By Hugo. Rossi

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Example text

What is the slope of the line y = −4x − 9? 4. Suppose y = 3x + 2. What is 2. Are the lines y = 2x + 1 and y = −2x − 4 perpendicular? 5. What is the minimum of f (x) = (x + 3)2 − 4? 3. When is the line ax + by = c parallel to the y-axis? To the x-axis? 6. What is the result of completing the square for f (x) = x 2 + 1? y if x increases by 3? 18 CHAPTER 1 PRECALCULUS REVIEW Exercises In Exercises 1–4, find the slope, the y-intercept, and the x-intercept of the line with the given equation. 1. y = 3x + 12 2.

19 51. If objects of weights x and w1 are suspended from the balance in Figure 14(A), the cross-beam is horizontal if bx = aw1 . If the lengths a and b are known, we may use this equation to determine an unknown weight x by selecting w1 such that the cross-beam is horizontal. If a and b are not known precisely, we might proceed as follows. First balance x by w1 on the left as in (A). Then switch places and balance x by w2 on the right as in (B). The average x¯ = 12 (w1 + w2 ) gives an estimate for x.

Because the sine function is periodic with period 2π , θ1 θ1 sin θ2 = − sin θ1 ⇔ θ2 = −θ1 + 2π k or θ2 = θ1 + π + 2π k where k is an integer. Taking θ2 = 4x and θ1 = 2x, we see that θ2 = θ1 + π θ2 = −θ1 sin 4x = − sin 2x −sin θ1 θ2 = −θ1 or θ2 = θ1 + π . 5π 3 2π 3 π 0 π 3 −1 4π 3 π 2 3π 2 2π x 3 1 π 2 π −1 2π −1 (A) y = cos x FIGURE 15 5π 3 x= and π , 2 3π 2 over [0, 2π ]. y = cos 2x • Shift to the left π/2 units: y = cos 2 x + Shift left π/2 units π 2 Expand vertically by a factor of 3 y 3 1 π 2π −3 −3 4π , 3 Compress horizontally by a factor of 2: 3 1 π, • y x 2π , 3 Solution The graph is obtained by scaling and shifting the graph of y = cos x in three steps (Figure 15): sin 4x + sin 2x = 0.

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