How to plot the function of the root of x. "Function" root of x ", its properties and graphs". Square root as an elementary function

Function graph and properties at = │Oh│ (module)

Consider the function at = │Oh│, where but- a certain number.

The scope of functions at = │Oh│, is the set of all real numbers. The figure shows respectively function graphs at = │NS│, at = │ 2x │, at = │NS/2│.

You can see that the graph of the function at = | Oh| is obtained from the function graph at = Oh if the negative part of the graph of the function at = Oh(it is below the O axis NS), reflect symmetrically this axis.

The schedule is easy to see properties functions at = │ Oh │.

At NS= 0, we get at= 0, that is, the origin of coordinates belongs to the graph of the function; at NS= 0, we get at> 0, that is, all other points of the graph lie above the O axis NS.

For opposite meanings NS, values at will be the same; axis O at this is the axis of symmetry of the graph.

For example, you can plot the function at = │NS 3 │. To compare functions at = │NS 3 │ and at = NS 3, we will compile a table of their values for the same values of the arguments.

From the table we see that in order to plot the function at = │NS 3 │, you can start by plotting the function at = NS 3. After that, it stands symmetrically to the O axis NS display that part of it that is below this axis. As a result, we get the graph shown in the figure.

Function graph and properties at = x 1/2 (root)

Consider the function at = x 1/2 .

The scope of this function is the set of non-negative real numbers, since the expression x 1/2 matters only for NS > 0.

Let's build a graph. To compile a table of its values, we use a micro calculator, rounding the values of the function to tenths.

After drawing points on the coordinate plane, and smoothly connecting them, we get function graph at = x 1/2 .

The plotted graph allows us to formulate some properties functions at = x 1/2 .

At NS= 0, we get at= 0; at NS> 0, we get at> 0; the graph goes through the origin; the rest of the graph points are located in the first coordinate quarter.

Theorem... Function graph at = x 1/2 symmetrical graph function at = NS 2, where NS> 0, relatively straight at = NS.

Proof... Function graph at = NS 2, where NS> 0, is the branch of the parabola located in the first coordinate quarter. Let the point R (but; b) is an arbitrary point of this graph. Then the equality is true b = but 2. Since, by condition, the number but is nonnegative, then the equality but= b 1/2. This means that the coordinates of the point Q (b; but) transform the formula at = x 1/2 to true equality, or otherwise, point Q (b; but at= x 1/2 .

It is also proved that if the point M (with; d) belongs to the graph of the function at = x 1/2 then point N (d; with) belongs to the graph at = NS 2, where NS > 0.

It turns out that every point R(but; b) function graph at = NS 2, where NS> 0 corresponds to a single point Q (b; but) function graph at = x 1/2 and vice versa.

It remains to prove that the points R (but; b) and Q (b; but) are symmetric with respect to the straight line at = NS... By dropping perpendiculars to the coordinate axes from points R and Q, we obtain points on these axes E(but; 0), D (0; b), F (b; 0), WITH (0; but). Dot R intersection of perpendiculars PE and QC has coordinates ( but; but) and therefore belongs to the straight line at = NS... Triangle PRQ is isosceles, since its sides RP and RQ equal │ b – but│ each. Straight at = NS halves as a corner DOF and angle PRQ and crosses the segment PQ at a certain point S... Therefore, the segment Rs is the bisector of the triangle PRQ... Since the bisector of an isosceles triangle is its height and median, then PQ ┴ Rs and PS = QS... This means that the points R (but; b) and Q (b; but) symmetric with respect to the straight line at = NS.

Since the graph of the function at = x 1/2 symmetrical graph function at = NS 2, where NS> 0, relatively straight at= NS, then the graph of the function at = x 1/2 is the branch of the parabola.

Consider the function y = √x. The graph of this function is shown in the figure below.

Function graph y = √x

As you can see, the graph resembles a rotated parabola, or rather one of its branches. We get a branch of the parabola x = y ^ 2. It can be seen from the figure that the graph only once touches the Oy axis, at a point with coordinates (0; 0).
Now it's worth noting the main properties of this function.