The Heaviside step function, H(x), is defined as follows: H(x)=1 for x 0 H(x)=0 for x <

Describe y = f(3x 2) as the result of first applying a stretch or compression and then applying a shift to f.

Describe y = 5(g(x) 8) as the result of first applying a stretch or compression and then applying a shift to g.

(6, 9). For each of the following functions, find a point on its graph: (a) g(2x) 5 (b) 3g(x)+1 (c) (g( 1 3 (x + 4)) 8) (d) 1 2 g(5x 15) 8

Table 6.24 gives values of x and f(x). Supply the values of each function shown. In some cases, there may not be enough information to fill in a box.

Table 6.25 gives values of function v. Create a table giving five values of the function w(t) = 40 2v(0.5t).

Figure 6.61 shows the graph of fx) = |x|. Graph each transformation of f(x): (a) g(x) = f(x)+1 (b) h(x) = f(x + 1) (c) j(x) = f(2x + 1) 3 (d) k(x) = 1 2 f(2x4)+1 (e) m(x) = 1 2 f(4x+12)3

Figure 6.62 shows the graph of fx)=2x. Graph each transformation of f(x): (a) f(3x + 6) (b) f(1 2 x 1) (c) 0.4f(x + 1) 2 (d) f( 1 2 x + 4) + 1

Using Figure 6.63, graph the following functions. (a) y = f(x)+2 (b) y = 2f(x) (c) y = f(x 3) (d) y = 1 2 f(x + 1) 3

Using Figure 6.64, sketch y = 2f(0.5x) + 20.

In Exercises 1011, use Figure 6.65 to graph the function.y = 2f(0.5x)

In Exercises 1011, use Figure 6.65 to graph the function.h(x) = f(2x 1)

Using Figure 6.66, graph each of the following functions on separate sets of axes, together with the graph of the original function. Label intercepts and asymptotes. (a) y = 3f(x) (b) y = f(x 1) (c) y = f(x) 1 (d) y = 2f(x) (e) y = 1 2 f(x + 2) 1 (f) y = f(x)

The graph of v has a horizontal asymptote at y = 4 and a vertical asymptote at t = 5. Give the asymptotes of w(t)=3 0.5v(2t).

The function p(t) has domain 1 t 12 and range 40 < p 160. What is the new domain and range of (a) p( 1 5 t + 4)? (b) 1 10 p(2t) + 50?

In Exercises 1516, sketch and label graphs of the functions using Figure 6.67. Briefly describe the relationship between your graph and the graph of f.h(x) = f(2x + 6).

In Exercises 1516, sketch and label graphs of the functions using Figure 6.67. Briefly describe the relationship between your graph and the graph of f.q(x) = f (2(x + 6)).

Figure 6.68 shows the graph of a function f. (a) Graph the function that is obtained by first shifting the graph of f vertically upward by 3 units and then stretching it vertically by a factor of 2. (b) Graph the function that is obtained by first stretching the graph of f vertically by a factor of 2 and then shifting up vertically by 3 units. (c) Compare your answers in parts (a) and (b).

Is y = f(2x 6) the function obtained from f by a horizontal compression by a factor of 1/2 followed by a horizontal shift of 6 units to the right? Explain why or why not.

Is the function y = 5(g(x)2) obtained from g by a vertical stretch by a factor of 5 followed by a vertical shift of 2 units downward? If not, what vertical shift is needed?

The Heaviside step function, H(x), is defined as follows: H(x)=1 for x 0 H(x)=0 for x < 0. Graph the Heaviside function, then sketch graphs of the following transformations of H: (a) y = H(x) 2 (b) y = H(x + 2) (c) y = 3H(x)+4

The graph of g is found by shifting the graph of f to the left by 4 units, then reflecting it vertically across the xaxis, then shifting it up by 2 units, then finally stretching it vertically by a factor of 3. Find a formula for g in terms of f.

The graph of g is the graph of f shifted right by 3 units, then reflected vertically across the x-axis, then shifted down by 2 units, then stretched vertically by a factor of 2. Find a formula for g in terms of f.

The graph of g is found by shifting the graph of f up by 2 units and left by 3 units. The graph of h is found by reflecting the graph of g horizontally, then reflecting it vertically and stretching it vertically by a factor of 2. Find a formula for h in terms of f.

In Problems 2425, the points (12, 20), (0, 6), (36, 2) lie on the graph of f.List three points on the graph of g(x) = 10 2f(3x).

In Problems 2425, the points (12, 20), (0, 6), (36, 2) lie on the graph of f.The graph of f is found by shifting the graph of h to the left by 3 units, then stretching it vertically by a factor of 2, then shifting it up by 6 units. Find three points on the graph of h.

Use shifts, reflections, and vertical stretches to graph each parabola without a calculator. Then write each equation in standard form: (a) y = (x + 3)2 4 (b) y = 2(x + 1)2 + 3

Sketch y = 160 4f(x/10) if f(x) = 20 2x.

