The figure shows the graphs y kx b. Linear function
Assignments for properties and graphs quadratic function cause, as practice shows, serious difficulties. This is rather strange, because the quadratic function is passed in the 8th grade, and then the whole first quarter of the 9th grade is "forced out" the properties of the parabola and its graphs are plotted for various parameters.
This is due to the fact that forcing students to build parabolas, they practically do not devote time to "reading" graphs, that is, they do not practice comprehending the information obtained from the picture. Apparently, it is assumed that, having built a 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 of mathematical mini-research is required, which, of course, most ninth-graders do not have. Meanwhile, the GIA proposes to determine the signs of the coefficients precisely according to 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 is called quadratic, its graph is a parabola. As the name suggests, the main term is ax 2... That is a should not be zero, other coefficients ( b and With) can be equal to zero.
Let's see how the signs of its coefficients affect the appearance of a parabola.
The simplest relationship 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
V in this case a = 0,5
And now for a < 0:
y = - 0.5x2 - 3x + 1
In this case a = - 0,5
Influence of the coefficient With is also easy enough to trace. Let's imagine that we want to find the value of the function at the point X= 0. Substitute zero in 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 a chart. 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 apex of the parabola. Its abscissa (coordinate along the axis X) is found by the formula x in = - b / (2a)... In this way, b = - 2х в... That is, we act as follows: on the chart we find the top of the parabola, we 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 = - 2х в identify the sign b.
Let's consider an example:
The branches are directed upwards, which means a> 0, the parabola crosses the axis at below zero means With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. Hence b = - 2х в = -++ = -. b < 0. Окончательно имеем: a > 0, b < 0, With < 0.
Linear function is called a function of the form y = kx + b given on the set of all real numbers. Here k- slope (real number), b – free term (real number), x Is the independent variable.
In a particular 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 a 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 the line along the Oy axis, counting from the origin.
The geometric meaning of the coefficient k – tilt angle a straight line to the positive direction of the Ox axis, is counted counterclockwise.
Linear function properties:
1) The domain 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 is a general function;
d) b = 0, k = 0, hence y = 0 - both even and odd function.
4) The linear function does not possess the periodicity property;
5) Intersection points with coordinate axes:
Ox: y = kx + b = 0, x = -b / k, hence (-b / k; 0)- the point of intersection with the abscissa axis.
Oy: y = 0k + b = b, hence (0; b)- the point of intersection with the ordinate axis.
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- is positive at x from (-b / k; + ∞),
y = kx + b- is negative at x from (-∞; -b / k).
b) k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b- is positive at x from (-∞; -b / k),
y = kx + b- is negative at x from (-b / k; + ∞).
c) k = 0, b> 0; y = kx + b is positive over the entire domain of definition,
k = 0, b< 0; y = kx + b is negative throughout the entire domain.
7) The intervals of monotonicity of the linear function depend on the coefficient k.
k> 0, hence y = kx + b increases over 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 build 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 an independent variable, k and b are any numbers.
The graph of a linear function is a straight line.
1. To build function graph, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two values of x, substitute them in the equation of the function, and from them calculate the corresponding values of y.
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). We connect them and get the 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
The coefficient b shows the shift 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 up 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 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 the 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 decrease. Coefficient b = 3, and the graphs, as in the previous case, intersect the OY axis at the point (0; 3)
Consider the graphs of the functions y = 2x + 3; y = 2x; y = 2x-3
Now in all equations of functions the coefficients k are equal to 2. And we got three parallel straight lines.
But the b coefficients are different, and these graphs intersect the OY axis at different points:
The graph of the function y = 2x + 3 (b = 3) crosses the OY axis at the 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) crosses the OY axis at the 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 has the form:
If k> 0 and b, then the graph of the function y = kx + b has the form:
If k, then the graph of the function y = kx + b has the form:
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 of 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. Separately, we 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 different meanings function that does not match the function definition.
4. The condition for the 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 the perpendicularity of two straight lines:
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 the OY axis. The abscissa of any point belonging to the OY axis is 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):
5. Monomial is called the product of numerical and alphabetic factors. Coefficient is called the numerical factor of the monomial.
6. To write a monomial in a standard form, you need to: 1) Multiply the numerical factors and put their product in the first place; 2) Multiply degrees 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. You can draw a straight line through any two points, and, moreover, only one.
11. Two lines either have only one common point, or do not have common points.
12. Two geometric shapes are said to be equal if they can be overlapped.
13. The point of a segment dividing it in half, that is, into two equal segments, is called the midpoint of the segment.
14. The ray emanating from the top of the angle and dividing it into two equal angles is called the bisector of the angle.
15. The flattened 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 more than 90 °, but less than 180 °, that is, more than a right angle, but less than a deployed angle.
19. Two corners in which one side is common, and the other two are extensions of one another, are called adjacent.
20. The sum of adjacent angles is 180 °.
21. Two corners are called vertical if the sides of one corner are extensions of the sides of the other.
22. The vertical angles are equal.
23. Two intersecting lines are called perpendicular (or mutually
perpendicular) if they form four right angles.
24. Two straight lines perpendicular to the third do not intersect.
25 factor a polynomial- means to represent it as a product of several monomials and polynomials.
26. Methods for factoring a polynomial:
a) removal of the common factor from the brackets,
b) using formulas for abbreviated multiplication,
c) the method of grouping.
27. To factor out a polynomial by factoring the common factor outside the parentheses, you need:
a) find this common factor,
b) put it outside the brackets,
c) divide each term of the polynomial by this factor and add the results obtained.
Equality tests for 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 equal.
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 equal.
3) If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal.
Educational minimum
1. Factorization by 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. Formulas for abbreviated multiplication:
(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 apex of the triangle with the middle of the opposite side is called median triangle.
4. The perpendicular drawn from the apex of the triangle to the straight line containing the opposite side is called height 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 the height.
7. Circumference called geometric figure, consisting of all points of the plane located at a given distance from this point.
8. The segment connecting the center with any point of the circle is called radius circles .
9. A segment connecting two points of a circle is called it chord.
The chord passing through the center of the circle is called diameter
10. Direct proportionality y = kx , where X - independent variable, To - a non-zero number ( To - coefficient of proportionality).
11. Graph of direct proportionality Is a straight line through the origin.
12. Linear function is called a function that can be specified by the formula y = kx + b , where X - independent variable, To and b - some numbers.
13. Linear function graph 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 the linear function and the signs of the coefficients k and b
1.Two straight lines in the plane are called parallel, if they don't overlap.
