帮我解析以下题目，特别是应用详细解释清楚---**Question 15** Determine the angle measure of $\frac{\pi}{3}$ radians as degrees. (K:1) Marked out of 1.00 a. 60° b. 90° c. 75° d. 150° **Question 16** Determine the primary trigonometric ratios for the angle $\frac{\pi}{4}$ radians. (C:1) Marked out of 1.00 a. $\sin\frac{\pi}{4} = \frac{1}{2}$, $\cos\frac{\pi}{4} = \frac{\sqrt{3}}{2}$, $\tan\frac{\pi}{4} = 1$ b. $\sin\frac{\pi}{4} = \frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{4} = \frac{1}{2}$, $\tan\frac{\pi}{4} = \sqrt{3}$ c. $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\tan\frac{\pi}{4} = 1$ d. $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\tan\frac{\pi}{4} = 0$ **Question 17** Determine the exact value of $\sin \pi/6$. (I:1) Marked out of 1.00 a. $\frac{\sqrt{3}}{2}$ b. $\frac{1}{2}$ c. $1$ d. $\frac{\sqrt{2}}{2}$ **Question 18** Determine the range of the function $y = 2\sin(\frac{x}{3}) - 1$ (A:1) Marked out of 1.00 a. [-2,2] b. [-1,3] c. [-2,1] d. [-3,1] **Question 19** Determine which graph represents the function $f(x) = 2\cos(\frac{x}{2}) + 1$. (A:1) Marked out of 1.00 Chart Description: The image contains four graphs, labeled a, b, c, and d. All graphs are on a grid with labeled horizontal (x) and vertical (y) axes. The x-axis appears to represent angle or phase, and the y-axis represents the function value. Graph a: - Type: Line chart (representing a trigonometric function). - X-axis: Labeled from 0 to 15, with tick marks at intervals of 1. Labels appear at 0, 5, 10, 15. - Y-axis: Labeled from -5 to 5, with tick marks at intervals of 1. Labels appear at -5, 0, 5. - The curve starts at y=3 at x=0, decreases to y=1 around x=1.5, reaches a minimum of y=-1 around x=3, increases to y=1 around x=4.5, reaches a maximum of y=3 around x=6, decreases to y=1 around x=7.5, reaches a minimum of y=-1 around x=9, increases to y=1 around x=10.5, reaches a maximum of y=3 around x=12, decreases to y=1 around x=13.5, reaches a minimum of y=-1 around x=15. Graph b: - Type: Line chart (representing a trigonometric function). - X-axis: Labeled from 0 to 15, with tick marks at intervals of 1. Labels appear at 0, 5, 10, 15. - Y-axis: Labeled from -5 to 5, with tick marks at intervals of 1. Labels appear at -5, 0, 5. - The curve starts at y=3 at x=0, decreases, crosses the x-axis, reaches a minimum of y=-1 around x=4.5, increases, crosses the x-axis, reaches a maximum of y=3 around x=10.5, decreases. Graph c: - Type: Line chart (representing a trigonometric function). - X-axis: Labeled from 0 to 15, with tick marks at intervals of 1. Labels appear at 0, 5, 10, 15. - Y-axis: Labeled from -5 to 5, with tick marks at intervals of 1. Labels appear at -5, 0, 5. - The curve starts at y=1 at x=0, increases to a maximum of y=3 around x=1.5, decreases to y=1 around x=3, reaches a minimum of y=-1 around x=4.5, increases to y=1 around x=6, reaches a maximum of y=3 around x=7.5, decreases to y=1 around x=9, reaches a minimum of y=-1 around x=10.5, increases to y=1 around x=12, reaches a maximum of y=3 around x=13.5, decreases to y=1 around x=15. Graph d: - Type: Line chart (representing a trigonometric function). - X-axis: Labeled from 0 to 15, with tick marks at intervals of 1. Labels appear at 0, 5, 10, 15. - Y-axis: Labeled from -5 to 5, with tick marks at intervals of 1. Labels appear at -5, 0, 5. - The curve starts at y=1 at x=0, decreases to a minimum of y=-1 around x=1.5, increases to y=1 around x=3, reaches a maximum of y=3 around x=4.5, decreases to y=1 around x=6, reaches a minimum of y=-1 around x=7.5, increases to y=1 around x=9, reaches a maximum of y=3 around x=10.5, decreases to y=1 around x=12, reaches a minimum of y=-1 around x=13.5, increases to y=1 around x=15. **Question 20** Given the parameters below, determine the correct