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Similar Question 1
<p>Write an equation for</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{\pi}{3}</code></p>
Similar Question 2
<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following sets of transformations. For each set of transformations, determine the equation of the resulting function and sketch its graph.</p><p>horizontal stretch by a factor of 2, reflection in the y-axis</p>
Similar Question 3
<p>Determine the amplitude and the period of each sinusoidal function. Then, transform the graph of <code class='latex inline'>y=\sin x</code> to sketch a graph of each function. </p><p><strong>(a)</strong> <code class='latex inline'>y=5\sin3x</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{y=-3\cos\frac{4}{3}x}</code></p><p><strong>(c)</strong> <code class='latex inline'>y=3\sin\pi x</code></p>
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Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'> \displaystyle y = \frac{1}{5}\sin 4(x + \frac{\pi}{2}) - 3 </code></p>
<p>Sketch a graph of each function below on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle \begin{array}{ccccccc} &(a) & y = \sin x + 4 & &(b) & y = \sin x -2 \\ &(c) & y = \cos x -5 & &(b) & y = \cos x + 1 \\ \end{array} </code></p>
<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle y = \cos x -5 </code></p>
<p>State each as a cosine graph</p><img src="/qimages/285" />
<p>State each as a cosine graph</p><img src="/qimages/284" />
<p>Determine an equation for each cosine function. </p><img src="/qimages/530" />
<p>State the period, amplitude, horizontal translation, and equation of the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 0.5\cos(4x)</code></p>
<p> Find the information that is listed below for each sine function then sketch it.</p> <ul> <li>Amplitude</li> <li>Any reflections</li> <li>Period</li> <li>Phase Shift</li> <li>Vertical Shift</li> </ul> <p><code class='latex inline'>y = 5\sin(x - \frac{\pi}{6}) + 8</code></p>
<p>For each of the following equations, state the parent function and the transformations that were applied. Graph the transformed function.</p><p><code class='latex inline'>\displaystyle f(x) = -2\sin(3x) + 1, 0 \leq x \leq 360^o </code></p>
<p>What is the amplitude? </p><img src="/qimages/980" />
<p>Write an equation for</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{3\pi}{4}</code></p>
<p>Sketch a graph of in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{\pi}{3}</code></p>
<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p> <ul> <li>Determine the equation of the model in the form <code class='latex inline'>y = a\sin kx</code>.</li> </ul>
<p>Determine the amplitude, the period, the phase shift, and the vertical translation for each function with respect to <code class='latex inline'>y=\cos x</code>. Then, sketch a graph of the function for two cycles. </p><p><strong>(a)</strong> <code class='latex inline'>\displaystyle{y=3\cos\left(x-\frac{\pi}{4}\right)+6}</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{y=-5\cos\left[\frac{1}{4}\left(x+\frac{4\pi}{3}\right)\right]-5}</code></p><p><strong>(c)</strong> <code class='latex inline'>\displaystyle{7\cos[3\pi(x-2)]}+7</code></p>
<p>Graph <code class='latex inline'>f(x) = \frac{1}{2}\cos(x - \frac{\pi}{4})+3</code> showing:</p> <ul> <li>symmetry, if any</li> <li>x intercept, if any</li> <li>y intercept, if any</li> <li>asymptotes, if any</li> <li>domain </li> </ul>
<p>Sketch a graph of </p><p>cosine function with an amplitude of 5.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>cosine function with a phase shift of <code class='latex inline'>\frac{3\pi}{2}</code></li> </ul>
<p>State the period, amplitude, and equation of the axis of the trigonometric function that produces each of the following tables of values. Then use this information to write the equation of the function.</p><img src="/qimages/497" />
<p>Determine a sinusodial equation for each of the following graphs.</p><img src="/qimages/500" />
<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p> <ul> <li>Write the caution of the function in the form <code class='latex inline'>y = \sin[k(x - d)]</code>.</li> </ul>
<p> Sketch each function and determine its period.</p><p><code class='latex inline'> \displaystyle y = |\sin x| </code></p><p><a href="https://youtu.be/cldcTIigYk4">HINT</a></p>
<p>Identify the key characteristics of <code class='latex inline'>y = -2\cos(4x+\pi)+4</code>, and sketch its graph. Check your graph with a graphing calculator.</p>
