5. Monomial is called the product of numeric and alphabetic factors. Coefficient is called the numerical factor of the monomial.
6. To write the monomial in standard form, you need: 1) Multiply the numerical factors and put their product in first place; 2) Multiply the powers with the same bases and put 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, it is necessary 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 the other polynomial and add the resulting products.
10. It is possible to draw a straight line through any two points, and only one.
11. Two lines either have only one common point or no common points.
12. Two geometric figures are called equal if they can be superimposed.
13. The point of the segment dividing 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 developed angle is 180°.
16. An angle is called a right angle if it is 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 greater than 90°, but less than 180°, i.e., more than a right angle, but less than a straight angle.
19. Two angles that have one side in common and the other two are extensions 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 extensions 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. Factorize a polynomial means to represent it as a product of several monomials and polynomials.
26. Methods for factoring a polynomial:
a) bracketing the common factor,
b) the use of abbreviated multiplication formulas,
c) grouping.
27. To factorize 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 results obtained.
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 angles adjacent to it of one triangle are respectively equal to a side and two angles adjacent to it 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 by abbreviated multiplication formulas:
a 2 - b 2 \u003d (a - b) (a + b)
a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)
a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)
2. Abbreviated multiplication formulas:
(a + b) 2 \u003d a 2 + 2ab + b 2
(a - b) 2 \u003d a 2 - 2ab + b 2
(a + b) 3 \u003d a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 \u003d a 3 - 3a 2 b + 3ab 2 - b 3
3. The line segment that connects 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 tall triangle.
5. In an isosceles triangle, the angles at the base are equal.
6. In an isosceles triangle, the bisector drawn to the base is the median and height.
7. Circle a geometric figure is called, consisting of all points of the plane located at a given distance from a given point.
8. A line segment joining the center with a point on the circle is called radius circles .
9. A line segment that connects two points on a circle is called chord.
The chord passing through the center of the circle is called diameter
10. Direct proportionality y = kx , where X is an independent variable, to is a non-zero number ( to is the coefficient of proportionality).
11. Direct proportionality graph is a straight line passing through the origin.
12. Linear function is a function that can be given by the formula y = kx + b , where X is an independent variable, to and b - some numbers.
13. Graph of a linear function- 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 a special case, if k = 0, we get 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 a 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 - the angle of inclination of the straight line to the positive direction of the Ox axis, is considered counterclockwise.
Linear function properties:
1) The domain of definition of a linear function is the entire real axis;
2) If k ≠ 0, then the range of the linear function is the entire real axis. If k = 0, then the range 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 is even;
b) b = 0, k ≠ 0, therefore y = kx is odd;
c) b ≠ 0, k ≠ 0, hence y = kx + b is a general function;
d) b = 0, k = 0, hence y = 0 is both an even and an odd function.
4) The linear function does not have the property of periodicity;
Ox: y \u003d kx + b \u003d 0, x \u003d -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 y-axis.
Note. If b = 0 and k = 0, then the function y = 0 vanishes for any value of x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any values of the variable x.
6) Intervals of sign constancy depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b - positive for x from (-b/k; +∞),
y = kx + b - is negative for x from (-∞; -b/k).
b) k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b - positive for x from (-∞; -b/k),
y = kx + b - is negative for x from (-b/k; +∞).
c) k = 0, b > 0; y = kx + b is positive over the entire domain,
k = 0, b< 0; y = kx + b отрицательна на всей области определения.
7) Intervals of monotonicity of a linear function depend on the coefficient k.
k > 0, hence y = kx + b increases over the entire domain,
k< 0, следовательно y = kx + b убывает на всей области определения.
8) The graph of a linear function is a straight line. To draw 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) Intersection points with coordinate axes:
Ox: 5x - 3 \u003d 0, x \u003d 3/5, therefore (3/5; 0) is the point of intersection with the abscissa axis.