equation of the trigonometric function. The function has an amplitude of 3, a period of $2\pi$, and a phase shift of $\frac{\pi}{4}$ to the right. The function is expressed in radians. (I:1) Marked out of 1.00 a. $f(x) = 3\sin(x+\frac{\pi}{4})$ b. $f(x) = 3\cos(x-\frac{\pi}{4})$ c. $f(x) = \sin(3x-\frac{\pi}{4})$ d. $f(x) = \cos(x-\frac{\pi}{4}) + 3$ **Question 21** A Ferris wheel has a radius of 20 m, makes a full rotation in one minute and the axle stands 25 m above the ground. Which equation models the height of a chair on the Ferris wheel that starts at the top, where x is in seconds? (K:1) Marked out of 1.00 a. $y = 10\cos(\frac{60}{2\pi}x) + 25$ b. $y = 20\cos(\frac{\pi}{30}x - \frac{\pi}{2}) + 25$ c. $y = 20\sin(60x + \frac{\pi}{2}) + 25$ d. $y = 20\sin(\frac{\pi}{30}x + \frac{\pi}{2}) + 25$ **Question 22** Identify the period of the function $f(x) = \sin(3x)$ (K:1) Marked out of 1.00 a. $2\pi$ b. $\pi/3$ c. $6\pi$ d. $2\pi/3$ **Question 23** Determine the solutions to the equation $\sin^2x - \sin x - 6 = 0$ within the interval $0 \leq x \leq 2\pi$. (I:1) Marked out of 1.00 a. The solutions are $\frac{\pi}{3}$ and $\frac{4\pi}{3}$. b. The solutions are $\frac{\pi}{6}$ and $\frac{5\pi}{6}$. c. There are no solutions in the interval $0 \leq x \leq 2\pi$. d. The solutions are $\frac{\pi}{4}$ and $\frac{3\pi}{4}$. **Question 24** Complete Marked out of 1.00 Which of the following matches the correct function types with the graphs? Chart Description: Type: Three 2D graphs plotted on coordinate axes. Graph A - f(x): Plotted on axes with X from approx -2 to 3 and Y from approx -30 to 20. The graph is a curve resembling a cubic polynomial. It decreases from the left, has a local minimum, increases to a local maximum, and then decreases to the right. It crosses the Y-axis around y = -15. Graph B - g(x): Plotted on axes with X from approx -4 to 6 and Y from approx -2 to 2. The graph is a periodic wave resembling a sinusoidal function. It passes through the origin (0,0), reaches a peak at approx (1, 1), crosses the x-axis at approx (2,0), reaches a trough at approx (3,-1), crosses the x-axis at approx (4,0), and reaches a peak at approx (5,1). The amplitude is 1, and the period is approx 4. Graph C - h(x): Plotted on axes with X from approx -2 to 2 and Y from approx 0 to 20. The graph is an increasing curve resembling an exponential growth function. It increases rapidly as x increases. It passes through the Y-axis at approx (0, 1). Options: a. Graph A is an odd degree function, Graph B shows a sinusoidal function with an amplitude of 1 and a period of 4π/3, and Graph C represents exponential growth with a base of 2. b. Graph A depicts only increasing intervals, Graph B depicts sinusoidal oscillation with a period of 4π and an amplitude of 1, and Graph C depicts a polynomial function with a linear rate of change. c. Graph A is a degree 3 function, Graph B is an odd cosine function, and Graph C grows exponentially. d. Graph A has a rate of change that is quadratic, Graph B has a sinusoidal rate of change with a period of π/2, and Graph C has a constant rate of change. **Question 25** Complete Marked out of 1.00 (K:1) The point (2,8) is on the graph y = x³. Find the corresponding coordinates of this point on the graph y = 4[½(x + 2)]³ - 5. Options: a. (-4/3,27) b. (8,27) c. (4,37) d. (4,27) **Question 26** Complete Marked out of 1.00 (C:1) Consider the family of polynomial functions with the zeros -2, 1 and 3. Determine which of the following polynomial functions passes through the point (2, -12). Options: a. f(x) = ½(x + 2)(x - 1)(x - 3) b. f(x) = (x - 2)(x + 1)(x + 3) c. f(x) = (x + 2)(x - 1)(x - 3) d. f(x) = 3(x + 2)(x - 1)(x - 3) **Question 27** Complete Marked out of 1.00 (C:1) The graph of which of the following rational functions