Graphs of sine and cosine functions are periodic waves that repeat at regular intervals. In the form \(f(x)=A\sin(B(x-c))+D\) or \(f(x)=A\cos(B(x-c))+D\), \(A\) changes the amplitude, \(B\) changes the period, \(c\) shifts the graph horizontally, and \(D\) shifts the midline. These problems include vertical stretches, reflections, period changes, horizontal shifts, vertical shifts, and writing equations from graph features.
Notes
Practice Problems
Graph the following Trigonometric Functions
\(\textbf{1)}\) \(f(x)=3\sin{(x)} \)
\(f(x)=3\sin(x)\)
\(\text{The parent function is }y=\sin(x).\)
\(\text{The amplitude is }|A|=3.\)
\(\text{The period is }\frac{2\pi}{|B|}=2\pi.\)
\(\text{The midline is }y=0.\)
\(\text{Graph }y=\sin(x)\text{ with a vertical stretch by }3.\)
\(f(x)=3\sin(x)\)
\(\text{The parent function is }y=\sin(x).\)
\(\text{The amplitude is }|A|=3.\)
\(\text{The period is }\frac{2\pi}{|B|}=2\pi.\)
\(\text{The midline is }y=0.\)
\(\text{Graph }y=\sin(x)\text{ with a vertical stretch by }3.\)
\(\textbf{2)}\) \(f(x)=\cos{(x)}+4 \) 
\(f(x)=\cos(x)+4\)
\(\text{The parent function is }y=\cos(x).\)
\(\text{The amplitude is }1.\)
\(\text{The period is }2\pi.\)
\(\text{The }+4\text{ shifts the graph up }4.\)
\(\text{The midline is }y=4.\)
\(f(x)=\cos(x)+4\)
\(\text{The parent function is }y=\cos(x).\)
\(\text{The amplitude is }1.\)
\(\text{The period is }2\pi.\)
\(\text{The }+4\text{ shifts the graph up }4.\)
\(\text{The midline is }y=4.\)
\(\textbf{3)}\) \(f(x)=3\sin{(2x)}-1 \) 
\(f(x)=3\sin(2x)-1\)
\(\text{The amplitude is }|A|=3.\)
\(\text{The period is }\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi.\)
\(\text{The }-1\text{ shifts the graph down }1.\)
\(\text{The midline is }y=-1.\)
\(\text{Graph a sine curve with amplitude }3\text{ and period }\pi.\)
\(f(x)=3\sin(2x)-1\)
\(\text{The amplitude is }|A|=3.\)
\(\text{The period is }\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi.\)
\(\text{The }-1\text{ shifts the graph down }1.\)
\(\text{The midline is }y=-1.\)
\(\text{Graph a sine curve with amplitude }3\text{ and period }\pi.\)
\(\textbf{4)}\) \(f(x)=-4\sin{(\frac{x}{2})}+4 \) 
\(f(x)=-4\sin\left(\frac{x}{2}\right)+4\)
\(\text{The amplitude is }|-4|=4.\)
\(\text{The negative sign reflects the graph over the }x\text{-axis.}\)
\(\text{The period is }\frac{2\pi}{\left|\frac{1}{2}\right|}=4\pi.\)
\(\text{The }+4\text{ shifts the graph up }4.\)
\(\text{The midline is }y=4.\)
\(f(x)=-4\sin\left(\frac{x}{2}\right)+4\)
\(\text{The amplitude is }|-4|=4.\)
\(\text{The negative sign reflects the graph over the }x\text{-axis.}\)
\(\text{The period is }\frac{2\pi}{\left|\frac{1}{2}\right|}=4\pi.\)
\(\text{The }+4\text{ shifts the graph up }4.\)
\(\text{The midline is }y=4.\)
\(\textbf{5)}\) \(f(x)=\sin{(x+\frac{\pi}{2})} \) 
\(f(x)=\sin\left(x+\frac{\pi}{2}\right)\)
\(\text{The parent function is }y=\sin(x).\)
\(\text{The graph is shifted left }\frac{\pi}{2}.\)
\(\text{The amplitude is }1.\)
\(\text{The period is }2\pi.\)
\(\text{The midline is }y=0.\)
\(f(x)=\sin\left(x+\frac{\pi}{2}\right)\)
\(\text{The parent function is }y=\sin(x).\)
\(\text{The graph is shifted left }\frac{\pi}{2}.\)
\(\text{The amplitude is }1.\)
\(\text{The period is }2\pi.\)
\(\text{The midline is }y=0.\)
\(\textbf{6)}\) \(f(x)=-2\cos{(\pi x)} \) 
\(f(x)=-2\cos(\pi x)\)
\(\text{The amplitude is }|-2|=2.\)
\(\text{The negative sign reflects the graph over the }x\text{-axis.}\)
\(\text{The period is }\frac{2\pi}{|\pi|}=2.\)
\(\text{The midline is }y=0.\)
\(\text{Graph a reflected cosine curve with amplitude }2\text{ and period }2.\)
\(f(x)=-2\cos(\pi x)\)
\(\text{The amplitude is }|-2|=2.\)
\(\text{The negative sign reflects the graph over the }x\text{-axis.}\)
\(\text{The period is }\frac{2\pi}{|\pi|}=2.\)
\(\text{The midline is }y=0.\)
\(\text{Graph a reflected cosine curve with amplitude }2\text{ and period }2.\)
\(\textbf{7)}\) \(f(x)=\cos{(\pi x+\frac{\pi}{4})} \) 
\(f(x)=\cos\left(\pi x+\frac{\pi}{4}\right)\)
\(f(x)=\cos\left(\pi\left(x+\frac{1}{4}\right)\right)\)
\(\text{The graph shifts left }\frac{1}{4}.\)
\(\text{The amplitude is }1.\)
\(\text{The period is }\frac{2\pi}{|\pi|}=2.\)
\(\text{The midline is }y=0.\)
\(f(x)=\cos\left(\pi x+\frac{\pi}{4}\right)\)
\(f(x)=\cos\left(\pi\left(x+\frac{1}{4}\right)\right)\)
\(\text{The graph shifts left }\frac{1}{4}.\)
\(\text{The amplitude is }1.\)
\(\text{The period is }\frac{2\pi}{|\pi|}=2.\)
\(\text{The midline is }y=0.\)
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\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
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\(\bullet\text{ Geometric Mean}\)
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\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
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\(\bullet\text{ Product-Sum Identities}\)
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\(\bullet\text{ Cofunction Identities}\)
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\(\bullet\text{ Proving Trigonometric Identities}\)
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\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
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\(\bullet\text{ Solving Trigonometric Equations}\)
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