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Graphing Trigonometric Functions>
Graphing Trigonometric Functions 2Graphing Trigonometric Functions 2
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Question 1 of 4
1. Question
Which graph best represents both of the trigonometric functions below?`color(red)(y=cosx)``y=secx`Hint
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General Form of a Cos Function
`y=``a` `\text(sin)` `b``x`Period Fomula
$$P_{\text{sin}}=\frac{2\pi}{\color{#00880A}{b}}$$First, identify the values of the `cos` function`y` `=` `a` `\text(cos)` `b``x` `y` `=` $$\color{#004ec4}{a}\;\text{cos}\;x$$ `a` `=` `1` `b` `=` `1` Next, solve for the period of the `cos` function$$P_{\text{cos}}$$ `=` $$\frac{2\pi}{\color{#00880A}{b}}$$ `=` $$\frac{2\pi}{\color{#00880A}{1}}$$ Substitute known values `=` `2pi` To graph the `cos` curve, it is better to divide the value of its period into for parts and have the curve meet the following conditionsCurve starts at the peak of the amplitude `(a)`
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (`P=2pi`)
Therefore, this will be the `cos` curve for `y=\text(sin) x` with a period of `2pi`Next, recall that `\text(sec) x=1/(\text(cos) x)`Given this, we can graph `y=\text(csc) x` with regards to the following:`=` `\text(Undefined values of) 1/(\text(cos) x) \text(will be asymptotes)` `=` `\text(Defined values of) 1/(\text(cos) x) \text(will be points of intersection)` Finally, graph `y=\text(sec) x`Asymptotes`1/(\text(cos) pi/2)` `=` `\text(undefined)` `1/(\text(cos) 3pi/2)` `=` `\text(undefined)` 
Points of intersection`1/(\text(cos) 0)` `=` `1` `1/(\text(cos) pi)` `=` `-1` `1/(\text(cos) 2pi)` `=` `1` 
Take note that the curves of `y=\text(sec) x` will keep approaching but will never intersect with the asymptotes.
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Question 2 of 4
2. Question
Graph the trigonometric function`y=3cos 2x`Hint
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General Form of a Cos Function
`y=``a` `\text(sin)` `b``x`Period Fomula
$$P_{\text{sin}}=\frac{2\pi}{\color{#00880A}{b}}$$First, identify the values of the function`y` `=` `a` `\text(cos)` `b``x` `y` `=` $$\color{#004ec4}{3}\;\text{cos}\;\color{#004ec4}{2}x$$ `a` `=` `3` `b` `=` `2` Next, solve for the period of the function$$P_{\text{cos}}$$ `=` $$\frac{2\pi}{\color{#00880A}{b}}$$ `=` $$\frac{2\pi}{\color{#00880A}{2}}$$ Substitute known values `=` `pi` To graph the `cos` curve, it is better to divide the value of its period into for parts and have the curve meet the following conditionsCurve starts at the peak of the amplitude `(a)`
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (`P=pi`)
Therefore, this will be the `cos` curve for `y=3\text(cos) 2x` with a period of `pi`
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Question 3 of 4
3. Question
Graph the trigonometric function`y=4cos 2/3x`Hint
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General Form of a Cos Function
`y=``a` `\text(sin)` `b``x`Period Fomula
$$P_{\text{sin}}=\frac{2\pi}{\color{#00880A}{b}}$$First, identify the values of the function`y` `=` `a` `\text(cos)` `b``x` `y` `=` $$\color{#004ec4}{4}\;\text{cos}\;\color{#004ec4}{\frac{2}{3}}x$$ `a` `=` `4` `b` `=` `2/3` Next, solve for the period of the function$$P_{\text{cos}}$$ `=` $$\frac{2\pi}{\color{#00880A}{b}}$$ `=` $$\frac{2\pi}{\color{#00880A}{\frac{2}{3}}}$$ Substitute known values `=` `(6pi)/2` `=` `3pi` To graph the `cos` curve, it is better to divide the value of its period into for parts and have the curve meet the following conditionsCurve starts at the peak of the amplitude `(a)`
Curve intercepts x-axis at 1st quarter
Curve reaches minimum amplitude at 2nd quarter
Curve intercepts x-axis again at 3rd quarter
Curve starts again at the period (`P=3pi`)
Therefore, this will be the `cos` curve for `y=4\text(cos) 2/3x` with a period of `3pi`
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Question 4 of 4
4. Question
Graph the trigonometric function within the domain `-pi/2≤x≤2pi``y=tan x`Hint
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Period Fomula
$$P_{\text{tan}}=\frac{\pi}{\color{#00880A}{b}}$$First, solve for the period of the function`b` `=` `1` $$P_{\text{tan}}$$ `=` $$\frac{\pi}{\color{#00880A}{b}}$$ `=` $$\frac{\pi}{\color{#00880A}{1}}$$ Substitute known values `=` `pi` Asymptotes will occur within the domain for the values of `tan` that are undefined`\text(tan) -pi/2` `=` `\text(undefined)` `\text(tan) pi/2` `=` `\text(undefined)` `\text(tan) 3pi/2` `=` `\text(undefined)` 
To graph the `tan` curve, start at the lower domain `(-pi/2)` and draw a curve going passing through `(0,0)` and heading up to the second asymptote `(pi/2)`. This will cover a period of the curve.Repeat the process until you reach the upper domain `(2pi)`.First curve
Second curve
Third curve
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