has a hole? Options: a. y = (2x - 3) / (x - 3) b. y = (2x - 6) / (x - 6) c. y = (2x - 6) / (x - 3) d. y = (2x - 6) / (2x) **Question 28** Complete Marked out of 1.00 (A:1) Identify which of the following functions has a horizontal asymptote at y=0. Options: a. k(x) = ½x² + 3x - 5 b. g(x) = 5 / (x + 1) c. h(x) = log₅x **Question 29** Complete Marked out of 1.00 (A:1) Which equation is a quartic function with zeros at -4, -1, 2, 3? Options: a. y = (x + 4)²(x + 1)(x - 2)(x - 3) b. y = (x - 4)(x - 1)(x + 2)(x + 3) c. y = (x + 4)(x + 1)(x - 2)(x - 3)² d. y = (x - 2)(x - 3)(x + 4)(x + 1) **Question 30** Complete Marked out of 1.00 (I:1) Identify the vertical and horizontal asymptotes of f(x) = (x - 4) / (2x + 1). Options: a. vertical: x = 4, horizontal: y = ½ b. vertical: x = ½, horizontal: y = ½ c. vertical: x = -½, horizontal: y = ½ d. vertical: x = -½, horizontal: y = -½ **Question 31** Complete Marked out of 1.00 (K:1) In which of the following functions is x-3 a factor? Options: a. f(x) = x³ - 10x² + 29x - 24 b. h(x) = 3x³ - 6x² + 3x + 9 c. j(x) = x³ - 9x² + 15x - 12 **Question 32** Complete Marked out of 1.00 (C:1) Identify the connection between the vertical asymptotes of a rational equation and the x-intercepts of the graph of the corresponding reciprocal function. Options: a. The real roots correspond to the horizontal asymptotes of the graph. b. The real roots correspond to the vertical asymptotes of the graph. c. The real roots are the same as the y-intercepts of the graph. d. The real roots are the same as the x-intercepts of the graph. **Question 33** Complete Marked out of 1.00 (C:1) Given the polynomial equation p(x) = x² - 4x + 3 and inequality p(x) ≤ 0, determine which of the following statements correctly describes the relationship between the solutions to the equation and the inequality. Options: a. The solutions to p(x) = 0 are the intervals where the polynomial is zero, and the solutions to p(x) ≤ 0 are the points where the polynomial is less than or equal to zero. b. The solutions to p(x) = 0 and p(x) ≤ 0 are identical, representing the same set of x-values. c. The solutions to p(x) = 0 are the points where the polynomial is equal to zero, and the solutions to p(x) ≤ 0 are the intervals where the polynomial is less than or equal to zero. d. The solutions to p(x) = 0 are the intervals where the polynomial is less than or equal to zero, and the solutions to p(x) ≤ 0 are the points where the polynomial is zero. Question 34 Complete Marked out of 1.00 Evaluate the polynomial inequality $x^2 - 5x + 6 > 0$. (A:1) a. $x < 2$ or $x > 3$ b. $2 \le x \le 3$ c. $2 < x < 3$ d. $x \le 2$ or $x \ge 3$ Question 35 Complete Marked out of 1.00 The formula for the average rate of change of a function f(x) over an interval [a, b] is identified as: (K:1) a. $\frac{f(b) + f(a)}{b + a}$ b. $f(a) - f(b)$ c. $\frac{f(a) - f(b)}{b - a}$ d. $\frac{f(b) - f(a)}{b - a}$ Question 36 Complete Marked out of 1.00 Identify which statement best describes the relationship between the instantaneous and average rates of change. (C:1) a. The instantaneous rate of change is the same as the average rate of change over any interval. b. The average rate of change is a specific case of instantaneous rate of change at a single point. c. The instantaneous rate of change can be found by taking the derivative of the function. d. The instantaneous rate of change can be approximated by the average rate of change over a small interval. Question 37 Complete Marked out of 1.00 Identify which one of the following functions is even. (A:1) a. $y = x^2 + x$ b. $y = 2^x$ c. $3x^2 + 4$ d. $y = \frac{1}{x}$ Question 38 Complete Marked out of 1.00 Identify which of the following statements correctly describes the result of multiplying two functions f(x) = x + 2 and g(x) = 