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 on 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 one common multiplier,
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 an isosceles triangle, the base angles 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 chord.
A chord passing through the center of a circle is called diameter
10. Direct proportionality y = kx , Where X – independent variable, To - Not equal to 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. Schedule 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.
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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 free term (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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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 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.
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.
A linear function is a function of the form y=kx+b, where x is the independent variable, k and b are any numbers.
The graph of a linear function is a straight line.
1. To build graph of a function, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the function equation, and use them to calculate the corresponding y values.
For example, to plot the function y= x+2, it is convenient to take x=0 and x=3, then the ordinates of these points will be equal to y=2 and y=3. We get points A(0;2) and B(3;3). Let's connect them and get a graph of the function y= x+2:
2.
In the formula y=kx+b, the number k is called the proportionality coefficient:
if k>0, then the function y=kx+b increases
if k
Coefficient b shows the displacement of the function graph along the OY axis:
if b>0, then the graph of the function y=kx+b is obtained from the graph of the function y=kx by shifting b units upward along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½ x+3; y=x+3
Note that in all these functions the coefficient k Above zero, and the functions are increasing. Moreover, the greater the value of k, the greater the angle of inclination of the straight line to the positive direction of the OX axis.
In all functions b=3 - and we see that all graphs intersect the OY axis at point (0;3)
Now consider the graphs of the functions y=-2x+3; y=- ½ x+3; y=-x+3
This time in all functions the coefficient k less than zero and functions are decreasing. Coefficient b=3, and the graphs, as in the previous case, intersect the OY axis at point (0;3)
Consider the graphs of the functions y=2x+3; y=2x; y=2x-3
Now in all function equations the coefficients k are equal to 2. And we got three parallel lines.
But the coefficients b are different, and these graphs intersect the OY axis at different points:
The graph of the function y=2x+3 (b=3) intersects the OY axis at point (0;3)
The graph of the function y=2x (b=0) intersects the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) intersects the OY axis at point (0;-3)
So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function y=kx+b looks like.
If k 0
If k>0 and b>0, then the graph of the function y=kx+b looks like:
If k>0 and b, then the graph of the function y=kx+b looks like:
If k, then the graph of the function y=kx+b looks like:
If k=0, then the function y=kx+b turns into the function y=b and its graph looks like:
The ordinates of all points on the graph of the function y=b are equal to b If b=0, then the graph of the function y=kx (direct proportionality) passes through the origin:
3. Let us separately note the graph of the equation x=a. The graph of this equation is a straight line parallel to the OY axis, all points of which have an abscissa x=a.
For example, the graph of the equation x=3 looks like this:
Attention! The equation x=a is not a function, so one value of the argument corresponds to different values of the function, which does not correspond to the definition of a function.
4. Condition for parallelism of two lines:
The graph of the function y=k 1 x+b 1 is parallel to the graph of the function y=k 2 x+b 2 if k 1 =k 2
5. The condition for two straight lines to be perpendicular:
The graph of the function y=k 1 x+b 1 is perpendicular to the graph of the function y=k 2 x+b 2 if k 1 *k 2 =-1 or k 1 =-1/k 2
6. Points of intersection of the graph of the function y=kx+b with the coordinate axes.
With OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero in the equation of the function instead of x. We get y=b. That is, the point of intersection with the OY axis has coordinates (0; b).
With OX axis: The ordinate of any point belonging to the OX axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero in the equation of the function instead of y. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b/k;0):