<p>Sketch a graph of </p><p>sine function with an amplitude of 3</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<p> Sketch each function and determine its period.</p><p><code class='latex inline'> \displaystyle y = | 2\sin x + 1| </code></p>
<p>Determine the amplitude and the period of each sinusoidal function. Then, transform the graph of <code class='latex inline'>y=\sin x</code> to sketch a graph of each function. </p><p><strong>(i)</strong> <code class='latex inline'>\displaystyle{y=\frac{1}{2}\cos\frac{1}{4}x}</code></p><p><strong>(ii)</strong> <code class='latex inline'>y=-1.5\sin0.2\pi x</code></p><p><strong>(iii)</strong> <code class='latex inline'>y=0.75\cos0.8\pi x</code></p>
<p> Find the information that is listed below for each sine function then sketch it.</p> <ul> <li>Amplitude</li> <li>Any reflections</li> <li>Period</li> <li>Phase Shift</li> <li>Vertical Shift</li> </ul> <p><code class='latex inline'>y = -2 \sin\frac{1}{4}(2x + \frac{\pi}{6}) + 1 </code></p>
<p>Consider the function <code class='latex inline'>y=3\cos[\pi(x+2)]-1</code> </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p>
<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle y = \sin x -2 </code></p>
<p>Determine an equation for each cosine function. </p><p><strong>(a)</strong></p><img src="/qimages/531" /><p><strong>(b)</strong></p><img src="/qimages/532" />
<p>For the following equation, what is the parent function and the transformation that are applied? </p><p><code class='latex inline'> y= \sin(3x) + 1 </code></p>
<p>Write an equation for</p><p>sine function with a phase shift of <code class='latex inline'>-\frac{\pi}{3}</code></p>
<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing technology.</p><p><code class='latex inline'>y = \displaystyle{\frac{1}{2}}\cos\left(\displaystyle{\frac{x}{2}}-\displaystyle{\frac{\pi}{12}}\right)-3</code></p>
<p> A ferris wheel has a radius of 10 m, and the bottom of the wheel passes 1 m above the ground. If the ferris wheel makes one complete revolution every 20 s, find an equation that gives the height above the ground of a person on the ferris wheel as a function of time. Assume the person gets on the bottom of the ferries wheel.</p>
<p>A sine function is transformed such that it has a single <code class='latex inline'>x</code>-intercept in the interval <code class='latex inline'>[0, \pi]</code>, a period of <code class='latex inline'>\pi</code>, and a <code class='latex inline'>y</code>-intercept of 3. </p><p> Determine an equation for a function that satisfies the properties given.</p>
<p> Sketch the following graph.</p><p><code class='latex inline'> \displaystyle y = 2\sin(2x ) </code></p>
<p>Determine the amplitude and the period of each sinusoidal function. Then, transform the graph of <code class='latex inline'>y=\sin x</code> to sketch a graph of each function. </p><p><strong>(a)</strong> <code class='latex inline'>y=5\sin3x</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{y=-3\cos\frac{4}{3}x}</code></p><p><strong>(c)</strong> <code class='latex inline'>y=3\sin\pi x</code></p>
<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following sets of transformations. For each set of transformations, determine the equation of the resulting function and sketch its graph.</p><p>horizontal compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{2}}</code>, horizontal translation <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code> to the left</p>
<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>sine function with a phase shift of <code class='latex inline'>-\frac{3\pi}{4}</code></li> </ul>
<p> Find the equation of cosine graph with its first maximum at <code class='latex inline'>x = \frac{2\pi}{3}</code> and first minimum at <code class='latex inline'> x = 3\pi</code>. The maximum value of the cosine curve is 10 and the minimum is 2.</p>
<p>Determine a sinusodial equation for each of the following graphs.</p><p> <img src="/qimages/498" /></p>
<p>Write an equation for </p><p>cosine function with an amplitude of 5.</p>
<p> Find the information that is listed below for each sine function then sketch it.</p> <ul> <li>Amplitude</li> <li>Any reflections</li> <li>Period</li> <li>Phase Shift</li> <li>Vertical Shift</li> </ul> <p><code class='latex inline'>y = \sin (4x - 6\pi)</code></p>
<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle y = \cos x + 1 </code></p>
