Trigonometry Week 1 Test.pdf - Trigonometry Week 1 Test...

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Unformatted text preview: 1/9/2021 Trigonometry Week 1 Test WEEK 1 TEST - SECTIONS 1.1 AND 1.4 ------- Take the Practice Tests, and then the Weekly Test Trigonometry Week 1 Test - Grade Report Score: 88% (88 of 100 pts) Submitted: Jan 9 at 2:28pm 1/26 1/9/2021 Trigonometry Week 1 Test Question: 1 Grade: 1.0 / 4.0 Choose the table and graph of y Select one table and one graph. Choice 2 = −x −3. Selected x y −1 4 0 3 1 4 2 7 x y −1 −4 0 −3 1 −4 2 −7 x y −1 −2 0 −1 1 −4 2 −6 Points No Yes +1 No Yes +1 2/26 1/9/2021 Trigonometry Week 1 Test No No Solution Choose several values for x. Here, −1, 0, 1 and 2 are chosen. Substitute each x -value into the equation to find the corresponding y -value. x y y = 2 −x −3 −1 −4 y = −(−1) 0 −3 y = 2 −(0) −3 = −3 1 −4 y = 2 −(1) −3 = −4 2 −3 = −4 3/26 1/9/2021 2 Trigonometry Week 1 Test −7 y = 2 −(2) −3 = −7 Plot each ordered pair from the table on a coordinate plane and then draw a line through the points. 4/26 1/9/2021 Trigonometry Week 1 Test Question: 2 Grade: 1.0 / 4.0 Find both the x- and y- intercepts and the graph of y = 2 −3x −3x+6. Select the correct set of intercepts. (1, 0), (−2, 0) and (0, 6) (50%) Select the correct graph. (50%) Solution Find the x -and y -intercepts. x -intercept: y 0 = = x − 1 x y -intercept: y = Substitute 0 for y and solve for x. 2 −3x −3x+6 −3(x − 1) (x + 2) = = 0 1 or x + 2 or x = = The equation is quadratic, so solve by factoring. 0 −2 2 −3(0) −3(0)+6 = 6 Substitute 0 for x and solve for y. There are two x − intercepts, (1, 0) and (−2, 0) , and one y − intercept, (0, 6) . Plot the x − and y − intercepts and then sketch a line through the intercepts. 5/26 1/9/2021 Trigonometry Week 1 Test Question: 3 If g(x) g(5) = Grade: 1.0 / 4.0 6x − ∣ 3 + x ∣, find g(5) and g(−6). 22 (50%) = g(−6) = -39 (50%) Solution Evaluate the function for x = 5 g(5) = 30− ∣ 8 ∣ = = g(−6) 6(5) − |3 + (5)| = So, g(5) 6(−6) − |3 + (−6)| = 22 and g(−6) Question: 4 = and x = = −6. 30 − 8) −36− ∣ −3 ∣ = = 22 −36 − 3 = −39 −39. Grade: 1.0 / 4.0 Find the domain of the function. f (x) = 5 x 9 (−∞, ∞) − − 2 + 3x − √ 13 (100%) Solution Since the x -values of a polynomial function can be any real number, the domain of f is all real numbers, (−∞, ∞) . 6/26 1/9/2021 Trigonometry Week 1 Test Question: 5 Grade: 1.0 / 4.0 If g (x) = 3x2 + 5x − 3 and f (x) = −2x2 −18x+7, find all real x -values such that g (x) = f (x) . x = 2/5;-5 (100%) If there are multiple solutions, separate the answers with semicolons (;). If the answer is not an integer, enter it as a fraction. Solution g (x) = f (x) 2 3x + 5x − 3 = 2 −2x −18x+7 Substitute 3x2 2 5x +23x − 10 = 0 Move all the terms to one side and simplify. (5x − 2) (x + 5) = 0 Factor. or x + 5 5x − 2 x = So, If g (x) = 0 2 ∕ 5 = x 2 3x + 5x − 3 = = Set each factor equal to 0. 0 Solve each equation. −5 and f (x) for g (x) and −2x2 −18x+7 for f (x) . + 5x − 3 = 2 −2x −18x+7 then g (x) = f (x) when x = 2 ∕ 5 or x = −5. 