2x - 1. (C:1) a. The product function f(x)g(x) will have the same x-intercepts of only f(x). b. The product function f(x)g(x) will be a linear function. c. The product function f(x)g(x) will have a combination of x-intercepts from both f(x) and g(x). d. The product function f(x)g(x) will have the same y-intercepts of f(x) and g(x). Question 39 Complete Marked out of 1.00 If $f(x) = 5x - 2$, find $f^{-1}(18)$. (K:1) a. 4 b. 88 c. 0.011 d. 3.2 Question 40 Complete Marked out of 1.00 Determine the interval for x where the following inequality holds true: $-2x + 4 > 3$ (A:1) a. $x \in$ of all real numbers b. $x < -\frac{1}{2}$ c. $x > \frac{1}{2}$ d. $x < \frac{1}{2}$ Question 41 Complete Marked out of 3.00 Fill in the blanks for the following polynomial expression: $y = 3(x + 2)(x - 5)^2$. (A:3) Write out for infinity and negative infinity. Degree: Leading Coefficient: End Behaviour: Zeros: Y-intercept: Question 42 Complete Marked out of 3.00 Determine the parameters of a different trigonometric function that is equal to $f(x) = 10sin[\frac{\pi}{6}(x + 3)] + 12$. (K:3) $f(x) = 10 \boxed{ } \boxed{ \frac{\pi}{6} } (x - \boxed{1}) + \boxed{12}$ | -13 | -12 | -11 | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | |-----|-----|-----|-----|----|----|----|----|----|----|----|----|----|---|---| | | | | | | | | | | | | | | | | sin tan Question 43 Complete Marked out of 3.00 Determine the value of $\theta$ for the equation $7cos\theta + 2cos^2\theta - 1 = -4$ on the interval of $0 \le x \le 2\pi$. Write your answer in exact ratios (use pi or $\pi$ and represent fractions as a/b). Fill in the blanks. $cos\theta =$ Relative angle $\theta$ (degrees) = Principle angle $\theta$ (degrees) : Question 44 Complete Marked out of 3.00 Given the polynomial function $f(x) = x^3 - x^2 - 5x + 6$, solve for when $f(x) \le 0$ and represent the solution on a number line. Roots are $x = \boxed{\text{,}} \boxed{\text{,}} \boxed{\text{}}$ (put in order from least to greatest) $f(x) \le 0$ at ($\boxed{\text{ }}$ $\boxed{\text{}}$) $\boxed{\text{and}}$ [$\boxed{\text{}}$ $\boxed{\text{}}$] Here is the extraction of the content from the image: **Question 45** Complete Marked out of 4.00 The displacement of an object, in meters, is given by $f(t) = 6t^3 - 3t^2 + t - 2$, where t is time in seconds. a) Find the average rate of change on the interval $1 \le t \le 4$ round to the nearest whole number. (2 marks) b) Estimate the instantaneous rate of change at t=3. round to the nearest whole number. (2 marks) (A:4) Fill in the blanks: Leave both answers to the whole number. The average rate of change on the interval $1 \le t \le 4$ is \_\_\_\_\_\_ meters per second. The estimated Instantaneous Rate of Change at t=3 is \_\_\_\_\_\_ meters per second. **Question 46** Complete Marked out of 3.00 Given $f(x) = 5x - 1$ and $g(x) = -3x + 2$, determine the following: (1:3) Fill in the blanks: do not include spaces, uses ^ to represent and exponent. A) $f(x) + g(x) =$ \_\_\_\_\_\_ B) $f(3) + g(3) =$ \_\_\_\_\_\_ C) $f(x)g(x) =$ \_\_\_\_\_\_ **Question 47** Complete Marked out of 2.00 1600g of radioactive material has a half-life of 8 hours. Determine how long it will take until there is only 300g left, in hours. Round to two decimal places Fill in the blanks: Time = \_\_\_\_\_\_ (A:2) **Question 48** Complete Marked out of 3.00 Describe the transformation from the graph of $y = \log_2 x$ to $y = -2\log_2(\frac{1}{4}x+3) - 1$. (C:3) a) The graph is stretched horizontally by a factor of \_\_\_\_\_\_ b) The graph is \_\_\_\_\_\_ over the \_\_\_\_\_\_ . c) The graph is stretched vertically by a factor of \_\_\_\_\_\_ d) The graph is shifted \_\_\_\_\_\_ unit to the \_\_\_\_\_\_ e) The graph is shifted \_\_\_\_\_\_ units \_\_\_\_\_\_ **Question 49** Complete Marked out of 2.00 