How to sketch a tangent graph with a transformed equation

Key takeaway: For tangent graphs, the hardest part is usually the spacing of the key points. Once you identify the period and vertical shift, the rest falls into place point by point.

Step 1: Read the transformation

Your equation is

$y=4\tan\left(\frac{3}{2}(\theta-\pi)\right)+7$

This has the form

$y=a\tan(b(\theta-c))+d$

where:

• $a=4$ gives a vertical stretch
• $b=\frac{3}{2}$ changes the period
• $c=\pi$ shifts right by $\pi$
• $d=7$ shifts up by 7

Step 2: Find the tangent key spacing

For tangent, the period is

$\frac{\pi}{|b|}=\frac{\pi}{3/2}=\frac{2\pi}{3}$

So the distance between consecutive vertical asymptotes is $\frac{2\pi}{3}$.

The central point occurs where the tangent input is 0:

$\frac{3}{2}(\theta-\pi)=0 \Rightarrow \theta=\pi$

Then

$y=4\tan(0)+7=7$

So the middle point is $(\pi,7)$.

Step 3: Use the standard tangent guide points

For tangent, the common guide values are at input angles:

$-\frac{\pi}{3},\,-\frac{\pi}{6},\,0,\,\frac{\pi}{6},\,\frac{\pi}{3}$

After multiplying by the factor and shifting, these become the points around $\theta=\pi$.

Your plotted point at $\theta=\pi$ with $y=7$ is correct. The point with $y=\frac{5\pi}{6}$ would not belong to this graph because the vertical shift is 7, not a multiple of $\pi$.

Step 4: What to actually plot

The graph should be centered at $(\pi,7)$, with asymptotes spaced $\frac{2\pi}{3}$ apart, and the curve rising through the center point because the coefficient is positive.

---

Pitfalls the pros know 👇 A frequent mistake is mixing up the input angle and the output value. The $x$-axis here is $\theta$, so values like $\frac{5\pi}{6}$ belong on the horizontal axis, not the vertical one.

Another easy error is treating the $+7$ as if it affects the period. It only shifts the graph up.

What if the problem changes? If the coefficient in front of tangent were negative, the graph would still have the same asymptote spacing, but it would fall through the center instead of rising. If the inside shift were $\theta+\pi$ instead of $\theta-\pi$, the whole graph would move left by $\pi$ rather than right.

Tags: tangent graph, period, vertical asymptote