7/26 1/9/2021 Trigonometry Week 1 Test Question: 6 Grade: 1.0 / 4.0 True or false? The following is a graph of a function. Your response False Solution The graph fails the vertical line test. Therefore, this is not a function. 8/26 1/9/2021 Trigonometry Week 1 Test Question: 7 Grade: 1.0 / 4.0 Choose the table and graph of y = 2+3x. Select one table and one graph. Choice Selected x y −1 5 0 2 1 −1 2 −4 Points No No x y −1 −5 0 −2 1 5 2 4 No Yes +1 9/26 1/9/2021 Trigonometry Week 1 Test x y −1 −1 0 2 1 5 2 8 Yes +1 No Solution Choose several values for x. Here, −1, x y y = 2+3x −1 −1 y = 2+3 (−1) 0 2 y = 2+3 (0) = 2 1 5 y = 2+3 (1) = 5 2 8 y = 2+3 (2) = 8 = 0, 1 and 2 are chosen. Substitute each x -value into the equation to find the corresponding y -value. −1 10/26 1/9/2021 Trigonometry Week 1 Test Plot each ordered pair from the table on a coordinate plane and then draw a line through the points. Question: 8 Grade: 1.0 / 4.0 Write the equation of a circle with center at (−4, 4) and radius 3 . Your response (x+4) 2 + (y−4) 2 = 9 Solution Substitute r = 3, h = −4, 2 2 (x − (−4)) + (y − (4)) Question: 9 and k = 2 (3) = 4 ⇒ into the equation of a circle with center at(h, k) , 2 2 (x+4) + (y−4) = (x − h) 2 + (y − k) 2 2 = r , and simplify. 9 Grade: 1.0 / 4.0 True or false? The equation x2 − 6 + 6y 2 = 4 is a function. Your response false Solution Solve the equation for y 2 2 x − 6 + 6y = 4 ⇒ 6y 2 = 2 4 − x + 6 ⇒ y 2 1 = 2 (10 − x ) 6 There are some x -values that will have two corresponding y -values, so x2 − 6 + 6y 2 = 4 is not a function. 11/26 1/9/2021 Trigonometry Week 1 Test Question: 10 Grade: 1.0 / 4.0 Determine the distance between the two points (−1, −1) and (5, 7). The distance between the points is 10 (100%) Solution Substitute the coordinates from the ordered pairs (points) into the Distance Formula d Let (x1 , y1 ) d = (−1, −1) and (x2 , y2 ) = (5, 7) − −−−−−−−−−−−−−−−−−−− 2 2 = √(x − x ) + (y − y ) 2 1 2 1 and simplify. . = − −−−−−−−−−−−−−−−−−−−− √(5 − (−1))2 + (7 − (−1))2 Substitute. = − −−−−−−−− √(6)2 + (8)2 Simplify within the parentheses. = − −−−−− √ 36 + 64 Evaluate the powers. = − − − √ 100 Add. = 10 Evaluate the square root. 12/26 1/9/2021 Trigonometry Week 1 Test Question: 11 Grade: 1.0 / 4.0 Find both the x- and y- intercepts and the graph of 4x+2y = 6. Select the correct intercepts. (0, 3 ) and ( 3 2 , 0) (50%) Select the correct graph. (50%) Solution Find the x - and y - intercepts. x -intercept: y -intercept: 4x+2(0) 4(0)+2y = = 6 6 ⇒ ⇒ The x − and y − intercepts are (0, x = y = 3 ) 3 2 3 and ( Substitute 0 for y and solve for x. Substitute 0 for x and solve for y. 3 2 , 0) . The equation is linear, so the graph forms a line. Plot the x − and y − intercepts and then sketch a line through the intercepts. 13/26 1/9/2021 Trigonometry Week 1 Test Question: 12 Write x2 + y 2 Grade: 1.0 / 4.0 + 8x − 6y = 11 in the standard form of a circle and then identify the circle's center and radius. The standard form equation of the circle is Your Answer: Correct Answer: (x+4) (x+4) 2 2 + (y−3) + (y−3) 2 2 = 36 = 36 Your response Correct response Enter the center as an ordered pair, including the comma. Enter the center as an ordered pair, including the comma. The center is at (-4,3) (100%) The center is at (-4,3) Your response Correct response Enter the radius. Enter the radius. The radius is 6 (100%) The radius is 6 Solution Group the x-terms and the y-terms, and move the constant term to the right side. 2 2 (x + 8x) + (y − 6y) = 11 Complete the square twice to write the x -terms as a perfect square trinomial and the y -terms as a perfect square trinomial. 