"Power Function" Concept and Properties of Functions PPT

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"Power Function" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Through specific examples, understand the concept of power functions and be able to find the analytical formula of power functions.

2. Combine the images of power functions y=x, y=x2, y=x3, y=x-1, y=x^(1/2) to understand their changing rules.

3. Be able to use the basic properties of power functions to solve related practical problems.

Power function PPT, part 2: independent preview

1. Definition of power function

1. (1) What is the difference between the function y=2x and y=x2?

Tip: In the function y=2x, the constant 2 is the base and the independent variable x is the exponent, so it is an exponential function; and in the function y=x2, the independent variable x is the base and the constant 2 is the exponent, so it is a power function.

(2) What are the common characteristics of the analytical expressions of functions y=x, y=x2, y=x3, y=x-1 and y=x^(1/2)?

Tip: The base is the independent variable, and the coefficient of the independent variable is 1; the exponent is a constant; the coefficient of the power xα is 1; there is only one term on the right side of the equal sign of the analytical expression.

2. Fill in the blanks

Generally, the function y=xα is called a power function, where x is the independent variable and α is a constant.

3. Do it

In the function y=1/x^4, y=3x2, y=x2+2x, y=1, the number of power functions is .

Analysis: The function y=1/x^4 =x-4 is a power function; the coefficient of x2 in the function y=3x2 is not 1, so it is not a power function; the function y=x2+2x is not y=xα(α∈R) form, so it is not a power function; the function y=1 is not the same function as y=x0=1 (x≠0), so y=1 is not a power function.

Answer:1

2. Image and properties of power function

1. In the same plane rectangular coordinate system, the image of the power function y=x, y=x2, y=x3, y=x^(1/2), y=x-1 is as shown in the figure below.

(1) Do their images all pass through the same fixed point?

Tip: Yes, they all pass the fixed point (1,1).

(2) Among the above five functions, which ones are increasing functions within (0,+∞)? Which ones are decreasing functions?

Tip: The functions that are increasing within (0,+∞) are: y=x, y=x2, y=x3,y= . The functions that are decreasing within (0,+∞) are: y=x-1 .

(3) Among the above five functions, which ones have a graph that is symmetric about the origin and are odd functions? Which ones have a graph that is symmetric about the y-axis and are even functions?

Tip: The graph is symmetric about the origin, and the odd functions are: y=x, y=x3, y=x-1; the graph is symmetric about the y-axis, and the even functions are: y=x2.

2. Fill in the form

Properties of power functions

3. Determine whether it is right or wrong:

(1) The image of a power function can appear in any quadrant of the plane rectangular coordinate system. ()

(2) The graph of the power function must pass through (0,0) and (1,1).()

Answer: (1)× (2)×

Power function PPT, the third part: inquiry learning

The concept of power function

Example 1 The function f(x)=(m2-m-5)xm-1 is a power function, and when x∈(0,+∞), f(x) is an increasing function. Try to determine the value of m.

Analysis: From f(x)=(m2-m-5)xm-1 is a power function, and when x>0 it is an increasing function. You can first use the definition of the power function to find the value of m, and then use monotonicity to determine The value of m.

Solution: According to the definition of power function,

Get m2-m-5=1,

Solve to get m=3 or m=-2.

When m=3, f(x)=x2 is an increasing function on (0,+∞);

When m=-2, f(x)=x-3 is a decreasing function on (0,+∞), which does not meet the requirements. Therefore, m=3.

Reflection: The basis for judging whether a function is a power function is whether the function is in the form of y=xα (α is a constant), that is: (1) the coefficient is 1; (2) the exponent is a constant; (3) nothing is added after term. On the contrary, if a function is a power function, the function must have this form.

Variation training 1: If the graph of the power function y=(m2-3m+3)x^(m^2 "-" m"-" 2) does not reach the origin, find the value of the real number m.

Solution: From the definition of the power function, we get m2-3m+3=1, and the solution is m=1 or m=2;

When m=1, m2-m-2=-2, the function is y=x-2, its graph does not reach the origin, and the conditions are met;

When m=2, m2-m-2=0, the function is y=x0, its graph does not reach the origin, and the conditions are met.

To sum up, m=1 or m=2.

Power function PPT, part 4: thinking methods

The “convexity” of power functions

(1) Definition of upper convex function and lower convex function

Suppose the function f(x) is defined on [a,b], if any two different points x1,x2,f (x_1+x_2)/2 ≥ in [a,b] (f"(" x_1 ")" +f"(" x_2 ")" )/2 are all true, then f(x) is said to be a convex function on [a,b], that is, Convex function.

Suppose the function f(x) is defined on [a,b], if any two different points x1,x2,f in [a,b] (x_1+x_2)/2 ≤ (f"(" x_1 ")" +f"(" x_2 ")" )/2 are true, then f(x) is said to be a downward convex function on [a,b], that is, Convex function.

From a geometric point of view, this definition is: Take any two points on the graph of function f(x). If the part of the function graph between these two points is always above the line segment connecting the two points, then this function It is an upward convex function; if the part of the function graph between these two points is always below the line segment connecting the two points, then this function is a downward convex function. Judging from the function graph, generally a quadratic function has a downward opening. is an upward convex function, and a quadratic function with an upward opening is a downward convex function.

(2) Convexity of power function

①Power function y=xα,x∈(0,+∞), when α>1, the function is a downward convex function;

②Power function y=xα,x∈(0,+∞), when 0<α<1, the function is an upward convex function;

③Power function y=xα,x∈(0,+∞), when α<0, the function is a downward convex function.

Power function PPT, part 5: practice in class

1. The power function y=kxα passes through the point (4,2), then the value of k-α is ()

A.-1 B.1/2 C.1 D.3/2

Analysis: The power function y=kxα passes through the point (4,2),

So {■(k=1"," @4^α=2"," )┤The solution is {■(k=1"," @α=1/2 "." )┤So k-α=1/ 2.

Answer:B

2. The images of the power function y=x2, y=x-1, y=x^(1/3), y=x^("-" 1/2) in the first quadrant are as follows: curve()

A.C2,C1,C3,C4

B.C4,C1,C3,C2

C.C3,C2,C1,C4

D.C1,C4,C2,C3

Analysis: The "high and low" relationship between the power function image on the right side of the straight line x=1 in the first quadrant is "referring to the height of the big picture", so the image of the power function y=x2 in the first quadrant is C1, y=x The image of -1 in the first quadrant is C4, the image of y=x^(1/3) in the first quadrant is C2, the image of y=x^(1/3) in the first quadrant for C3.

Answer:D

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Update Time: 2024-11-16

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