Determine the value of x for the following equation: $2^{x+1} = 15$. Round your answer to two decimal places. Fill in the blanks: x = \_\_\_\_\_\_ (K:2) **Question 50** Complete Marked out of 2.00 The population of a city increases by 5% every year. If the current population is 100,000, what will the population be in 10 years? Fill in the blanks: Leave your answer to a whole number, do not include any spaces or commas. The population of the city in 10 years will be \_\_\_\_\_\_ (I:2) **Question 51** Complete Marked out of 2.00 What is the exact value of $\cos \frac{3\pi}{4}$ ? $\cos \frac{3\pi}{4} = \frac{\_\_\_\_\_\_}{\_\_\_\_\_\_}$ (C:2) **Question 52** Complete Marked out of 3.00 Given $f(x) = \frac{1}{3}\sin(2x-\frac{\pi}{3})$, determine the amplitude, period, and phase shift. Amplitude: \_\_\_\_\_\_ Period: \_\_\_\_\_\_ Phase Shift: \_\_\_\_\_\_ **Question 53** Complete Marked out of 2.00 Verify the identity $\sin(2x) = 2\sin(x)\cos(x)$. (I:2) Fill in the blanks: Do not include any spaces in your answer. $\sin(2x) =$ \_\_\_\_\_\_ **Question 54** Complete Marked out of 4.00 The function $f(x) = ax^3 - 3x^2 + bx - 30$ has three factors. Two of these factors are x-5 and x+2. Determine the values of a and b. Fill in the blanks: a = \_\_\_\_\_\_ b = \_\_\_\_\_\_ (A:4) **Question 55** Complete Marked out of 2.00 Determine the value of x algebraically. Leave your answer to two decimal places. $\frac{2x+1}{x-3} = \frac{4x-5}{2x+3}$ Fill in the blanks: x = \_\_\_\_\_\_ (I:2) **Question 56** Complete Marked out of 2.00 Find the instantaneous rate of change of the tangent line to the curve $y = x^3 - 3x^2 + 2$ at x=2. Fill in the blanks: do not include any spaces in your final answer. Instantaneous rate of change = \_\_\_\_\_\_ (A:2) **Question 57** Complete Marked out of 4.00 A company's revenue R(x) in thousands of dollars is modeled by the function $R(x) = -2x^2 - 12x + 50$, where x is the number of units sold. The cost function C(x) in thousands of dollars is given by $C(x) = -x^2 + 6x + 30$. (A:4) Fill in the blanks: leave no spaces. use ^ to represent an exponent. A) Find the profit function P(x) (revenue - cost): P(x) = \_\_\_\_\_\_ B) Determine the number of units sold that maximizes profit. The number of units sold to maximize profit is \_\_\_\_\_\_ . The maximum profit is $ \$ $ \_\_\_\_\_\_ thousand. Question 58 Complete Marked out of 5.00 Expand and simplify the following: log₂ √( (x² + 5)(2x - 3) ) / ( 4√x⁴ ) using laws of logarithms. (1:5) Fill in the blanks. Use ^ to represent exponents. No spaces. Simplify the Expression Inside the Square Root: = Apply the Square Root to the Entire Fraction: = Apply the Logarithm to the Simplified Expression: = Apply the Power Rule to the Logarithm of the Square Root: = Apply the Product Rule: = --- Question 59 Complete Marked out of 5.00 Consider the function y = (x² - 5x + 6) / (x² - 3x + 2). A) Find the x-intercept (1 mark) x-intercept = ( [blank] , [blank] ) B) Find the y-intercept (0.5 mark) y-intercept = ( [blank] , [blank] ) C) State the vertical asymptote (0.5 mark) Vertical Asymptote = x = [blank] D) State the horizontal asymptote (0.5 mark) Horizontal Asymptote = y = [blank] E) Determine if there is a hole or not. If there is a hole, state the coordinate. If there is no hole, don't write anything. (0.5 mark) Hole coordinate; if applicable = ( [blank] , [blank] ) F) State the intervals of increasing slopes (1.5 marks) Intervals of Increasing = ( [blank] , [blank] ) U ( [blank] , [blank] ) U ( [blank] , [blank] ) G) State the intervals of decreasing slopes (0.5 mark) Intervals of Increasing = ( [blank] , [blank] ) U ( [blank] , [blank] ) U ( [blank] , [blank] ) (1:5) [Options for filling blanks] -∞ | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 3 | 4 | 5 | ∞ | (blank)