2 2 (x +8x) + (y −6y) 2 2 (x +8x + 16) + (y −6y + 9) = = 11 Complete the square twice. 11 + 16 + 9 2 2 (x+4) + (y−3) The standard form equation is (x+4)2 = + (y−3) Factor each perfect square trinomial and simplify the right side. 36 2 = 36 , so the center is (−4, 3) and the radius is 6 . 14/26 1/9/2021 Trigonometry Week 1 Test Question: 13 Grade: 1.0 / 4.0 Write the equation of a circle where A (1, 6) and B (3, 8) are the endpoints of a diameter. Your response (x−2) 2 + (y−7) 2 = 2 Solution Find the radius r and the coordinates of the center (h, k) , then substitute those values into the standard form equation of a circle, (x − h) 2 + (y − k) 2 2 = r . Use the Distance Formula to find the length of AB, the circle‘ s diameter, and divide the diameter by 2 to find the radius. Then, use the Midpoint Formula to find the midpoint of AB, the circle‘s center. For both formulas, let A (1, 6) be (x1 , y1 ) and B (3, 8) be (x2 , y2 ) . x1 = 1, x2 = 3, y1 = 6, y2 = 8 AB = − −−−−−−−−−−−−−−−−− √(1 − (3))2 + ((6) − 8)2 = √(−2)2 + (−2)2 = −−−− √4 + 4 = – √8 − −−−−−−−−−−− So the radius is center= ( – √8 2 1+(3) Substitute r 2 = , – = √ 2. 6+8 2 – √ 2, h ) = (2, 7) = 2, and k = – 2 (√ 2) 2 2 (x − (2)) + (y − (7)) = 7 ⇒ into the equation of a circle with center at (h, k) , (x − h) 2 + (y − k) 2 = 2 r , and simplify. 2 2 (x−2) + (y−7) = 2 15/26 1/9/2021 Trigonometry Week 1 Test Question: 14 Grade: 1.0 / 4.0 Find the domain of the function. − −−−−−−− f (r) = √ −11r + 55 Your response (−∞, 5] Solution The domain of f is all real numbers except those r -values that make the radicand < 0. So, set the expression in the radicand ≥ 0 and solve for r. −11r + 55 ≥ 0 ⇒ r ≤ 55 11 = 5 Reverse the inequality symbol when dividing by −11. The domain of f is all real numbers less than or equal to 5, (−∞, 5] 16/26 1/9/2021 Trigonometry Week 1 Test Question: 15 Grade: 0.0 / 4.0 Use symmetry and intercepts to sketch the graph of y = ∣ 4x ∣ +2. Your response Solution Test the equation for each type of symmetry. Respect to the x -Axis −y y = = Respect to the y-Axis ∣ 4x ∣ +2 y − ∣ 4x ∣ −2 Not equivalent to y So, the graph of y = = y ∣ 4x ∣ +2. ∣ 4x ∣ +2 = = Respect to the Origin ∣ 4 (−x) ∣ +2 −y ∣ 4 (x) ∣ +2 y Equivalent to y = ∣ 4x ∣ +2. = = ∣ 4 (−x) ∣ +2 − ∣ 4 (x) ∣ −2 Not equivalent to y = ∣ 4x ∣ +2. is symmetric with respect to the y-axis. Find the intercepts. x -intercept: 0 -intercept: y y = = ∣ 4x ∣ +2 ∣ 4(0) ∣ +2 ⇒ = no solution 2 Substitute 0 for y and solve for x . Substitute 0 for x and simplify to find y. So, the graph does not cross the x -axis, and it crosses the y-axis at 2. Find some additional points on the graph, on one side of the line of symmetry. Then, use the fact that for each (x, y) on a graph that is symmetric with respect to the y-axis, (x, y) is also on the graph. Plot the points and sketch the graph. 