Unit 2: Functions and Graphs

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1 AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to eactly one element in the range. The domain is the set of all possible inputs for the function. On a graph these are the values of the independent variable (most commonly known as the values). The range is the set of all possible outputs for the function. On a graph these are the values of the dependent variable (most commonly known as the y values). We use the notation f() to represent the value (again, in most cases, a y - value) of a function at the given independent value of. For any value of, (, f ( )) is a point on the graph of the function f( ). E. 1 Given f ( ) to epress the domain and range., graph the function and determine the domain and range. Use interval notation

2 AMHS Precalculus - Unit 17 E. Given f ( ) to epress the domain and range., graph the function and determine the domain and range. Use interval notation E. 3 For the function f ( ) 4, find and simplify: a) f ( 3) b) f ( h ) E. 4 For f( ), 0 1, 0 find: a) f (1) b) f ( 1) c) f ( ) d) f (3)

3 AMHS Precalculus - Unit 18 E. 5 The graph of the function f is given: a) Determine the values: f ( ) f (0) b) Determine the domain: c) Determine the range: f () f (4) E. 6 The graph of the function f is given: a) f ( 3) f (0) f (4) b) For what numbers is f( ) 0? c) What is the domain of f? d) What is the range of f? e) What is (are) the -intercept(s)? f) What is the y - intercept? g) For what numbers is f( ) 0? h) For what numbers is f( ) 0?

4 AMHS Precalculus - Unit 19 Vertical Line Test for a Function: An equation is a function iff every vertical line intersects the graph of the equation at most once. E. 7 Determine which of the curves are graphs of functions: a) b) c) Domain (revisited) Rule for functions containing even roots (square roots, 4 th roots, etc): E. 1 Determine the domain and range of f ( ) 4 3 E. Determine the domain of f ( t) t t 15

5 AMHS Precalculus - Unit 0 Rule for functions containing fractional epressions: E. 3 Determine the domain of h ( ) E. 4 Determine the domain of g ( ) 1 15 E. 5 Determine the domain of h ( ) 3

6 AMHS Precalculus - Unit 1 Intercepts (revisited) The y -intercept of the graph of a function is(0, f (0)). The - intercept(s) of the graph of a function f() is/are the solution(s) to the equation f( ) 0. These - values are called the zeros of the function f(). E. 1 Find the zeros of f ( ) (3 1)( 9) E. Find the zeros of f ( ) 5 6 E. 3 Find the zeros of 4 f ( ) 1 E. 4 Find the - and y - intercepts (if any) of the graph of the function 1 f ( ) 4

7 AMHS Precalculus - Unit E. 5 Find the - and y - intercepts (if any) of the graph of the function f ( ) 4( ) 1 E. 6 Find the - and y - intercepts (if any) of the graph of the function f( ) 4 16 E. 7 Find the - and y - intercepts (if any) of the graph of the function 3 f ( ) 4

8 AMHS Precalculus - Unit 3 Transformations Horizontal and Vertical shifts Suppose y f () is a function and c is a positive constant. Then the graph of 1. y f () c is the graph of f shifted vertically up c units.. y f () c is the graph of f shifted vertically down c units. 3. y f ( c ) is the graph of f shifted horizontally to the left c units. 4. y f ( c ) is the graph of f shifted horizontally to the right c units. E. 1 Consider the graph of a function y f () shown on the coordinates. Perform the following transformations. y f ( ) 3 y f ( ) y f ( 1) y f ( 3)

9 AMHS Precalculus - Unit 4 Suppose y f () is a function. Then the graph of 1. y f () is the graph of f reflected over the -ais.. y f ( ) is the graph of f reflected over the y -ais. E. Consider the graph of a function y f (). Sketch y f ( ) 3 Common (Parent) Functions f ( ) f ( )

10 AMHS Precalculus - Unit 5 f ( ) f ( ) 3 3 f ( ) 1 f( ) f ( ) or

11 AMHS Precalculus - Unit 6 Combining common functions with transformations Sketch the graphs of the following functions. Determine the domain and range and any intercepts. E. 1 f ( ) 1 E. f ( ) 1 E. 3 3 f ( ) ( ) 1 E. 4 f ( ) 1 3

12 AMHS Precalculus - Unit 7 Symmetry (revisited) Tests for Symmetry The graph of a function f is symmetric with respect to: 1. the y -ais if f ( ) f ( ) for every in the domain of the f().. The origin if f ( ) f ( ) for every in the domain of the f(). If the graph of a function is symmetric with respect to the y -ais, we say that f is an even function. If the graph of a function is symmetric with respect to the origin, we say that f is an odd function. In eamples 1-3, determine whether the given function y f () is even, odd or neither. Do not graph. E f ( ) E. f ( ) 3 E. 3 f ( )

13 AMHS Precalculus - Unit 8 Transformations Vertical Stretches and Compressions Suppose y f () is a function and c a positive constant. The graph of y cf () is the graph of f 1. Vertically stretched by a factor of c if c 1. Vertically compressed by a factor of c if 0 c 1 E.1 Given the graph of y f () a) Sketch y f ( ) b) y 1 f ( ) E. Sketch the graph of the following functions. Include any intercepts. f ( ) 1 f ( ) 3( 1)

14 AMHS Precalculus - Unit 9 Quadratic Functions A quadratic function y f () is a function of the form constants. f ( ) a b c where a 0, b and c are The graph of any quadratic function is called a parabola. The graph opens upward if a 0 and downward if a 0. The domain of a quadratic function is the set of real numbers (, ). A quadratic function has a verte (which serves as the minimum or maimum of the function depending on the value of a ), a line of symmetry, and may have zero, one or two - intercepts. E. 1 Sketch the graph of f ( ) ( 1) 3. Determine any intercepts.

