first quarter of the 9th grade they “torment” the properties of the parabola and build its graphs for various parameters.
This is due to the fact that when forcing students to construct parabolas, they practically do not devote time to “reading” the graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, after constructing a dozen or two graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and appearance graphic arts. In practice this does not work. For such a generalization, serious experience in mathematical mini-research is required, which most ninth-graders, of course, do not possess. Meanwhile, the State Inspectorate proposes to determine the signs of the coefficients using the schedule.
We will not demand the impossible from schoolchildren and will simply offer one of the algorithms for solving such problems.
So, a function of the form y = ax 2 + bx + c called quadratic, its graph is a parabola. As the name suggests, the main term is ax 2. That is A should not be equal to zero, the remaining coefficients ( b And With) can equal zero.
Let's see how the signs of its coefficients affect the appearance of a parabola.
The simplest dependence for the coefficient A. Most schoolchildren confidently answer: “if A> 0, then the branches of the parabola are directed upward, and if A < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой A > 0.
y = 0.5x 2 - 3x + 1
In this case A = 0,5
And now for A < 0:
y = - 0.5x2 - 3x + 1
In this case A = - 0,5
Impact of the coefficient With It's also pretty easy to follow. Let's imagine that we want to find the value of a function at a point X= 0. Substitute zero into the formula:
y = a 0 2 + b 0 + c = c. It turns out that y = c. That is With is the ordinate of the point of intersection of the parabola with the y-axis. Typically, this point is easy to find on the graph. And determine whether it lies above zero or below. That is With> 0 or With < 0.
With > 0:
y = x 2 + 4x + 3
With < 0
y = x 2 + 4x - 3
Accordingly, if With= 0, then the parabola will necessarily pass through the origin:
y = x 2 + 4x
More difficult with the parameter b. The point at which we will find it depends not only on b but also from A. This is the top of the parabola. Its abscissa (axis coordinate X) is found by the formula x in = - b/(2a). Thus, b = - 2ax in. That is, we proceed as follows: we find the vertex of the parabola on the graph, determine the sign of its abscissa, that is, we look to the right of zero ( x in> 0) or to the left ( x in < 0) она лежит.
However, that's not all. We also need to pay attention to the sign of the coefficient A. That is, look at where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine the sign b.
Let's look at an example:
The branches are directed upwards, which means A> 0, the parabola intersects the axis at below zero, that is With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. So b = - 2ax in = -++ = -. b < 0. Окончательно имеем: A > 0, b < 0, With < 0.