<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'> \displaystyle y = \frac{1}{3} \cos(3x + \frac{\pi}{2}) + 1 </code></p>
<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>cosine function with a phase shift of <code class='latex inline'>\frac{5\pi}{6}</code></li> </ul>
<p>Determine the amplitude, the period, the phase shift, and the vertical translation for each function with respect to <code class='latex inline'>y=\sin x</code>. Then, sketch a graph of the function for two cycles. </p><p><strong>(a)</strong> <code class='latex inline'>\displaystyle{y=3\sin\left(x+\frac{\pi}{4}\right)-1}</code></p><p><strong>(b)</strong> <code class='latex inline'>\displaystyle{-2\sin\left[\frac{1}{2}\left(x-\frac{5\pi}{6}\right)\right]+4}</code></p><p><strong>(c)</strong> <code class='latex inline'>y=2\sin[2\pi(x+3)]-2</code></p>
<p>Write an equation for each function.</p> <ul> <li>sine function with a phase shift of <code class='latex inline'>\frac{\pi}{2}</code></li> </ul>
<p>Write an equation for each function.</p> <ul> <li>cosine function with a phase shift of <code class='latex inline'>\frac{3\pi}{2}</code></li> </ul>
<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'> \displaystyle y = \frac{1}{5} \sin(3x + \frac{\pi}{4}) + 10 </code></p>
<p>State each as a cosine graph</p><img src="/qimages/286" />
<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p> <ul> <li>What is the period of the model?</li> </ul>
<p>The frequency of a periodic function is defined as the number of cycles complete in 1 s and is typically measured in hertz (Hz). It is the reciprocal of the period of a periodic function.</p> <ul> <li>One of the A-notes from a flute vibrates <code class='latex inline'>440</code> times in 1 s. It is said to have a frequency of <code class='latex inline'>440</code> Hz. What is the period of the A-note?</li> </ul>
<p>Determine an equation in the form <code class='latex inline'>y=a\sin kx</code> or <code class='latex inline'>y=a\cos kx</code> for each graph. </p><p><strong>(a)</strong></p><img src="/qimages/525" /><p><strong>(b)</strong></p><img src="/qimages/526" />
<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'> \displaystyle y = -\frac{1}{2}\cos 4x - 3 </code></p>
<p>Write an equation for </p><p>cosine function with an amplitude of 2, reflected in the x-axis.</p>
<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing technology.</p><p><code class='latex inline'>y = 0.5\sin\left(\displaystyle{\frac{x}{4}}-\displaystyle{\frac{\pi}{16}}\right)-5</code></p>
<p> Graph <code class='latex inline'>y = (\sin x)^2, -2\pi \leq x \leq 2\pi.</code></p><p>(a) determine the period;</p><p>(b) graph the function for one complete period.</p>
<p> Given the function <code class='latex inline'>y = \sin 2\pi x</code></p><p>(a) determine the period.</p><p>(b) graph the function for one complete period.</p>
<p>Find an expression that describes the location of each of the following values for <code class='latex inline'>y = cos \theta</code>, where <code class='latex inline'>n \in \mathbb{I}</code> and <code class='latex inline'>\theta</code> is in radians.\</p><p>(a) <code class='latex inline'>\theta-intercepts</code></p><p>(b) maximum values</p><p>(c) minimum values</p>
<p>The frequency of a periodic function is defined as the number of cycles complete in 1 s and is typically measured in hertz (Hz). It is the reciprocal of the period of a periodic function.</p> <ul> <li>The sound can be modelled using a sine function of the form <code class='latex inline'>y = \sin kx</code>. What is the value of <code class='latex inline'>k</code>.</li> </ul>
<p>The Round—Up is a popular ride at many theme parks. It consists of a large wheel that starts in a horizontal position. As the ride picks up rotational speed, the wheel tilts up to an angle of about <code class='latex inline'>\displaystyle \frac{\pi}{4}</code></p><img src="/qimages/2166" /><p> Determine a model for the Round-Up using a cosine function.</p>
<p>A crank 20 cm long, starting at <code class='latex inline'>60^{\circ}</code> above from the horizontal, sweeps out a positive angle at the uniform rate of 10 rad/s.</p><p>(b) How far is the end of the crank from the horizontal line after 0.1 s?</p>
<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>cosine function with a phase shift of <code class='latex inline'>\pi</code></li> </ul>
<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>Determine the amplitude of the function </p>