17/26 1/9/2021 Trigonometry Week 1 Test Question: 16 Grade: 0.0 / 4.0 Find the domain of the function. g (p) = p−5 2 p +3p−18 Your response (−6, 3) Solution The domain of g is all real numbers except those p - values that make the denominator = solve for p. 2 p +3p − 18 = = −6 or So, set the expression in the denominator = 0 and 0 (p + 6) (p − 3) p 0 p = = 0 3 The domain of g is all real numbers except −6 and 3, R ∖ {−6, 3} . 18/26 1/9/2021 Trigonometry Week 1 Test Question: 17 Grade: 1.0 / 4.0 If F (x) = ⎧ 6 ⎪ if x ≤ −10 ⎨ ∣5x − 6∣ if −10 < x < 6 if x ≥ 6 ⎩ ⎪ find F F F 2 5x + 5 and F (12) . (−10) = 6 (50%) (12) = 725 (50%) (−10) Solution Evaluate the function for x = −10 and x = 12 When x = −10, F (x) = 6. So, F (−10) = 6 When x So, F = 12, F (x) (−10) = 6 Question: 18 = and F 2 5x + 5. (12) = So, F 725 (12) = 2 5(12) + 5 = 725 . Grade: 0.0 / 4.0 is the midpoint between R(x, y) and S (4, 11). Find the coordinates of R. Enter the answer as an ordered pair, including the comma. The coordinates of R are (1/2,7) (0%) M (−3, 3) Solution Substitute the x -coordinate from the midpoint and the x -coordinate from S into the equation xM −3 = x+4 −6 = x + 4 Multiply both sides by 2. −10 = x Subtract 4 from both sides. 2 = y+11 x +x 1 2 2 . Substitute. Substitute the y -coordinate from the midpoint and the y -coordinate from S into the equation yM 3 = = y +y 1 2 2 . Substitute. 2 6 = y + 11 Multiply both sides by 2. −5 = y Subtract 11 from both sides. 19/26 1/9/2021 Trigonometry Week 1 Test Question: 19 Grade: 1.0 / 4.0 If f (x) = 3 ⎧ ⎪ if x ≤ −8 ⎨ ∣2x − 3∣ if −8 < x < 7 if x ≥ 7 ⎩ ⎪ 2x + 2 find f (7) and f (−12) . f (7) = 16 (50%) f (−12) = 3 (50%) Solution Evaluate the function for x = 7 and x = −12. When x = 7, f (x) = 2x + 2. So, f (7) = 2(7) When x = −12, f (x) = 3. So, f (−12) = 3 So, f (7) = 16 and f (−12) = 3. Question: 20 + 2 = 16 Grade: 1.0 / 4.0 True or false? The equation x − 3 + 3y = 2 is a function. Your response true Solution Solve the equation for y 1 x − 3 + 3y = 2 ⇒ 3y = 2 − x + 3 ⇒ y = (5 − x ) 3 Each x -value has one corresponding y -value, so x − 3 + 3y = 2 is a function. 20/26 1/9/2021 Trigonometry Week 1 Test Question: 21 Grade: 1.0 / 4.0 Identify the type of symmetry, if any, for the graph of y = 5 6x . Your response Symmetric with respect to the origin Solution Test for symmetry with respect to the x -axis. −y y 5 6x = 5 −6x = Substitute −y for y . Divide both sides by −1 to solve the equation for y . The equations are not equivalent, so the graph is not symmetric with respect to the x -axis. Test for symmetry with respect to the y -axis. y 5 6(−x) = y 5 −6x = Substitute −x for x . Simplify. The equations are not equivalent, so the graph is not symmetric with respect to the y -axis. Test for symmetry with respect to the origin. −y = 5 6(−x) Substitute −x for x and −y for y . −y = 5 −6x Simplify. y = 5 6x Divide both sides by −1 to solve the equation for y . The equations are equivalent, so the graph of y = 5 6x is symmetric with respect to the origin. 