MATH 150 Pre-Calculus Fall, 2014, WEEK 6

Home , Constant function, Quadratic function

JoungDong Kim

Week 6: 4A, 4B, 4C, 4D Chapter 4. Functions Chapter 4A. Introduction to Functions

Deﬁnitions

A function is a relation in which no two diﬀerent ordered pairs have the same ﬁrst element. •

A function can also be deﬁned as a rule that assigns exactly one element in set B to each element • in set A.

Set A, the set of all ﬁrst coordinates, is called the domain of the function. • Set B, the set of all second coordinates, is called the range. •

Note. There is only one output (y-value) for each input (x-value).

1 Ex1) Function or Not? a) (2, 3), ( 1, 3), (5, 3), (8, 3) { − }

b) (1, 5), (2, 7), (3, 9), (1, 4) { }

Notation. y = f(x) : x is input, f(x) or y is output.

Evaluating Functions To evaluate the function f, we ﬁnd the output for a given input.

Ex2) For f(x)=2x2 x + 1, ﬁnd f( 3), f(a + h), and f( x). − − −

2 f(x + h) f(x) Ex3) Evaluate − , h = 0 for the function f(x)=2x2 x + 1. h 6 −

f(x + h) f(x) Ex4) Evaluate − , h = 0 for the function f(x)= x2 x 1. h 6 − − − DIY.

3 Domain of a Function The domain of a function is the set of all inputs for the function.

In Sum, The domain of most algebraic functions is the set of all real numbers EXCEPT:

1. any real numbers that cause the denominator of a function to be 0.

2. any real numbers that cause us to take the square root (or any radical with an even index) of a negative number. and more...

4 Ex5) Find the domain of each function. 1 a) f(x)= x2 x −

b) g(x)= √2x 3 −

t c) h(t)= √t +1

5 Ex6) Find the domain and range of each function. a) f(x)= x2 5x +6 −

b) g(x)= √x2 5x +6 −

6 √x +2 Ex7) Find the domain of f(x)= . x 1 −

7 Applying Functions Ex8) If x and y are the lengths of the adjacent sides of a rectangle whose parameter is p, write y as a function of x.

Ex9) An open box is constructed from a 12-inch square piece of cardboard by cutting squares of equal length from each corner and turning up the sides. Write a function for the volume of the cardboard box in terms of the length of the cutout square. Restrict the domain appropriately.

8 Chapter 4B - Graphs of Functions

Deﬁnition If f is a function with domain A, then the graph of f is the set of ordered pairs (x, f(x)), where x A. ∈ In other words, the graph of f is the set of all points (x, y) such that y = f(x).

Note. y = mx + b: linear function, straight line.

Ex10) Graph the function f(x)= √x + 3, and ﬁnd the domain, range, x-intercept, and y-intercept.

9 Find domain and range from Graphs; To ﬁnd the domain of f, vertical line scanning the x-axis. To ﬁnd the range of f, horizontal line scanning the y-axis.

10 Ex11) Find x- and y-intercepts of the graph of f(x)= x3 9x. −

11 Vertical Line Test A set of points in the coordinate plane represents a function if and only if no vertical line intersects the graph in more than one point.

12 Catalog of Basic Functions

1. Constant Function f(x)= c, c : real constant

2. Identity Function f(x)= x

3. Squaring Function f(x)= x2

13 4. Cubing Function f(x)= x3

5. Square Root Function f(x)= √x

6. Absolute Value Function f(x)= x | |

7. Reciprocal 1 f(x)= x

14 Piecewise Functions

 x, x< 0 Ex12) Graph f(x)= by plotting points.   x2, x 0 ≥  

15 Ex13) Graph the piecewise function

 x, if x 0 f(x)= ≥   x, if x< 0 −  

16 Ex14) Graph the piecewise function

2  x if x> 1   f(x)=  3 if x =1   x + 1 if x< 1  −  

17 Increasing, Decreasing, and Constant

Deﬁnition. A function f is increasing on an interval I if and only if for every x1 < x2 I, ∈ f(x1) < f(x2).

Deﬁnition. A function f is decreasing on an interval I if and only if for every x1 < x2 I, ∈ f(x1) > f(x2).

Deﬁnition. A function f is constant on an interval I if and only if for every x1, x2 I, f(x1)= ∈ f(x2).

18 Ex15) Use the graph to ﬁnd the following for each function.

Domain • Range • Intercepts • Intervals where f is increasing, decreasing, and constant •

19 Chapter 4C. Transformations of Functions

Vertical and Horizontal Shifting Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if it is negative.

Ex16) Use the graph of f(x)= x2 to sketch the graph of each function. a) g(x)= x2 +3 b) h(x)= x2 2 −

Vertical Shifts of Graphs Suppose c> 0. To graph y = f(x)+ c, shift the graph of y = f(x) upward c units. To graph y = f(x) c, shift the graph of y = f(x) downward c units. −

20 Horizontal Shifts of Graphs Suppose c> 0. To graph y = f(x c). shift the graph of y = f(x) to the right c units. − To graph y = f(x + c). shift the graph of y = f(x) to the left c units.