<p> Sketch the following graph.</p><p><code class='latex inline'> \displaystyle y = -3\sin( \frac{5}{2}x ) </code></p>
<p>Graph <code class='latex inline'>f(x) = 3\sin(2(x+1))-1</code> showing:</p> <ul> <li>State the transformation mapping</li> </ul>
<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p> <ul> <li>What is the amplitude of the model?</li> </ul>
<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p><p>Is the function <code class='latex inline'>y = \sin x</code> even, odd, or neither?</p>
<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following sets of transformations. For each set of transformations, determine the equation of the resulting function and sketch its graph.</p><p>vertical stretch by a factor of 3, horizontal translation <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code> to the right</p>
<p>A popular attraction in many water them parks is the wave pool. Suppose that the wave generator moves back and forth 20 complete cycles every minute to produce waves measuring 1.2 m from the top of the crest to the bottom of the trough.</p> <ul> <li>Sketch a graph of the model over two cycles.</li> </ul>
<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'> \displaystyle y = 20 \sin 3x + 10 </code></p>
<p> Graph each of the following function for one complete period.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccc} &(a) & y = \sin \frac{\pi}{2}x &(b) & y = \sin 4\pi x \\ \end{array} </code></p>
<p>State each as a cosine graph</p><img src="/qimages/283" />
<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing technology.</p><p><code class='latex inline'>y = 5\cos\left(x+\displaystyle{\frac{\pi}{4}}\right)-2</code></p>
<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p> <ul> <li>Determine the period of the function.</li> </ul>
<p>Sketch the following graphs.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccc} &(a) & y = -2\sin x &(b)& y = 3 \sin(x ) &(c) & y = -10 \sin( x ) \\ \end{array} </code></p>
<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p><p>Is the function <code class='latex inline'>y = \cos x</code> even, odd, or neither?</p>
<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p> <ul> <li>Determine the amplitude of the function.</li> </ul>
<p>Consider the function <code class='latex inline'>\displaystyle{y=4\sin\left[3\left(x+\frac{\pi}{3}\right)\right]}</code> </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p>
<p> Answer the following for each graph.</p> <ul> <li>amplitude</li> <li>period</li> <li>Maximum and Minimum value</li> </ul> <p><code class='latex inline'> \displaystyle \begin{array}{ccccc} &(a) & y = 5\cos 3x &(b)& y = -3 \cos \frac{7}{2}x &(c) &y = -2 \cos(2x + \frac{\pi}{12}) \\ \end{array} </code></p>
<p>State the period, amplitude, and equation of the axis of the trigonometric function that produces each of the following tables of values. Then use this information to write the equation of the function.</p><p> <img src="/qimages/496" /></p>
<p>Sketch a graph of </p><p>cosine function with an amplitude of 2, reflected in the x-axis.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions. Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = 4\sin{x} +3</code></p>
<p> Sketch the following graph.</p><p><code class='latex inline'> \displaystyle y = 3\sin(\frac{1}{2}x ) </code></p>
<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions. Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = -\sin\left(\displaystyle{\frac{1}{4}}x\right)</code></p>
<p>Determine an equation for each cosine function. </p><img src="/qimages/529" />
<p>Sketch a graph of </p><p>sine function with an amplitude of 4, reflected in the x-axis.</p><p>on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p>
<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p> <ul> <li>Graph the function over two cycles.</li> </ul>
<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'> \displaystyle y = - \cos(2x + \frac{\pi}{8}) + 2 </code></p>
<p>A sine function has a maximum value of 7, a minimum value of -1, a phase shift of <code class='latex inline'>\displaystyle{\frac{3\pi}{4}}</code> rad to the left, and a period of <code class='latex inline'>\displaystyle{\frac{\pi}{2}}</code>.</p><p>Write an equation for the function.</p>
<p> Sketch the following graph.</p><p><code class='latex inline'> \displaystyle y = -3\sin( \frac{2}{3}x ) </code></p>