21/26 1/9/2021 Trigonometry Week 1 Test Question: 22 Grade: 1.0 / 4.0 If w(x) = 6x2 − 2x, find w(3 + t). Enter the expression in simplest form. w(3 + t) 2 6t + 34t + 48 = (100%) Solution Evaluate the function for x w(3 + t) = 3 + t. = 6(3 + t) = 6(3 + t) (3 + t) − 2(3 + t) = 6(9 + 6t + t = 54 + 36t + 6t = 6t So, w(3 + t) 2 2 − 2(3 + t) 2 ) − 2(3 + t) FOIL 2 Distribute. − 6 − 2t Combine the like terms +34t+48 = 6t 2 +34t+48 Question: 23 Expand the power. . Grade: 1.0 / 4.0 If g (x) = 4x2 +3x − 10, find all real x -values such that g (x) = 0. If there are multiple solutions, separate the answers with semicolons (;). If an answer is not an integer, enter it as a fraction. x = 5/4;-2 (100%) Solution Substitute 0 for g (x) and solve for x. 4x − 5 x = = 0 = 2 4x +3x − 10 Substitute. 0 = (4x − 5) (x + 2) Factor. 0 or x + 2 Set each factor equal to 0. 5 ∕ 4 So, if g (x) = x = = 0 −2 2 4x +3x − 10, Solve each equation. then g (x) = 0 when x = 5 ∕ 4 or x = −2. 22/26 1/9/2021 Trigonometry Week 1 Test Question: 24 Grade: 1.0 / 4.0 Identify the center, radius, and graph of (x − 3)2 2 + (y + 3) = 25. Your response Correct response Enter the center as an ordered pair, including the comma. Enter the center as an ordered pair, including the comma. The center is at (3,-3) (100%). The center is at (3,-3). Your response Correct response Enter the radius exactly. Do not use a decimal approximation. Enter the radius exactly. Do not use a decimal approximation. The radius is 5 (100%). Select the graph of (x − 3)2 The radius is 5. + (y + 3) 2 = 25. Your Answer: Correct Answer: Solution From the equation, h = 3, k = −3, and r2 = 25, so r = 5. Therefore, the circle’s center is at (3, −3) and the radius is 5. Plot the points 5 units above, below, to the left and to the right of (3, −3) , and then sketch the circle through those points. 5 units above (3, −3) ⇒ (3, 2) 5 units below (3, −3) units left of (3, −3) 5 units right of (3, −3) 5 ⇒ (3, −8) ⇒ (−2, −3) ⇒ (8, −3) 23/26 1/9/2021 Trigonometry Week 1 Test 24/26 1/9/2021 Trigonometry Week 1 Test Question: 25 Grade: 1.0 / 4.0 If a person drives 370 miles at an average of 40 miles per hour, then their distance d from the destination (in miles) is a function of the number of hours h driven. Express this function as an equation, table, and graph. Your response The function's equation is d (h) Correct response = 370 − 40h The function's equation is d (h) (100%) Your response The function can be represented by a table. = 370 − 40h Correct response The function can be represented by a table. h d (h) h d (h) 0 370 0 370 2 290 (50%) 2 290 5 170 (50%) 5 170 Select the correct graph for d (h) Your Answer: Correct Answer: 25/26 1/9/2021 Trigonometry Week 1 Test Solution The distance traveled after h hours is 40h. So, the distance d from the destination, which is 370 miles at h Choose appropriate values for h and substitute into the equation to find the ordered pairs for the table. d (0) = 370 − 40 (0) = 370 d (2) = 370 − 40 (2) = 290 d (5) = 370 − 40 (5) = 170 = 0, is d (h) = 370 − 40h. So, a corresponding table is h d (h) 0 370 2 290 5 170 The equation d (h) = 370 − 40h describes a line with slope −40 and y -intercept 370. 26/26 ...
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