Coordinating Period, Amplitude, and Midline

Section 2.5 Coordinating Period, Amplitude, and Midline

In this section, we revisit how to transform graphs of functions, now looking specifically at trigonometric functions.

T: Be able to determine the equation of a trigonometric function given its graph.
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Definition 2.5.1.

For a function $f\text{,}$ the period is the smallest positive value $t$ such that for all $x$ in the domain, $f(x+t)=f(x)\text{.}$

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Informally, the period of a trigonometric function is the distance it takes to complete one cycle, before it begins to repeat.

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Investigation 2.5.1.

In Desmos, at: https://www.desmos.com/calculator/lujykscfnv, you will find the following graphs: orange) $y=a\sin(b(x-c))+d\text{,}$ and purple) $y=a\cos(b(x-c))+d\text{.}$ You should have sliders for $a\text{,}$ $b\text{,}$ $c\text{,}$ and $d\text{.}$ To begin, set $a=1\text{,}$ $b=1\text{,}$ $c=0\text{,}$ and $d=0\text{.}$

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Understanding the cosine graph.
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1. Turn on the cosine graph (purple).
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2. Let $a=5\text{.}$ How has the cosine changed compared to when $a=1\text{?}$ What specific part of the graph measures 5 units? This is called the amplitude. Include a sketch that shows the amplitude.
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3. As you slide the value of $b\text{,}$ does the cosine wave (purple) change?
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4. Set $b=1\text{.}$ On the cosine function, what is the interval starting from $x=0$ in order for the graph to make one complete cycle and return to its original starting position? (This is the period.)
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5. Repeat Item 1.d with $b=4\text{,}$ with $b=2\text{,}$ with $b=2\pi\text{,}$ $b=\frac{\pi}{2}\text{,}$ and with $b=\frac{\pi}{4}\text{.}$
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  Table 2.5.2.
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  $b$ 1 2 4 $2 \pi$ $\frac{\pi}{2}$ $\frac{\pi}{4}$
  
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6. You cannot directly see $b$ on the cosine wave. Instead, $b$ affects the period. If you know $b\text{,}$ how can you determine the period of the cosine wave? In reverse, if you know the cosine wave period, how can you find $b\text{?}$
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7. Slide the value of $c\text{.}$ How does the cosine wave change?
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8. Slide the value of $d\text{.}$ How does the cosine wave change? The line $y=d$ is called the midline of the cosine wave. Explain why this name is used.
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Understanding the sine graph.The constants $a\text{,}$ $b\text{,}$ $c\text{,}$ and $d$ operate in the same way in the sine function, $f(x)=a \sin(b(x-c)) + d\text{,}$ but you begin with the base function $f(x)= \sin(x)\text{.}$
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Problem 2.5.3.

Predict what each of the following graphs will look like and sketch. Use Desmos to check your answer. Be sure to label the amplitude, period, midline and endpoints.

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$y=2\cos(x-2 \pi)+3$ on $-2\pi \leq x \leq 2 \pi$

Figure 2.5.4.
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$y=\cos(\pi x)-2$ on $-4 \leq x \leq 4$

Figure 2.5.5.
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Problem 2.5.6.

Predict what each of the following graphs will look like and sketch. Use Desmos to check your answer. Be sure to label the amplitude, period, midline and endpoints.

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$y=14\sin(2\pi x)+4$ on $-1 \leq x \leq 2$
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Figure 2.5.7.
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$y=-\frac{1}{2}\sin(\frac{\pi}{2}(x-1))+2$ on $-4 \leq x \leq 4$

Figure 2.5.8.
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Problem 2.5.9.

Refer to Figure 2.5.10. Note that $P=(\frac{\pi}{4},-1)\text{,}$ $Q=(\frac{5\pi}{4}, -1)\text{,}$ and $R=(\frac{9\pi}{4},-1)\text{.}$

A graph of a periodic function on a coordinate grid. The horizontal axis is marked from approximately negative four to seven, and the vertical axis ranges from about negative eight to zero. Three points labeled P, Q, and R appear at successive peaks of the graph, evenly spaced along the horizontal axis. — Figure 2.5.10.
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Write a function equation for $f$ of the form $f(x)=a \sin (b(x-c))+d\text{.}$
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Write a function equation for $f$ of the form $f(x)=a \cos (b(x-c))+d\text{.}$
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Problem 2.5.11.

Refer to Figure 2.5.12.

A graph of a periodic function drawn on a coordinate grid. The horizontal axis is labeled from approximately negative four to four, and the vertical axis ranges from about negative fourteen to six. The graph consists of repeating downward-opening parabolic arcs. Local maxima occur near x equals negative three, negative one, one, and three, with corresponding y-values near five. Local minima occur midway between these points at y-values near negative fourteen. — Figure 2.5.12.
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Write a function equation for $f$ of the form $f(x)=a \sin (b(x-c))+d\text{.}$
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Write a function equation for $f$ of the form $f(x)=a \cos (b(x-c))+d\text{.}$
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Problem 2.5.13.

For each graph below, write a function either of the form $f(x)=a \sin (b(x-c))+d$ or $f(x)=a \cos (b(x-c))+d\text{.}$

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$A graph of a trigonometric function drawn on a coordinate grid. The horizontal axis is labeled in radians and includes values such as negative six, negative four, negative two, zero, two, four, six, and eight. The vertical axis ranges approximately from zero at the top to negative two at the bottom. The graph has a maximum at $(-pi/2, 0)$ and another maximum at $(3pi/2, 0)\text{.}$$

Figure 2.5.14.
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$A graph of a periodic trigonometric function on a coordinate grid. The horizontal axis is marked with values including zero and five, and the vertical axis is marked with values including zero and negative five. The graph consists of repeating smooth curves with a maximum at $(0,3)$ and a minimum at $(1,-7)\text{.}$$

Figure 2.5.15.
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\(b\)	1	2	4	\(2 \pi\)	\(\frac{\pi}{2}\)	\(\frac{\pi}{4}\)
period