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Session 9

Session 9. Agenda: Questions from 4.1-4.3? 5.1 – Characteristics of Functions and Parent Functions 5.2 – Evaluating Functions 5.3 – Transformations of Functions Things to do before our next meeting. Questions?. 5.1 – Characteristics of Functions and Parent Functions.

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Session 9

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  1. Session 9 Agenda: • Questions from 4.1-4.3? • 5.1 – Characteristics of Functions and Parent Functions • 5.2 – Evaluating Functions • 5.3 – Transformations of Functions • Things to do before our next meeting.

  2. Questions?

  3. 5.1 – Characteristics of Functions and Parent Functions • A function, f, is a rule that assigns to each value in a set A a unique value in set B. • Set A is called the domain of the function. • Set B is called the range of the function. • In a function, no value in the domain can be assigned to more than one value in the range. • Function notation: We write f(a)=b, if the element a in the domain is assigned to the element b in the range.

  4. Functions can be represented by sets of ordered pairs where the first coordinates (x-values) are the elements in the domain and the second coordinates (y-values) are their corresponding elements in the range. • For example, consider the set of ordered pairs {(1,1), (2,7), (3,7), (5,9)}. Does this set of ordered pairs represent a function? If not, why not? If so, what are the domain and range of the function? • Using function notation, we would write f(1)=1, f(2)=7, f(3)=7, and f(5)=9.

  5. Consider the set of ordered pairs {(1,4), (3,7), (4,8), (1,9), (2,12)}. Does this set represent a function? If not, why not? If so, what are the domain and range of the function?

  6. Most of the time when dealing with functions, instead of listing ordered pairs, we express the function in terms of the rule that assigns the domain values to the range values. For example: are all examples of functions. • The domain is the set of all x-values for which the function is defined. • The range is the set of all values of f(x) that are attained. These are the y-values. • All the points (x, f(x)) for which the function is defined form the graph of the function in the coordinate plane.

  7. Given a graph, use the Vertical Line Test to determine if the graph represents a function: A graph represents a function if no vertical line intersects the graph at more than one point. • Determine whether each of the following graphs represent functions or not.

  8. Given the following graphs of functions, determine the domain and range from the graph.

  9. The x-intercepts of a graph are the points where the graph touches or crosses the x-axis. • The y-intercept of a graph is the point where the graph touches or crosses the y-axis. • Note that it is possible for graphs to not have x-intercepts or a y-intercept. • Determine the x and y-intercepts of the graphs on the previous slide.

  10. A function is said to be increasing on an interval if f(a)<f(b) whenever a<b on the interval.In other words, as x increases, so does y. This means the graph is rising from left to right. • A function is said to be decreasing on an interval if f(a)>f(b) whenever a<b on the interval.In other words, as x increases, y decreases. This means the graph is falling from left to right. • A function is said to be constant on an interval if f(a)=f(b) for any a and b on that interval. • A function has a maximum value if the function has a point (a, f(a)), where f(a)≥f(x) for all x in the domain. • A function has a minimum value if the function has a point (a, f(a)), where f(a)≤f(x) for all x in the domain.

  11. Determine the intervals where the following functions are increasing, decreasing, and/or constant. Determine the maximum and minimum values of the function if they exist.

  12. To determine if an equation represents a function (without using a graph), try to solve the equation for y. If it is possible to solve for y uniquely, then the equation represents a function. • Determine if the following equations represent functions.

  13. It is important to be able to recognize and graph the basic parent functions that are used often in mathematics. Make sure you are comfortable with the graphs below.

  14. 5.2 – Evaluating Functions • Evaluating functions involves determining the function value (y-value) for a given x-value in the domain. • Given the function , evaluate the following.

  15. For the same function below, find and simplify

  16. For the function below, find and simplify

  17. The difference quotient for a function f(x) is: • Find and simplify the difference quotient for the function

  18. Find and simplify the difference quotient for the function

  19. 5.3 – Transformations of Functions • Vertical Shifts • Given the graph of f(x), the graph of f(x)+c, c>0, is obtained by shifting the graph of f(x) UP c units. • Given the graph of f(x), the graph of f(x)-c, c>0, is obtained by shifting the graph of f(x) DOWN c units. • Horizontal Shifts • Given the graph of f(x), the graph of f(x+c), c>0, is obtained by shifting the graph of f(x)c units to the LEFT. • Given the graph of f(x), the graph of f(x-c), c>0, is obtained by shifting the graph of f(x)c units to the RIGHT.

  20. Identify the parent function and describe how the following graphs would be obtained from the graph of the parent function. Sketch a graph of the function.

  21. Vertical Scaling (Stretching/Shrinking) • Given the graph of f(x), the graph of cf(x), for c > 1, is found by vertically stretching the graph of f(x) by a factor of c. • Given the graph of f(x), the graph of cf(x), for 0 < c < 1, is found by vertically shrinking (or compressing) the graph of f(x) by a factor of c. • Reflections • Given the graph of f(x), the graph of –f(x) is found by reflecting the graph of f(x) about the x-axis. • Given the graph of f(x), the graph of f(-x) is found by reflecting the graph of f(x) about the y-axis.

  22. Identify the parent function and describe how the following graphs would be obtained from the graph of the parent function. Sketch a graph of the function.

  23. Sketch a graph of the following function using transformations.

  24. Given the graph of f(x) below, sketch a graph of

  25. The graph of f(x)=1/x is shifted to the right 5, reflected about the y-axis, shrunk by a factor of ⅛, shifted down 4 units, and then reflected across the x-axis. Write the function rule for the resulting graph.

  26. Things to Do Before Next Meeting: • Work on Sections 5.1-5.3 until you get all green bars! • Write down any questions you have. • Continue working on mastering 4.1-4.3. After you have all green bars on 4.1-4.3, retake the Chapter 4 Test until you obtain at least 80%.

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