<p>Recall the quotient identity <code class='latex inline'>\tan x = \frac{\sin x}{\cos x}</code>. Use the graphs of <code class='latex inline'>y = \sin x</code> and <code class='latex inline'>y= \cos x</code> in conjunction with the quotient identity to explain the shape of the graph of <code class='latex inline'>y = \tan x</code>.</p>
<p>The voltage of the electricity supply in North America can remodelled using a sine function. The maximum value of the voltage is about 120 V. The frequency is 60 Hz.</p> <ul> <li>Graph the model over two cycles.</li> </ul>
<p>What is the maximum value? the minimum value? </p><img src="/qimages/980" />
<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following sets of transformations. For each set of transformations, determine the equation of the resulting function and sketch its graph.</p><p>vertical compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{2}}</code>, vertical translation 3 units up</p>
<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>a) Determine the vertical translation of the function.</p><p>b) What is the desired period of the function?</p>
<p>A cosine function has a maximum value of 1, a minimum value of —5, a phase shift of 2 rad to the right, and a period of 3. </p><p> Write an equation for the function.</p>
<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions. Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = \sin\left(4x+\displaystyle{\frac{2\pi}{3}}\right)</code></p>
<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p> <ul> <li>Determine the vertical translation.</li> </ul>
<p>Write an equation for</p><p>cosine function with a phase shift of <code class='latex inline'>\frac{5\pi}{6}</code></p>
<p> Answer the following for each graph.</p> <ul> <li>amplitude</li> <li>period</li> </ul> <p><em>(a)</em> <code class='latex inline'> \displaystyle y = 5\sin(x ) </code></p><p><em>(b)</em> <code class='latex inline'> \displaystyle y = -3 \sin(\frac{7}{2}x ) </code></p>
<p>Consider the function <code class='latex inline'>\displaystyle{=4\sin\left[2\left(x+\frac{2\pi}{3}\right)\right]-5}</code>. </p><p><strong>(a)</strong> What is the amplitude?</p><p><strong>(b)</strong> What is the period?</p><p><strong>(c)</strong> Describe the phase shift.</p><p><strong>(d)</strong> Describe the vertical translation.</p><p><strong>(e)</strong> Describe the <code class='latex inline'>x</code>-intercepts of the function. Explain your answer.</p>
<p> Sketch <code class='latex inline'>y = -\sin x</code>. Then sketch <code class='latex inline'>y = \sin(-x)</code>. What do you notice?</p>
<p>State the transformations that were applied to the parent function <code class='latex inline'>f(x) = \sin{x}</code> to obtain each of the following transformed functions. Then graph the transformed functions.</p><p><code class='latex inline'>f(x) = \sin(x-\pi) - 1</code></p>
<p> Answer the following for each graph.</p> <ul> <li>amplitude</li> <li>period</li> </ul> <p><code class='latex inline'> \displaystyle y = -2 \sin(3x ) </code></p>
<p>Write an equation for </p><p>sine function with an amplitude of 4, reflected in the x-axis.</p>
<p>What is the vertical translation? </p><img src="/qimages/980" />
<p>A popular attraction in many water them parks is the wave pool. Suppose that the wave generator moves back and forth 20 complete cycles every minute to produce waves measuring 1.2 m from the top of the crest to the bottom of the trough.</p> <ul> <li>Model the displacement, <code class='latex inline'>d</code> , in metres, of the wave from the mean pool level as a function of time, <code class='latex inline'>t</code>, in seconds.</li> </ul>
<p>Model the graph shown using a sine function</p><img src="/qimages/527" /><p><strong>(a)</strong> From the graph, determine the amplitude, the period, the phase shift, and the vertical translation,</p><p><strong>(b)</strong> Write an equation for the function.</p>
<p>State the amplitude, period, phase shift, and vertical translation.</p><p><code class='latex inline'> \displaystyle y = \sin(2x + \frac{\pi}{8}) </code></p>
<p>Model the graph shown using a cosine function.</p><img src="/qimages/528" /><p><strong>(a)</strong> From the graph, determine the amplitude, the period, the phase shift, and the vertical translation,</p><p><strong>(b)</strong> Write an equation for the function.</p>
<p>State the period, amplitude, horizontal translation, and equation of the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y=\sin\left(x-\displaystyle{\frac{\pi}{4}}\right)+3</code></p>
<p>Write an equation for</p><p>cosine function with a phase shift of <code class='latex inline'>\frac{4\pi}{3}</code></p>