15 AMHS Precalculus - Unit 30 The standard form of a quadratic function is parabola and his the line of symmetry. f ( ) a( h) k where ( hk, ) is the verte of the E. Rewrite the quadratic function f ( ) 3 in standard form by completing the square. Determine any intercepts, the verte, the line of symmetry and sketch the graph. E. 3 Rewrite the quadratic function f ( ) in standard form by completing the square. Determine any intercepts, the verte, the line of symmetry and sketch the graph.

16 AMHS Precalculus - Unit 31 E. 4 Complete the square to find all the solutions to the equation a b c 0 The verte of any parabola of the form b b a a. f ( ) a b c is (, f ( )) b b E. 5 Find the verte of the quadratics from eamples and 3 directly by using (, f ( )) a a. E. 6 Find the verte from eample by using the - intercepts and the line of symmetry.

17 AMHS Precalculus - Unit 3 E.7 Find the intercepts and verte of the function 1 f ( ) 1 E. 8 Find the maimum or the minimum of the function. 1. f ( ) f ( ) 6 3 E.9 Determine the quadratic function whose graph is given.

18 AMHS Precalculus - Unit 33 Freely Falling Object - Suppose an object, such as a ball, is either thrown straight upward or downward with an initial velocity v 0 or simply dropped ( v 0 0 ) from an initial height s 0. Its height, st () as a 1 function of time t can be described by the quadratic function s() t gt v t s 0 0 Gravity on earth is 3 ft / sec or 9.8 m / sec. Also, the velocity of the object while it is in the air is v() t gt v 0 E. 10 An arrow is shot vertically upward with an initial velocity of 64 ft / sec from a point 6 feet above the ground. 1. Find the height st () and the velocity vt () of the arrow at time t 0.. What is the maimum height attained by the arrow? What is the velocity of the arrow at the time it attains its maimum height? 3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time? E. 11 The height above the ground of a toy rocket launched upward from the top of a building is given by s( t) 16t 96t What is the height of the building?. What is the maimum height attained by the rocket? 3. Find the time when the rocket strikes the ground. What is the velocity at this time?

19 AMHS Precalculus - Unit 34 Horizontal Stretches and Compressions Suppose y f () is a function and c a positive constant. The graph of y f ( c) is the graph of f 1. Horizontally compressed by a factor of 1 c if c 1. Horizontally stretched by a factor of 1 c if 0 c 1 E.1 Given the graph of y f () c) Sketch y f ( ) d) 1 y f ( ) E. Consider the function f ( ) 4 a) On the same ais, sketch f ( ), f and 1 f( ). Identify any intercepts of each function.

20 AMHS Precalculus - Unit 35 b) On the same ais, sketch f ( ), f and 1 f( ). Identify any intercepts of each function. List the transformations on f ( ) required to sketch f ( ) 1

21 AMHS Precalculus - Unit 36 Silly String Activity Objective: The use a quadratic function to model the path of silly string. Materials: Can of silly string, tape measure, stopwatch, clear overhead transparency, TI84 Personnel: Timekeeper, Silly-String operator, assistant Calculate the initial velocity v0 of the silly string as it eits the can. 1. Hold the can of silly string 1 foot above the ground. Have the timekeeper start the stopwatch and say go. At this time, shoot a short burst of silly string towards the ceiling. Have the class keep a casual eye on the maimum height the silly string achieves. When the silly string hits the floor, have the timekeeper stop the stopwatch and record the elapsed time.. Measure the maimum height of the silly string observed by the class. Use the position equation 1 s() t gt v0t s0 with g = 3 ft / sec to calculate v 0. ( s 0 = 1, get t from the timekeeper. This represents the time it took for the silly string to reach the ground, i.e. st () =0) Now that we know gv, 0 and s0 we can set up a position equation to model the height of the silly string as a function of time. Use this equation to determine the maimum height (the verte holds this info) of the silly string. How does this compare to the actual height observed by the class. What factors might have caused it to be different? Now we are going to get the assistant to lean over the can of silly string (with the clear overhead transparency protecting the face) in its original position 1 foot above the ground and see if the assistant can move fast enough to avoid getting silly string in the face. Calculate the time it would take for the silly string to reach the assistant s face (set st () = the height of the assistant s face and solve fort ) Once the reaction time for the assistant has been calculated and discussed, see if the assistant can actually react that quickly, i.e. avoid silly string in the face. To date, it has never been done. Enjoy!

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