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 exactly 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 x 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( x) to represent the value (again, in most cases, a y - value) of a function at the given independent value of x. For any value of x, ( x, f ( x)) is a point on the graph of the function f( x ). Ex. 1 Given f ( x) to express the domain and range. x, graph the function and determine the domain and range. Use interval notation
2 AMHS Precalculus - Unit 17 Ex. Given f ( x) to express the domain and range. x, graph the function and determine the domain and range. Use interval notation Ex. 3 For the function f ( x) x x 4, find and simplify: a) f ( 3) b) f ( x h) Ex. 4 For x, x 0 f( x) x 1, x 0 find: a) f (1) b) f ( 1) c) f ( ) d) f (3)
3 AMHS Precalculus - Unit 18 Ex. 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) Ex. 6 The graph of the function f is given: a) f ( 3) f (0) f (4) b) For what numbers x is f( x) 0? c) What is the domain of f? d) What is the range of f? e) What is (are) the x -intercept(s)? f) What is the y - intercept? g) For what numbers x is f( x) 0? h) For what numbers x is f( x) 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. Ex. 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): Ex. 1 Determine the domain and range of f ( x) 4 x 3 Ex. Determine the domain of f t t t ( ) 15
5 AMHS Precalculus - Unit 0 Rule for functions containing fractional expressions: 5x Ex. 3 Determine the domain of hx ( ) x 3x 4 Ex. 4 Determine the domain of gx ( ) x 1 x 15 Ex. 5 Determine the domain of hx ( ) 3 x x
6 AMHS Precalculus - Unit 1 Intercepts (revisited) The y -intercept of the graph of a function is(0, f (0)). The x - intercept(s) of the graph of a function f( x) is/are the solution(s) to the equation f( x) 0. These x - values are called the zeros of the function f( x ). Ex. 1 Find the zeros of f ( x) x(3x 1)( x 9) Ex. Find the zeros of f x x x ( ) 5 6 Ex. 3 Find the zeros of f x 4 ( ) x 1 Ex. 4 Find the x - and y - intercepts (if any) of the graph of the function 1 f ( x) x 4
7 AMHS Precalculus - Unit Ex. 5 Find the x - and y - intercepts (if any) of the graph of the function f x ( ) 4( x ) 1 Ex. 6 Find the x - and y - intercepts (if any) of the graph of the function f( x) x 4 x 16 Ex. 7 Find the x - and y - intercepts (if any) of the graph of the function 3 f ( x) 4 x
8 AMHS Precalculus - Unit 3 Transformations Horizontal and Vertical shifts Suppose y f ( x) is a function and c is a positive constant. Then the graph of 1. y f ( x) c is the graph of f shifted vertically up c units.. y f ( x) c is the graph of f shifted vertically down c units. 3. y f ( x c) is the graph of f shifted horizontally to the left c units. 4. y f ( x c) is the graph of f shifted horizontally to the right c units. Ex. 1 Consider the graph of a function y f ( x) shown on the coordinates. Perform the following transformations. y f ( x) 3 y f ( x) y f ( x 1) y f ( x 3)
9 AMHS Precalculus - Unit 4 Suppose y f ( x) is a function. Then the graph of 1. y f( x) is the graph of f reflected over the x -axis.. y f ( x) is the graph of f reflected over the y -axis. Ex. Consider the graph of a function y f ( x). Sketch y f ( x ) 3 Common (Parent) Functions f ( x) x f ( x) x
10 AMHS Precalculus - Unit 5 f ( x) x f ( x) 3 x f ( x) 3 x 1 f( x) x f ( x) x or x
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. Ex. 1 f ( x) x 1 Ex. f ( x) 1 x Ex. 3 f x 3 ( ) ( x ) 1 Ex. 4 f ( x) x 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 -axis if f ( x) f ( x) for every x in the domain of the f( x ).. The origin if f ( x) f ( x) for every x in the domain of the f( x ). If the graph of a function is symmetric with respect to the y -axis, 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 examples 1-3, determine whether the given function y f ( x) is even, odd or neither. Do not graph. Ex f ( x) x x x Ex. f ( x) x 3 Ex. 3 f ( x) x x
13 AMHS Precalculus - Unit 8 Transformations Vertical Stretches and Compressions Suppose y f ( x) is a function and c a positive constant. The graph of y cf ( x) 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 Ex.1 Given the graph of y f ( x) a) Sketch y f ( x) b) 1 y f ( x ) Ex. Sketch the graph of the following functions. Include any intercepts. f ( x) x 1 f ( x) 3( x 1)
14 AMHS Precalculus - Unit 9 Quadratic Functions A quadratic function y f ( x) is a function of the form constants. f ( x) ax bx 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 vertex (which serves as the minimum or maximum of the function depending on the value of a ), a line of symmetry, and may have zero, one or two x - intercepts. Ex. 1 Sketch the graph of f ( x) ( x 1) 3. Determine any intercepts.
15 AMHS Precalculus - Unit 30 The standard form of a quadratic function is parabola and x his the line of symmetry. f ( x) a( x h) k where ( hk, ) is the vertex of the Ex. Rewrite the quadratic function f ( x) x x 3 in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph. Ex. 3 Rewrite the quadratic function f ( x) 4x 1x 9 in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph.
16 AMHS Precalculus - Unit 31 Ex. 4 Complete the square to find all the solutions to the equation ax bx c 0 The vertex of any parabola of the form is ( b b, f ( )) a a. f ( x) ax bx c Ex. 5 Find the vertex of the quadratics from examples and 3 directly by using ( b b, f ( )) a a. Ex. 6 Find the vertex from example by using the x - intercepts and the line of symmetry.
17 AMHS Precalculus - Unit 3 Ex.7 Find the intercepts and vertex of the function 1 f x x x ( ) 1 Ex. 8 Find the maximum or the minimum of the function. 1. f x x x ( ) f x x x ( ) 6 3 Ex.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 ( v0 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 v0 Ex. 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 maximum height attained by the arrow? What is the velocity of the arrow at the time it attains its maximum height? 3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time? Ex. 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 maximum 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 ( x) is a function and c a positive constant. The graph of y f ( cx) 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 Ex.1 Given the graph of y f ( x) c) Sketch y f ( x) d) 1 y f ( x) Ex. Consider the function f x ( ) x 4 a) On the same axis, sketch f ( x), f ( x) and 1 f( x ). Identify any intercepts of each function.
20 AMHS Precalculus - Unit 35 b) On the same axis, sketch ( ), f x f x and 1 f( x ). Identify any intercepts of each function. List the transformations on f ( x) x required to sketch f ( x) x 1
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