<p> For each of the following, state the amplitude, period, phase shift, and vertical translation. Sketch the function.</p><p><code class='latex inline'> \displaystyle y = 2 \cos 3x + 10 </code></p>
<p>A crank 20 cm long, starting at <code class='latex inline'>60^{\circ}</code> above from the horizontal, sweeps out a positive angle at the uniform rate of 10 rad/s.</p> <ul> <li>At what times is the crank 12 cm from the horizontal line?</li> </ul>
<p>Predict the maximum value, the minimum value, and the values of x where they occur for each function on the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p><p><code class='latex inline'>\displaystyle y = \sin x + 4 </code></p>
<p>State the period, amplitude, horizontal translation, and equation of the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 5\cos\left(-2x+\displaystyle{\frac{\pi}{3}}\right)-2</code></p>
<p>Write an equation for</p><p>sine function with an amplitude of 3</p>
<p>State the period, amplitude, horizontal translation, and equation of the axis for each of the following trigonometric functions.</p><p><code class='latex inline'>y = 2\sin(3x)-1</code></p>
<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>sine function with a period of <code class='latex inline'>\frac{\pi}{2}</code></li> </ul>
<p>A function is said to be even if <code class='latex inline'>f(-x) = f(x)</code> for all values of . A function is said to be odd if <code class='latex inline'>f(-x) = -f(x)</code> for all values of <code class='latex inline'>x</code>.</p> <ul> <li>Is the function <code class='latex inline'>y = \tan x</code> even, odd, or neither?</li> </ul>
<p>Sketch each graph for <code class='latex inline'>0 \leq x \leq 2\pi</code>. Verify your sketch using graphing technology.</p><p><code class='latex inline'>y = -\cos\left(0.5x-\displaystyle{\frac{\pi}{6}}\right)+3</code></p>
<p>For each of the following, state the amplitude, period, phase shift, and vertical translation. For each graph, state it as a sine function. Note that the period is set as a multiple of <code class='latex inline'>\pi</code></p><img src="/qimages/282" />
<p>The Octopus ride at an amusement park completes one revolution every 60 s. The cars reach a maximum of 4 m above the ground and a minimum of 1 m above the ground. The height, h , in metres, above the ground can remodelled using a sine function of the from <code class='latex inline'>h = a\sin(kt) + c</code>, where <code class='latex inline'>t</code> represents the time, in seconds.</p><p>(d) Determine the value of <code class='latex inline'>k</code> that results in the period desired in part (c).</p>
<p>Find the equation of the following function in terms of </p><p>a) a sine function and </p><p>b) a cosine function</p><img src="/qimages/2107" />
<p>Sketch the graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>sine function with a phase shift of <code class='latex inline'>-6\pi</code></li> </ul>
<p>Sketch a graph in the interval <code class='latex inline'>x \in [-2\pi, 2\pi]</code>.</p> <ul> <li>cosine function with a phase shift of <code class='latex inline'>\frac{4\pi}{3}</code></li> </ul>
<p>A cosine function has a maximum value of <code class='latex inline'>7</code> and minimum value of <code class='latex inline'>-3</code>.</p> <ul> <li>Graph the function over two cycles.</li> </ul>
<p> Sketch the following graphs.</p><p><code class='latex inline'> \displaystyle \begin{array}{cccc} &(a) & y = \sin(3x ) &(b) &y = \sin(5x ) \\ &(c) & y = \sin( \frac{1}{4}x ) &(d)&y = \sin( \frac{3}{5}x ) \\ \end{array} </code></p>
<p>The trigonometric function <code class='latex inline'>f(x) = \cos{x}</code> has undergone the following sets of transformations. For each set of transformations, determine the equation of the resulting function and sketch its graph.</p><p>horizontal stretch by a factor of 2, reflection in the y-axis</p>
<p>A sine function has an amplitude of 3 and a period of <code class='latex inline'>\pi</code>.</p><p><strong>(a)</strong> Write the equation of the function in the form <code class='latex inline'>y = a\sin kx</code>.</p><p><strong>(b)</strong> Graph the function over two cycles.</p>
<p>One cycle of a sine function begins at <code class='latex inline'>x = - \frac{\pi}{6}</code> and ends at <code class='latex inline'>x = \frac{\pi}{3}</code>.</p> <ul> <li>Determine the phase shift of the function.</li> </ul>
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