Let h(t) = t 2. Parts (a)(c) investigate the effects of changing the order of horizontal transformations. (a) Give a formula in vertex form for the quadratic function obtained by first shifting the function h(t) to the right 6 units and then compressing horizontally by a factor of 1/2. (b) Give a formula in vertex form for the quadratic function obtained by first compressing the function h(t) horizontally by a factor of 1/2 and then shifting to the right 6 units. (c) If you first horizontally compress h(t) by a factor of 1/2, by how much and in what direction would you next need to shift horizontally in order to get the same result as in part (a)?

Which (if any) of the following transformations of f is not the same as the others? (a) Shift f up by 3, then stretch it vertically by a factor of 2, then reflect it vertically. (b) Reflect f vertically, then shift it down by 3, then stretch it vertically by a factor of 2. (c) Stretch f vertically by a factor of 2, then shift it up by 6, then reflect it vertically. (d) Reflect f vertically, then stretch it vertically by a factor of 2, then shift it down by 6.

Let f(x) = ex and g(x)=5ex2. If g(x) = kf(x), find k.

The log function has the property that the graph resulting from a horizontal stretch can also be obtained by a vertical shift. (a) Graph f(x) = log x and g(x) = log(10x) and determine the vertical shift. (b) Explain how you could have predicted the answer to part (a) from the properties of logarithms. (c) If h(x) = log(ax), what is the vertical shift k making h(x) = log(x) + k?

Applying a horizontal stretch by a factor of k (where k is a constant such that k > 1) to f(x) = ln x is equivalent to applying what shift to f? Give both the amount and the direction of the shift.

Shifting g(x) = ex to the right k units (where k is a constant such that k > 0) is equivalent to applying what stretch or compression to g? Be sure you describe the amount and direction of the stretch or compression needed.

(a) The point (2, 5) lies on the graph of y = r(x). What are the coordinates of the transformed point on the graph of y = 3(r(x) + 2)? (b) Express the function y = 3(r(x)+2) as the result of first applying a vertical stretch or compression and then applying a vertical shift to the function r. What is the stretch/compression? What is the shift?

(a) Describe the graph of y = 2f(x)+8 as the result of applying a stretch or compression to the graph of f, then applying a shift. (b) Now describe this graph as the result of applying a shift to the graph of f, then applying a stretch or compression.

(a) Describe the graph of y = f  1 3 x + 4 as the result of applying a stretch or compression to f, then applying a shift. (b) Describe this graph as the result of applying a shift to f, then applying a stretch or compression.

The function d(t) graphed in Figure 6.69 gives the winter temperature in F at a high school, t hours after midnight. (a) Describe in words the heating schedule for this building during the winter months. (b) Graph c(t) = 142 d(t). (c) Explain why c might describe the cooling schedule for summer months.

Let D(t) denote the depth, in meters, of the water at a fixed location in a bay t hours from noon. Find a formula for each function below by applying transformation(s) to D(t). (a) f(t), the depth of the tide in kilometers.(b) g(x), the depth of the tide in kilometers x minutes from noon. (c) j(x), the depth of the tide in kilometers x minutes from 2 pm.

Let f be defined by the graph in Figure 6.70. Find formulas (in terms of f) for the following transformations of f and sketch a graph of each. Show that these transformations lead to different outcomes. (a) First shift the graph of f upward by 3 units, then reflect it across the x-axis. (b) First reflect the graph of f across the x-axis. Then, shift it upward by 3 units.

(a) Graph h(x) = 2x2 8x 8. (b) Compare the graphs of h(x) and f(x) = x2. How are these two graphs related? Be specific.

For Problems 4142, find a formula for the family of functions obtained from f(x) by: (a) Stretching vertically (b) Stretching vertically and shifting horizontally. Compare the graph of a typical member of the family and the graph of f(x). Include intercepts and asymptotes in your discussion. Are the two families the same?f(x) = x

For Problems 4142, find a formula for the family of functions obtained from f(x) by: (a) Stretching vertically (b) Stretching vertically and shifting horizontally. Compare the graph of a typical member of the family and the graph of f(x). Include intercepts and asymptotes in your discussion. Are the two families the same?f(x)=2x

In Problems 4345, you are given that g(x) = r f (sx) + j, where r, s, j are constants.For the linear function f(x) = b + mx, is g also linear? If so, write g in the form g(x) = B + Mx and state the values of B and M.

In Problems 4345, you are given that g(x) = r f (sx) + j, where r, s, j are constants.For the exponential function f(x) = aekx, is g also exponential? If so, write g in the form g(x) = AeKx and state the values of A and K.

In Problems 4345, you are given that g(x) = r f (sx) + j, where r, s, j are constants.For the quadratic function f(x) = a(x h) 2 + k, is g also quadratic? If so, write g in the form g(x) = A(x H) 2 + K and state the values of A, H, and K.