1. Analyze families of functions
Objective:
Analyze families of functions
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Thursday, January 03, 2019

2. f (x) = mx + b m and b are real numbers
Linear Functions
f (x) = mx + b m and b are real numbers
Domain: All real numbers
Range: All real numbers
Graph is a non-vertical line with slope m and y-intercept b
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Thursday, January 03, 2019

3. Linear Functions
d) Increasing if m > 0 decreasing if m < 0 constant if m = 0
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Thursday, January 03, 2019

4. f (x) = b b is a real number
Constant Function
f (x) = b b is a real number
Domain: All real numbers
Range: b
Graph is a horizontal line with y-intercept b
An even function whose graph is constant over its domain.
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Thursday, January 03, 2019

5. y b x
f (x) = b
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Thursday, January 03, 2019

6. Domain : All real numbers Range: All real numbers
Identity Function
f (x) = x
Domain : All real numbers
Range: All real numbers
Graph is a line with slope 1 and y-intercept 0
An odd function whose graph is increasing over its domain.
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Thursday, January 03, 2019

7. y b
(1, 1) x
(-1, -1)
f (x) = x
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Thursday, January 03, 2019

8. Domain: All real numbers Range: Nonnegative real numbers
Polynomial Functions
Quadratic Function:
Domain: All real numbers
Range: Nonnegative real numbers
Graph is a parabola whose intercept is (0, 0)
An even function whose graph is decreasing for x < 0 and increasing for x > 0.
f (x) = x2
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Thursday, January 03, 2019

9. J Reasons
Thursday, January 03, 2019

10. Domain: All real numbers Range: All real numbers
Cubic Function
f (x) = x3
Domain: All real numbers
Range: All real numbers
The intercept of the graph is (0, 0)
An odd function whose graph is increasing over its domain
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Thursday, January 03, 2019

11. J Reasons
Thursday, January 03, 2019

12. Domain: All nonnegative real numbers
Square Root Function
Domain: All nonnegative real numbers
Range: All nonnegative real numbers
The intercept of the graph is (0, 0)
Neither even nor odd function whose graph is increasing for x > 0
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Thursday, January 03, 2019

13. J Reasons
Thursday, January 03, 2019

14. Domain: All real numbers Range: All nonnegative real numbers
Absolute Value Function
Domain: All real numbers
Range: All nonnegative real numbers
The intercept of the graph is (0, 0)
An even function whose graph is decreasing for x < 0 and increasing for x > 0
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Thursday, January 03, 2019

15. J Reasons
Thursday, January 03, 2019

16. Domain: All real numbers Range: The set of integers
Greatest-integer Function
Domain: All real numbers
Range: The set of integers
y-intercept: 0; x-intercepts: [0, 1)
Neither even nor odd function whose graph is constant [k, k + 1) for k an integer.
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Thursday, January 03, 2019

17. y 3 x -2 2
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Thursday, January 03, 2019

18. 6) Rational Function: R(x) = where p and q are polynomial functions and q is not the zero polynomial.
Domain is all real numbers except those for which the denominator q is 0
If, as x   or as x  -, the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R.
If, as x approaches some number c, the values |R(x)|  , then the line x = c is a vertical asymptote of the graph of R.
If an asymptote is neither horizontal nor vertical it is called oblique (slant).
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Thursday, January 03, 2019

19. Domain is all real numbers.
7) Exponential Function y = bx where base b is a positive real number and the exponent is a variable.
For b> 1
Domain is all real numbers.
Range is all real numbers greater than zero.
The x-intercept is none
The y-intercept is (0,1)
The behavior is continuous, one-to-one, and increasing.
Horizontal asymptote is y = 0 (negative x-axis).
Vertical asymptote is none
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Thursday, January 03, 2019

20. For 0<b<1 Domain is all real numbers
Exponential Function y = bx where base b is a positive real number and the exponent is a variable.
For 0<b<1
Domain is all real numbers
Range is all real numbers greater than zero
The x-intercept is none
The y-intercept is (0,1)
The behavior is continuous, one-to-one, and increasing.
Horizontal asymptote is y = 0 (positive x-axis).
Vertical asymptote is none.
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Thursday, January 03, 2019

21. 8) Logarithmic Function y = logb x where b > 0 and b ≠1
Domain is all real numbers greater than zero.
Range is all real numbers.
The x-intercept is (1,0).
The y-intercept is none
Horizontal asymptote is none
Vertical asymptote is x = 0
The behavior is continuous, one-to-one, and increasing for x >0
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Thursday, January 03, 2019

22. Natural logarithmic functions y = ln x
Domain is all real numbers greater than zero
Range is all real numbers
The x-intercept is (1, 0).
The y-intercept is none
Horizontal asymptote is none.
Vertical asymptote is x = 0 (y-axis).
The behavior continuous, one-to-one, and increasing for x >0.
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Thursday, January 03, 2019

23. Trigonometric Functions
Trigonometric Functions: sine, cosine, tangent, cotangent, secant, and cosecant. 
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Thursday, January 03, 2019

24. Piecewise functions
Piecewise functions is a function f(x) defined piecewise, that is f(x) is given by different expressions on various intervals.
Thursday, January 03, 2019
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25. Family of functions (graphs)
Group of graphs that displays one or more similar characteristics. y = x 2 y =3 x 2 y = x 2 + 2x + 3
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Thursday, January 03, 2019

26. Parent graph
basic graph that is transformed to create other members in a family of graphs. y = x 2 y =3 x 2 y = x 2 + 2x + 3
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Thursday, January 03, 2019