Ex17) Use the graph of f(x)= x2 to sketch the graph of each function a) g(x)=(x + 4)2 b) h(x)=(x 2)2 −

21 Ex18) Sketch the graph of f(x)= √x 3 + 4. −

22 Ex19) Sketch the graph of f(x)=(x 1)2 + 2. −

23 Reflecting Graphs To graph y = f(x), reflect the graph of y = f(x) in the x-axis. To graph y = −f( x), reflect the graph of y = f(x) in the y-axis. −

Ex20) Sketch the graph of each function. a) f(x)= x2 b) g(x)= √ x − −

24 Vertical Stretching and Shrinking The y-coordinate of y = cf(x) at x is the same as the corresponding y-coordinate of y = f(x) multiplied by c. Multiplying the y-coordinates by c has the eﬀect of vertically stretching or shrinking the graph by a factor of c.

Vertical Stretching and Shrinking of Graphs. To graph y = cf(x); If c> 1, stretch the graph of y = f(x) vertically by a factor of c. If 0

Ex21) Use the graph of f(x)= x2 to sketch the graph of each function. 1 a) g(x)=3x2 b) h(x)= x2 3

25 Ex22) Sketch the graph of the function f(x)=1 2(x 3)2 − −

26 Ex23) Sketch the graph of the function f(x)= x3 2 − −

27 1 Ex24) Sketch the graph of the function f(x)= +3 x 2 −

28 Ex25) Sketch the graph of the function f(x)= 3 x +1 2 − | | −

29 Even and Odd Functions Let f be a function. f is even if f( x)= f(x) for all x in the domain of f. − f is odd if f( x)= f(x) for all x in the domain of f. − −

Note. The graph of an even function is symmetric with respect to the y-axis, The graph of an odd function is symmetric with respect to the origin.

30 Ex26) Determine whether the functions are even, odd, or neither even nor odd. a) f(x)= x5 + x

b) g(x)=5x4

31 Chapter 4D. Maximum/Minimum Function Values.

Quadratic Functions A quadratic function is a function f of the form

f(x)= ax2 + bx + c where a, b, and c are real numbers and a = 0. The graph of a quadratic function is parabola. 6

Standard form of a Quadratic Function A quadratic function f(x)= ax2 + bx + c can be expressed in the standard form

f(x)= a(x h)2 + k − by completing the square. The graph of f is a parabola with vertex (h, k): the parabola opens upward if a> 0 or downward if a< 0.

32 Ex27) Let f(x)=2x2 12x + 23, express f in standard form and sketch the graph of f. −

33 Maximum or Minimum Value of a Quadratic Function Let f be a quadratic function with standard form f(x)= a(x h)2 + k. The maximum or minimum value of f occurs at x = h. − If a> 0, then the minimum value of f is f(h)= k. If a< 0, then the maximum value of f is f(h)= k.

Ex28) Find the maximum or minimum value of f(x)=5x2 30x + 49. −

34 Ex29) Write f(x) = 2x2 + 10x 7 in standard form. Draw the graph and ﬁnd vertex, axis − − of symmetry, domain, range, the maximum or minimum value, and the intervals of increasing and decreasing.

35 If we interested only in ﬁnding the maximum or minimum value, then a formula is available for doing so. This formula is obtained by completing the square for the general quadratic function as follows;

2 This equation is in standard form with h = b and k = b + c. Since the maximum or minimum − 2a − 4a value occurs at x = h, we have the following result Maximum or Minimum value of a Quadratic Function.

The maximum or minimum value of a quadratic function f(x)= ax2 + bx + c occurs at

b x = . −2a b If a> 0, then the minimum value of f is f . −2a b If a< 0, then the maximum value of f is f . −2a

36 Ex30) Let f(x)= 2x2 +4x 5. Find the axis of symmetry and the coordinates of the vertex. − −

37 Ex31) If f(3) = 5, f( 5) = 5, and f(x) is a quadratic function, what is the axis of symmetric? −

38 Zeros of Quadratic Functions 2 By a zero of a quadratic function f(x)= ax + bx + c, we mean a number x0 such that

2 f(x0)= ax0 + bx0 + c =0

Ex32) Find the zeros of f(x)= x2 5x + 6. −

39 Deﬁnition.(Discriminant) The expression b2 4ac is called the discriminant of the function 2 − f(x)= ax + bx + c, notated D = b2 4ac −

b2 4ac > 0 : two zeros −

b2 4ac = 0 : one zero −

b2 4ac < 0 : no zeros −

Ex33) Calculate the discriminant of f(x)= x2 4x + 10, and plot the function. −

40 Ex34) If the vertex of the quadratic function f(x) lies at the point (3, 6) and f(1) = 8, write the function f(x) and ﬁnd f(5).

41 Ex35) Most cars get their best gas mileage when traveling at a relatively modest speed. The gas mileage M for a certain new car is modeled by the function 1 M(s)= s2 +3s 31, 15 s 70 −28 − ≤ ≤ where s is the speed in mi/h and M is measured in mi/gal. What is the car’s best gas mileage, and what speed is it attained?

42 Local Maximum and Minimum Values

Observe the graph below. If we were walking along the graph, moving from left to right, we would be traveling downhill at the beginning. We would reach a low-point and begin traveling uphill until we reached a high point from which we would travel downhill again. After the next low point we continue uphill for the rest of the way. Critical points occur where the graph turns around. These are not necessarily the lowest or highest functional values for the entire graph, but they are high points or low points for a particular area of the graph. These points are called generically local extreme points, and may either be local maximum or local minimum points.

Ex36) Find the local maximum and minimum values of the function f(x)= x3 8x + 1. −