3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.

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1 PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III Vertical shift up by 4 units What is the equation of the transformed graph? Solve for x: x + x 8 x x < 0 Use absolute value notation to write an inequality that represents the statement: x is within units of on the real line 4 Solve the inequality 4x 5 Express your answer in interval notation 5 Use the definition of absolute value to write fx) = x 1) as a piecewise-defined function which does not contain absolute values Consider the function fx) = 4x 8x + 1 If possible, solve the following problems algebraically Check your results graphically a) Complete the square to write this polynomial in standard form b) Draw a complete graph, using the standard form found in a) c) State which transformations shifts, stretches, etc) that must be applied to the graph of y = gx) = x to obtain the graph of y = fx) 7 Let fx) = Determine: x + 1 if x < 0 x 1 if 0 x and gx) = x if x 1 x 1 if 1 < x a) f g)x) ) f b) x) g 8 Let fx) = x and gx) = x + x a) Determine the domain and range of both f and g 1

2 b) Determine fgx)), including its domain c) Determine gfx)), including its domain 9 Let P x) = x 5 x 4 9x + 7 a) Find all rational roots of P x) b) Find all real linear factors of P x) c) Find any irreducible quadratic factors of P x) d) Solve P x) 0 10 Let rx) = x + x x x 4x 5 a) What is the domain of this function? b) Find all zeros of this function x-intercepts) c) Find all vertical asymptotes of this function d) Determine where r is positive, and where it is negative e) Determine any horizontal and slant asymptotes f) Draw a complete graph of y = rx) g) Solve rx) < 0 11 Solve for x : 48 = x x x Use the unit circle definition of the standard trigonometric functions to explain the following: a) sin t + cos t = 1 b) sinπ t) = sin t c) tant + π) = tan t d) The range of ft) = sin t is [ 1, 1] k + 1)π e) The domain of gt) = tan t is t for integer k, ie all odd multiples of π f) ht) = cos t decreases for t 0, π) Where does it increase? g) sec t tan t = 1 1 For the function fx) = cos x π ), determine the following: a) Domainf) b) Rangef) c) A complete period for f d) The amplitude of f

3 e) The phase shift of f f) The zeros of f in the interval [ π, π] g) The graph of f, showing two complete periods 14 Solve the following equations exactly for x [ π, π]: a) sin x 1 b) sec x = 4 c) tan x = 1 d) sin x) 1 = 0 15 Prove that sinx + h) sin x h ) ) cos h 1 sin h = sin x + cos x, for all h 0 h h 1 Find δ such that sin x cos x = sinx + δ) for all x 17 Find A such that sin x + cos x = A sin x + cos 1 ), for all x 18 Find A and δ such that sin x + cos x = A sinx + δ) for all x 19 Solve the following equations exactly: a) sin t + tan t + cos t =, 0 t π b) sec θ 1 ) cos θ ) = 0, 0 θ π c) cosx) + cos x + 1 = 0 d) 4 sin x cos x =, π < x < π e) sinx) cosx) = 0 f) sec t =, 0 t π g) sin t cos t = 1 Hint: Write this in the form A sint + α) = 1 ) h) tan t tan t = 1, π t π

4 Solutions 1 y = x ) + 4 4, 0) 1, ) x 4, 1 ] [, ) 5 fx) = x + 5 if x < 5 x 5 if 5 x a) fx) = 4x 1) b) Your graph should be a parabola with vertex at 1, ), which opens up The graph crosses the x-axis at x = 1 ± c) Vertical stretch by a factor of 4; vertical shift units down; horizontal shift 1 unit to the right x + 1 if x < 0 7 a) f g)x) = x 1 if 0 x 1 0 if 1 < x ) f x+1 if x < 0 x 1 x b) = if 0 x 1 x g 1 if 1 < x 8 a) Note: gx) = x + 1 ), so Domainf) = [0, ), Rangef) = [, ), Domaing) =, ), Rangeg) = [0, ) b) fgx)) = gx) = x + 1, < x < x ) 5 c) gfx)) = = x 5 x + 5 4, x 0 9 a) x = b) x ), x ), x + ) c) x + ) d) [, ] [, ) 10 a) x 1, 5 b) x = 1 ± 1 c) Vertical asymptote x = 5 Note that x = 1 is a removeable discontinuity 4

