Quadratic and Cubic Functions. Quadratic and Cubic Functions Draw 4x 2 Graph
Sections: Mathematics
Topic:“Plotting a square function containing a module”.
(Using the example of the graph of the function y = x 2 - 6x + 3.)
Target.
- Investigate the location of the function graph on the coordinate plane, depending on the module.
- Develop skills in plotting a function containing a module.
During the classes.
1. The stage of updating knowledge.
a) Checking homework.
Example 1. Build a graph of the function y = x 2 - 6x + 3. Find the zeros of the function.
Solution.
2. Coordinates of the vertex of the parabola: x = - b / 2a = - (-6) / 2 = 3, y (3) = 9 - 18 + 3 = - 6, A (3; -6).
4. Zeros of the function: y (x) = 0, x 2 - 6x + 3 = 0, D = 36 - 43 = 36 - 12 = 24, D> 0,
x 1,2 = (6 ±) / 2 = 3 ±; B (3 -; 0), C (3 +; 0).
Graph in Fig. 1.
Algorithm for constructing a graph of a square function.
1. Determine the direction of the "branches" of the parabola.
2. Calculate the coordinates of the vertex of the parabola.
3. Write down the equation of the axis of symmetry.
4. Calculate multiple points.
b) Consider the construction of graphs of linear functions containing the module:
1.y = | x |. Function graph in Figure 2.
2.y = | x | + 1. The graph of the function in Figure 3.
3.y = | x + 1 |. Function graph Figure 4.
Conclusion.
1. The graph of the function y = | x | + 1 is obtained from the graph of the function y = | x | parallel translation to the vector (0; 1).
2. The graph of the function y = | x + 1 | is obtained from the graph of the function y = | x | parallel translation by vector (-1; 0).
2.Opiratsionno-executive part.
Stage research work... Group work.
Group 1. Build graphs of functions:
a) y = x 2 - 6 | x | + 3,
b) y = | x 2 - 6x + 3 |.
Solution.
1. Build a graph of the function y = x 2 -6x + 3.
2. Display it symmetrically about the Oy axis.
Graph in Figure 5.
b) 1. Construct a graph of the function y = x 2 - 6x + 3.
2. Display it symmetrically about the Ox axis.
Function graph in Figure 6.
Conclusion.
1. The graph of the function y = f (| x |) is obtained from the graph of the function y = f (x), mapping relative to the axis Oy.
2. The graph of the function y = | f (x) | is obtained from the graph of the function y = f (x), mapping relative to the Ox axis.
Group 2: Build graphs of functions:
a) y = | x 2 - 6 | x | + 3 |;
b) y = | x 2 - 6x + 3 | - 3.
Solution.
1. The graph of the function y = x 2 + 6x + 3 is displayed relative to the Oy axis, the graph of the function y = x 2 - 6 | x | + 3.
2. The resulting graph is displayed symmetrically about the Ox axis.
Function graph in Figure 7.
Conclusion.
Graph of the function y = | f (| x |) | is obtained from the graph of the function y = f (x), by sequential display relative to the coordinate axes.
1. The graph of the function y = x 2 - 6x + 3 is displayed relative to the Ox axis.
2. The resulting graph is transferred to the vector (0; -3).
Function graph in Figure 8.
Conclusion. The graph of the function y = | f (x) | + a is obtained from the graph of the function y = | f (x) | by parallel translation to the vector (0, a).
Group 3: Plot function graph:
a) y = | x | (x - 6) + 3; b) y = x | x - 6 | + 3.
Solution.
a) y = | x | (x - 6) + 3, we have a set of systems:
We build a graph of the function y = -x 2 + 6x + 3 at x< 0 для точек у(0) = 3, у(- 1) = - 4.
Function graph in Figure 9.
b) y = x | x - 6 | + 3, we have a set of systems:
We build a graph of the function y = - x 2 + 6x + 3 at x 6.
2. Coordinates of the vertex of the parabola: x = - b / 2a = 3, y (3) = 1 2, A (3; 12).
3. Equation of the axis of symmetry: x = 3.
4. Several points: y (2) = 11, y (1) = 3; y (-1) = - 4.
We build a graph of the function y = x 2 - 6x + 3 at x = 7 y (7) = 10.
Graph in Fig. 10.
Conclusion. When solving this group of equations, it is necessary to consider the zeros of the moduli contained in each of the equations. Then build a graph of the function on each of the intervals obtained.
(When plotting these functions, each group examined the effect of the module on the appearance of the function graph and made appropriate conclusions.)
Got a pivot table for graphs of functions containing a module.
A table for plotting the graphs of functions containing a module.
Group 4.
Plot a function graph:
a) y = x 2 - 5x + | x - 3 |;
b) y = | x 2 - 5x | + x - 3.
Solution.
a) y = x 2 - 5x + | x - 3 |, we pass to the set of systems:
We build a graph of the function y = x 2 -6x + 3 at x 3,
then the graph of the function y = x 2 - 4x - 3 for x> 3 along the points y (4) = -3, y (5) = 2, y (6) = 9.
Function graph in Figure 11.
b) y = | x 2 - 5x | + x - 3, we pass to the set of systems:
We build each graph on the corresponding interval.
Function graph in Figure 12.
Conclusion.
We found out the influence of the module in each term on the type of the graph.
Independent work.
Plot a function graph:
a) y = | x 2 - 5x + | x - 3 ||,
b) y = || x 2 - 5x | + x - 3 |.
Solution.
The previous graphs are displayed relative to the Ox axis.
Group 5
Plot the function: y = | x - 2 | (| x | - 3) - 3.
Solution.
Consider the zeros of two modules: x = 0, x - 2 = 0. We obtain intervals of constant sign.
We have a set of systems of equations:
We build a graph for each of the intervals.
Graph in Figure 15.
Conclusion. The two modules in the proposed equations have significantly complicated the construction of a general graph, consisting of three separate graphs.
The students recorded the performances of each of the groups, wrote down their conclusions, and participated in independent work.
3. Assignment at home.
Build graphs of functions with different module locations:
1.y = x 2 + 4x + 2;
2.y = - x 2 + 6x - 4.
4. Reflective - evaluative stage.
1. Grades for a lesson are made up of marks:
a) for work in a group;
b) for independent work.
2. What was the most interesting moment in the lesson?
3. Is your homework difficult?
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The function y = x ^ 2 is called a quadratic function. The graph of a quadratic function is a parabola. General form parabola is shown in the figure below.
Quadratic function
Fig 1. General view of the parabola
As you can see from the graph, it is symmetrical about the Oy axis. The axis Oy is called the axis of symmetry of the parabola. This means that if you draw a straight line parallel to the Ox axis above this axis. Then it will cross the parabola at two points. The distance from these points to the Oy axis will be the same.
The axis of symmetry divides the graph of the parabola into two parts, as it were. These parts are called the branches of the parabola. And the point of the parabola that lies on the axis of symmetry is called the apex of the parabola. That is, the axis of symmetry passes through the apex of the parabola. The coordinates of this point (0; 0).
Basic properties of a quadratic function
1. For x = 0, y = 0, and y> 0 for x0
2. The quadratic function reaches its minimum value at its vertex. Ymin at x = 0; It should also be noted that the function does not have a maximum value.
3. The function decreases in the interval (-∞; 0] and increases in the interval)
