Properties of function y=x2
We plotted the graph by points. It looked like this (see Fig. 1):
Fig. 1. Graph of the function
Note that the value of the function at any point is non-negative - this is logical, because squaring cannot result in a negative number, which means that everything under the axis cannot belong to our graph.
Note also that this graph is symmetric with respect to the ordinate axis. There is a special name for such functions: they are called even functions. Even functions have a graph symmetric with respect to , and this is due to the fact that , like for our function.
Note also that our graph decreases before 0, and increases after 0 (see Fig. 2).
Fig. 2. Decreasing and increasing of the graph of the function
Accordingly, the greater (at ), the greater and the greater (at ), the greater .
Properties of the function y=k*x2 when k > 0
Now let's try to change , for now we'll take it only positive.
For example, take k=2 and k=0.5. If we plot by points, we can see that at x=0; y=0 (see Fig. 3).
Figure 3. Graphs of the functions (blue graph) and (red)
Although we only gave three examples, we can already draw some conclusions about the function , at . First, the graph of such a function lies in the upper half-plane. And this is logical: if the square of a number (a non-negative number) is multiplied by a positive number , the result is non-negative.
Property 1. The set of values of function at is all non-negative numbers ().
Further, we note that if we substitute into this function, we still get . Therefore we can formulate another property.
Property 2. The graph at is symmetric about the -axis (see Fig. 4).
Fig. 4. The parity of functions
Such functions are called even functions.
Learn more about function properties
So, we are talking about a function . Let us distinguish several properties of it.
1. Parity. A function is said to be even if for any of its domain . For example, is an even function.
The graph of an even function is always symmetric about the ordinate axis (see Fig. 5).
Fig. 5. Symmetry of the graph of an even function with respect to the ordinate axis
Similarly, a function is called odd if for any of its domain . In particular, is an odd function. The graph of an odd function is symmetric with respect to the origin.
2. Monotonicity. A function is said to increase (strictly) on some interval if for any . But if for all such : , the function decreases on this interval.
For example. . Then if , then (since ). So the function is increasing for positive . Similarly, we prove that it decreases for negative .
The graphs and the equation of the function show that its smallest value is reached at zero and is equal to zero (indeed, the square of any other number is strictly positive). There is no largest value: the more we take modulo , the more we get.
Property 3. The graph at increases for positive X and decreases for negative X (see Fig. 6).
Fig. 6. Decreasing and increasing of the function
Transformation of the graph of the function y=k*x2 by changing k
What happens to the graph if we increase the number ? If we plot by points, then if we increase by a factor of two, for example, the ordinate of a point with the same abscissa will increase by the same factor of two. That is, in theory, the graph should become narrower. And vice versa, if decreases (leaving it positive), it will become wider (see Fig. 3). This hypothesis can be checked visually (see Fig. 7 and Fig. 8).
Figure 7. As decreases, the graph expands
Fig. 8. As increases, the graph narrows
Thus, keeping this property in mind, you can always plot the graph of function depending on by narrowing or expanding the graph of function .
Problem (k > 0)
What is the largest and smallest value of the function on the segment (See Figure 9.)
Fig. 9. The graph of the function
Everything is clear with the smallest value: the graph (and the property) show that the smallest value is . But what about the largest value? We can see that the further the abscissa is from zero, the greater the value of the function. This makes sense: the greater the value of the function, the greater the modulus of the abscissa. So the greatest value is reached at point and it is .
Properties of the function y=k*x2 when k < 0
Now let's turn to negative . We can draw the function at by points. Or you can take, for example, the graph of , which we already know how to draw, and then multiply each value by . How do you draw a graph? Multiplying by changes the sign of the ordinate, so essentially we just mirror the graph with respect to the abscissa axis (see Figure 10).
Fig. 10. Construction of the function at
1. Property 1. The set of values of the function at is all nonpositive numbers () (see Fig. 11).
The graph of the function is in the lower half-plane.
Figure 11. The function takes non-positive values at
2. Property 2. The graph at is symmetric about the axis - the function is even ().
3. Property 3. The graph at increases for negative and decreases for positive (see Figure 12).
Fig. 12. Decreasing and increasing of the function at
4. Transformations of the graph of function at : the graph narrows as decreases and widens as increases (see Fig. 13).
Fig. 13. Transformations of the graph of the function at
Problem (k < 0)
At what value of does the graph of the function pass through point ? Draw this graph.
Substitute the coordinates of the point into the equation. We obtain:.
Now plot (see Fig. 14).
Figure 14. graph of the function
You can check yourself from the graph - the graph passes through point .
Note that the negativity of the number was obvious at once: since our function takes a negative value at least at one point, it must be negative.
Solve the equation: .
Solve this equation graphically by plotting the left and right parts of the equation (see Figure 15):
Figure 15. solving the equation graphically
You can see from the graph that this equation has two solutions: and . Note that if the root of is easy to see from the graph, then is more complicated - it is a noninteger. Therefore usually in such cases the hypothesis is made that is the root, and then the test is performed by substitution.
In this lesson we found out what the graph of the function looks like and how it changes as changes. We found out what properties this graph has:
1. value area: at , at .
2. Parity (the graph is symmetric about the axis).
3. Monotonicity: at it increases at negative and decreases at positive; at it increases at positive and decreases at negative.