5 ) 1 1 d) Positive on, 1 1, 1 + ) 1 5, ); Negative on, 1 ) 1 ) 1 + 1, 5 e) Slant asymptote y = x + or y = x) f) Be sure to include at least 10 < x < 15, noting the vertical asymptote, and the hole in the graph at x = 1 g), 1 ) ) , 5 11 x = 1, ± 1 a) The point cos t, sin t) is on the unit circle x + y = 1 b) π t and t are complementary angles, that is, they have the same reference angle c) The straight line with angle t cuts two places on the unit circle, giving the same value for tant + π) and tan t d) sin t is just the y-coordinate on the unit circle e) At odd multiples of π, the line is vertical, and so the slope tan t) is undefined f) cos t is the x-coordinate, which decreases from 1 to -1 as t goes from 0 to π As t goes from π to π, this coordinate increases from -1 back to 1 g) Divide the Pythagorean identity in part a) by cos t, and use the definitions of tan t and sec t 1 a) Domainf) =, ) b) Rangef) = [ 4, ] c) [0, π] d) 5 14 a) e) π 4 f) Two solutions are α = π cos 1 1 ) = π tan 1 ), and β = 5π 4 1 cos 1 1 ) = π tan 1 ), which satisfy π < α, β < π, so the full set is α π, β π, α π, β π, α, β, α + π, β + π b) c) [ 5π ], π 11π [ 7π, 11π ], 7π, 5π, π, π, 5π, 7π, 11π } 5π 4, π 4, π 4, 7π 4 5 }

6 d) ± π } 8, ±π 8, ±5π 8, ±7π 8, ±9π 8, ±11π 8, ±1π 8, ±15π 8 15 Use the formula : sinθ + φ) = sin θ cos φ + cos θ sin φ 1 δ = π cos 1 = tan 1 1 ) π 4 17 A = 18 A =, δ = sin 1 = tan 1 ) 19 a) b) π 4, 5π } 4 π, 11π } c) π + πn, π + πn, π + πn, 4π d) 11π, 5π, 5π, π, π, π, 7π, 4π e) x = π 1 + πn, n an integer + πn, n an integer f) π 4, 5π 4 g) π ) sin 1 + πn = π ) tan 1 + πn, and 17 ) 5π 1 4 sin 1 + πn = 5π ) 1 4 tan 1 + πn, n an integer 17 ) ) h) Let α = tan 1, β = tan 1 The solutions are α π, α π, α, α + π, β π, β, β + π, β + π }

7 Completing the Square A very useful tool in your algebra arsenal is completing the square, that is, to re-write a quadratic expression as follows: ax + bx + c = ax h) + k We start by dividing by a 0: ax + bx + c = a [x + ba x ] + c, then use the identity α + β) β = α + αβ to write a [ x + b a x] + c = [ x ) ) ] = a + b a b + c a = a ) x + b a b + c 4a = a ) x + b a b 4ac 4a example: x x + 5 = x ) x + 5 [ = x ) ] ) = x 4) = x ) example: 1 x + 4x + = 1 x + 8x ) + = 1 [ x + 4) 4 ] + = 1 x + 4) 8 + = 1 x + 4) 5 example: 7 5x x = x 5x + 7 = [x + 5 ] x + 7 [ = x + 5 ) 5 ] ) = x + 5 )

8 = x + 5 ) example: Find the centre and radius of the circle with equation x x + y + 4y = 1 x x + y + 4y = x ) + y + ) = 1, from which x ) + y + ) = 5, giving a centre of,-), radius 5 Quadratic Equations As you proceed through your mathematics coursework, from 1 Analytical Geometry - Calculus I, through Calculus II, Calculus III and 5 Introduction to Ordinary Differential Equations, you will be surprised to find out how often we need to solve quadratic equations, of the form ax + bx + c = 0, a 0 To derive the quadratic equation, we complete the square as previously, obtaining 0 = ax + bx + c = a x + b ) b 4ac, a 4a from which Taking square roots, a x + b ) = b 4ac a 4a x = b a ± b 4ac = b ± b 4ac 4a a Sinusoidals Another useful tool is to re-write sums of sines and cosines as single values, as in a sin ωt + b cos ωt = A sinωt + δ), using the identity sinθ + φ) = sin θ cos φ + cos θ sin φ), where A = a + b and cos δ = a A, sin δ = b A 8