Oy: y = -3, therefore (0; -3) - point of intersection with the y-axis;
6) y = 5x - 3 is positive for x from (3/5; +∞),
y = 5x - 3 - negative for x from (-∞; 3/5);
7) y = 5x - 3 increases over the entire domain of definition;
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Tasks on the properties and graphs of a quadratic function, as practice shows, cause serious difficulties. This is rather strange, because the quadratic function is passed in the 8th grade, and then the entire first quarter of the 9th grade is "extorted" by the properties of the parabola and its graphs are built for various parameters.
This is due to the fact that forcing students to build 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, having built two dozen graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and the appearance of the graph. In practice, this does not work. For such a generalization, serious experience in mathematical mini-research is required, which, of course, most ninth-graders do not have. Meanwhile, in the GIA they propose to determine the signs of the coefficients precisely according to the schedule.
We will not demand the impossible from schoolchildren and simply offer one of the algorithms for solving such problems.
So, a function of the form y=ax2+bx+c is called quadratic, its graph is a parabola. As the name implies, the main component is ax 2. That is a should not be equal to zero, the remaining coefficients ( b and With) can be equal to zero.
Let's see how the signs of its coefficients affect the appearance of the parabola.
The simplest dependence for the coefficient a. Most schoolchildren confidently answer: "if a> 0, then the branches of the parabola are directed upwards, and if a < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой a > 0.
y = 0.5x2 - 3x + 1
In this case a = 0,5
And now for a < 0:
y = - 0.5x2 - 3x + 1
In this case a = - 0,5
Influence of coefficient With also easy enough to follow. 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=s. That is With is the ordinate of the point of intersection of the parabola with the y-axis. As a rule, 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=x2+4x+3
With < 0
y = x 2 + 4x - 3
Accordingly, if With= 0, then the parabola will necessarily pass through the origin:
y=x2+4x
More difficult with the parameter b. The point by 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 \u003d - b / (2a). In this way, b = - 2ax in. That is, we act as follows: on the graph we find the top of the parabola, 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, this is not all. We must also pay attention to the sign of the coefficient a. That is, to see where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine sign b.
Consider an example:
Branches pointing upwards a> 0, the parabola crosses the axis at below zero means 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.
Linear function is called a function of the form y = kx + b, defined on the set of all real numbers. Here k– angular coefficient (real number), b – free member (real number), x is an independent variable.
In a particular case, if k = 0, we obtain a constant function y=b, whose graph is a straight line parallel to the Ox axis, passing through the point with coordinates (0;b).
If a b = 0, then we get the function y=kx, which is in direct proportion.
b – segment length, which cuts off the line along the Oy axis, counting from the origin.
The geometric meaning of the coefficient k – tilt angle straight to the positive direction of the Ox axis is considered to be counterclockwise.
Linear function properties:
1) The domain of a linear function is the entire real axis;
2) If a k ≠ 0, then the range of the linear function is the entire real axis. If a k = 0, then the range 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, Consequently, y = b is even;
b) b = 0, k ≠ 0, Consequently y = kx is odd;
c) b ≠ 0, k ≠ 0, Consequently y = kx + b is a general function;
d) b = 0, k = 0, Consequently y = 0 is both an even and an odd function.
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, Consequently (-b/k; 0)- point of intersection with the abscissa axis.
Oy: y=0k+b=b, Consequently (0;b) is the point of intersection with the y-axis.
Note.If b = 0 and k = 0, then the function y=0 vanishes for any value of the variable X. If a b ≠ 0 and k = 0, then the function y=b does not vanish for any value of the variable X.
6) The intervals of sign constancy depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b- positive at x from (-b/k; +∞),
y = kx + b- negative at x from (-∞; -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 at x from (-b/k; +∞).
c) k = 0, b > 0; y = kx + b positive throughout the domain of definition,
k = 0, b< 0; y = kx + b is negative throughout the domain of definition.
7) Intervals of monotonicity of a linear function depend on the coefficient k.
k > 0, Consequently y = kx + b increases over the entire domain of definition,
k< 0 , Consequently y = kx + b decreases over the entire domain of definition.
8) The graph of a linear function is a straight line. To draw 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.