Review for test 2. Graphs of functions (vertical shifts, horizontal shifts, compression, stretching): Given the graph of y=f(x), and c >0

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1 Review for test 2 Graphs of functions (vertical shifts, horizontal shifts, compression, stretching): Given the graph of y=f(x), and c >0 a) The graph of y=f(x-c) is obtained by b) The graph of y=f(x+c) is obtained by c) The graph of y=cf(x) while c>1 is obtained by..a factor d) The graph of y=cf(x) while 0< c <1 is obtained by..a factor e) The graph of y=f(cx) while c>1 is obtained by..a factor f) The graph of y=f(cx) while c>1 is obtained by..a factor g) The graph of y= - f(x) is obtained by.. h) The graph of y=f(-x) obtained by.. Remark: a), b),g) keep the range of the function but change the domain c), d) keep the domain of the functions but change the range Quadratic functions y=f(x)= where a is not zeros 1) f(x)=a(x-h) 2 +k is called "the standard form of the function" 2) let b 2-4ac i) if <0 then =0 has NO real root ii) if =0 then 0 has EXACTLY one real root given by iii) if > 0 then 0 has EXACTLY two real roots given by x= 3) V(h,k) is called the vertex of the graph of the functions in fact : h= and k= =f( and hence f(x)=a(x ) if a>0 then f(-b/2a) is the minimum value of f(x) 5) if a<0 then f(-b/2a) is the maximum value of f(x) Operations on functions

2 A) Given two functions f(x) with a domain D 1 and g(x) with a domain D 2, then the following functions have a domain D 1 D 2 1) (f g)(x)= f(x) g(x) 2) f(x)g(x) B) f(x)/g(x) is defined for all x such that f(x) is defined and g(x) is not zero Composition of functions f o g(x)=f[g(x)] and note that f o g(x) g o f(x) One-to one and inverse functions f(x) with a domain D is called a one-to-one function if or Horizontal line test: f(x) is one-to-one if every horizontal line intersects the graph of f(x) at most one point - if f(x) is an increasing function OR an decreasing function the f(x) is one-to one When f(x) is one-to-one function then it has its inverse denoted by f -1 (x) a) f o f -1 (x)=f -1 of(x)=x b) f -1 (x) is NOT c) domain of f -1 = range of f d) range of f -1 =domain of f Practice exercises -5.Explain how the graph of the function compares to the graph of y = f(x). For example, for the equation y = 2f(x + 3), the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2. y = - f(-2(x + 5) ) +10 a) Shifted 5 units to the left, reflect through the y axis, stretched vertically a factor 2, and then shift up 10 units. b) Shifted 5 units to the left, stretched horizontally a factor 2, reflected through the y axis, compressed vertically a factor 2, reflected though the x-axis, and then shifted up 10 units c) Shifted 5 units to the left, compressed vertically by factor of -1/2, and then shift up 10 units

3 -4. Find the point in the second quadrant at which the graph of the quadratic function intersects the graph of the linear function -3. Find the point in the first quadrant at which the graph of the quadratic function the point (x,y)= intersects the graph of the linear function The graphs intersect at -2.Let y = f(x) be a function with domain D = [-5, 6] and range R = [-4, 8], P is on the graph of its. Find the domain D, and corresponding P on the graph of given function and range R for the function. Assume f(-5) = 8 and f(6) = -4. P (0,4/3) : y = f(x + 2) 5-1.Find the standard equation (y = a(x - h)^2 + k) of a parabola that has a vertical axis and satisfies the given conditions. Be sure to write your answer in the specified format. Vertex (4, -3), x-intercept Express f(x) in the form a(x - h) 2 + k. where 1.The point (a,b) is on the graph of the one-to-one function y = f(x). For each of the following functions, enter the ordered pair that corresponds to the transformation of (a,b). For example, the graph of y = f(x) + 1 is obtained by translating the graph of y = f(x) up one unit so the corresponding point on the new graph is (a,b + 1). (, ) (, ) (, ) 2. Suppose you are given functions and which are inverses of each other, and with domain and range all real numbers. Suppose further that you know. Mark the following True or False. a) b) c)

4 d) 3. The graph of a one-to-one function f is shown. Assume a = 21, b = -7, c = 7. (a) Find the domain D and range R of the function f. (b) Find the domain D 1 and range R 1 of the inverse function f Let h(x) = 8 - x. Use h, the table, and the graph to evaluate the expression. x f(x) it has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 80,000 km 2 in 80 days. Earlier, we found that the function g(t) = 1000t represents the contaminated area as a function of time, the number of days since contamination. From geometry, you can create a function r = f(a) that expresses the radius of a circle as a function of area. (a) Express the radius as a function of A (upper case!).

5 (b) what does the composite function model? (linear, quadratic, or neither) 6. The red graph above is of the function f(x), and the blue graph is g(x). Use the graphs to compute the following function values. The spacing between each gridline is one unit. (a) f(g(3)) = (b) f(g(0)) = 7. Consider the functions below. (a) Find (f g)(x) and the domain of f g. (b) Find (g f)(x) and the domain of g f. 8. Given 10 1 a) Find f(1) b) Find f -1 ( ) c) Find (f -1 of)( )? 9.Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions : x-intercepts -2 and 8, lowest point has y-coordinate Find the vertex of the quadratic function y = x(x - 2) 11. If a ball is thrown directly upward with a velocity of 21 ft/s, its height (in feet) after t seconds is given by y = 21t - 16t 2. What is the maximum height attained by the ball? 12.Let P be a point on the graph of y = x 2 with x-coordinate 4. Find the point Q also on the graph of y = x 2 such that the slope of the line passing through both P and Q is 7.

6 13.Find the standard equation (y = a(x - h)^2 + k) of the parabola shown in the figure. Be sure to answer in the specified format. 14.If a cylinder has constant length 10 inches, its volume becomes a function of its radius r,. Suppose the cylinder is inflated, and the growing radius r is given by the function where r is measured in inches and t is the number of seconds you have been inflating the cylinder. Compute the composition (g f) as a function of t. 15.Determine whether f is even, odd, or neither even nor odd : f(x) = 3x 5-2x Find the roots of the quadratic equation (x - 4)(x - 6) = 3