Names Date Math 00 Worksheet. Consider the function f(x) = x 6x + 8 (a) Complete the square and write the function in the form f(x) = (x a) + b. f(x) = x 6x + 8 ( ) ( ) 6 6 = x 6x + + 8 = (x 6x + 9) 9 + 8 = (x ) (b) Sketch the graph of g(x) = x and the function f(x) from part (a). y x
y=x^-6x+8 y=x^ - - - 0 6 - - (c) Explain how you would use the graph of x to graph the function f(x) = x 6x + 8 from question (a). From what you got in (a), you can notice that f(x) = g(x ). Therefore, to obtain the graph of f(x), you must shift the graph of x three units to the RIGHT and unit DOWN. (d) Consider the function f(x) = x 6x + 8 as above. Let P be the point (, ). In the table below, fill in the y-coordinate of the point Q for the given value of x and compute the corresponding slope m P Q of the secant P Q. x Q = (x, f(x)) m P Q (, ). (., ) (, ) 0 (0, ) Remember that the slope between two points P = (x, y ) and Q = (x, y ) is m P Q = y y x x.
x Q = (x, f(x)) m P Q (, 0) 0+ =. (.,.7).7+. = 0. (, 0) 0+ = 0 (0, 8) 8+ 0 = 9 = (e) Draw the corresponding secants on the graph above and estimate the slope of the tangent to the curve at the point where x =. Draw the tangent on the curve. The slope of the tangent line to the point where x = is 0 since the tangent line is horizontal. - 0 6 - The dashed line on the graph is the tangent line. - (f) Write the equation for the tangent line at the point x =. The equation of the line is y =.
. The height (in feet) of a particle moving back and forth in a straight line is given by h(t) = t t + at time t (in seconds). (a) What is the average velocity of the particle over the time period t = 0 to t =? First, h() = 96 and h(0) =. Then we have Avg. Veloctiy = change in height change in time = h() h(0) 0 = 96 = 9 = 9 feet per second (b) What is the average velocity of the particle over the time period t = to t =? Now, h() = 96 and h() =. So we have Avg. Veloctiy = change in height change in time = h() h() = 96 = 6 = feet per second (c) How can you estimate the instantaneous velocity at time? To estimate the instantaneous velocity at time t = we continue the process we started in parts a and b; i.e. we continue estimating the average velocity over smaller time intervals, such as the interval from [, ], then [.7, ],..., until the average velocity does not change. This limit is our estimate for the instantaneous velocity.. Consider the function g(x) = x + +. (a) Write the function as a piecewise defined function: g(x) = when x < when x { x when x < 0 Remember that x = x when x 0. { (x + ) when x + < 0 Therefore, x + = x + when x + 0. { (x + ) when x < We can rewrite this as x + = x + when x. To get g(x) we just must add to the above piece wise function, giving { x + when x < g(x) = x + when x
(b) Draw a graph of g(x) on the axes below. y x (c) Do you notice any geometric differences between the behavior of the graphs of f(x) (question ) at the point (, ) and g(x) (question ) at the point (, )? The function g(x) has a point (corner) at (, ) whereas the function f(x) is smoother at (, ).. (a) On the graphs below, plot and label the functions f(x) = sin(x) and g(x) = cos(x). (Label the axes.) y x
. 0. y=sin(x) - -. - -0. 0 0.. -0. - y=cos(x) -. (b) What are the domains and ranges of the sine and cosine functions? The domains of both functions are (, ), and the ranges are [, ]. Review Sheet Fill in the following table of values of the given trigonometric functions: θ 0 cos(θ) 6 sin(θ) tan(θ) sec(θ) csc(θ) cot(θ) θ 0 cos(θ) - 0 6 sin(θ) 0 0 tan(θ) 0 0 undefined sec(θ) - undefined csc(θ) undefined undefined cot(θ) undefined undefined 0