๐ Understanding the Arctangent Function
The arctangent function, denoted as $arctan(x)$ or $tan^{-1}(x)$, is the inverse of the tangent function. It returns the angle whose tangent is $x$. The basic graph of $arctan(x)$ has a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$ and a domain of all real numbers. Let's explore how transformations affect this graph.
- ๐ Definition: The arctangent function, $y = arctan(x)$, gives the angle $y$ such that $tan(y) = x$, where $-\frac{\pi}{2} < y < \frac{\pi}{2}$.
- ๐ History: The concept of inverse trigonometric functions developed alongside trigonometry itself, with mathematicians seeking to solve for angles given the ratios of sides in triangles.
- ๐ Key Principles: Transformations like shifts, stretches, and reflections can be applied to $arctan(x)$, changing its position, shape, and orientation on the coordinate plane.
๐ Graphing $y = arctan(x)$
The basic arctangent function, $y = arctan(x)$, has the following characteristics:
- ๐ Horizontal Asymptotes: It has horizontal asymptotes at $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$.
- ๐ค๏ธ Passing Through Origin: It passes through the origin (0, 0).
- โ๏ธ Increasing Function: It is an increasing function for all real numbers.
๐งช Transformations of $y = arctan(x)$
Let's explore how transformations affect the graph of $y = arctan(x)$:
- โ๏ธ Horizontal Shift: $y = arctan(x - h)$ shifts the graph horizontally. If $h > 0$, the graph shifts to the right; if $h < 0$, the graph shifts to the left.
- โ๏ธ Vertical Shift: $y = arctan(x) + k$ shifts the graph vertically. If $k > 0$, the graph shifts upward; if $k < 0$, the graph shifts downward.
- ัะฐัััะถะตะฝะธะต Vertical Stretch/Compression: $y = a \cdot arctan(x)$ stretches the graph vertically if $|a| > 1$ and compresses it if $0 < |a| < 1$. It also reflects the graph across the x-axis if $a < 0$.
- โ๏ธ Horizontal Stretch/Compression: $y = arctan(bx)$ compresses the graph horizontally if $|b| > 1$ and stretches it if $0 < |b| < 1$. It also reflects the graph across the y-axis if $b < 0$.
๐ข Worked Example 1: $y = arctan(x - 2) + 1$
This graph is a horizontal shift of the basic $arctan(x)$ two units to the right and a vertical shift one unit upward.
- โก๏ธ Horizontal Shift: The term $(x - 2)$ shifts the entire graph 2 units to the right.
- โฌ๏ธ Vertical Shift: The '+ 1' shifts the entire graph 1 unit up.
- asymptote Asymptotes: The horizontal asymptotes shift from $y = \pm \frac{\pi}{2}$ to $y = \frac{\pi}{2} + 1$ and $y = -\frac{\pi}{2} + 1$.
๐งฎ Worked Example 2: $y = 2 \cdot arctan(x)$
This graph represents a vertical stretch of the basic $arctan(x)$ by a factor of 2.
- ๐ Vertical Stretch: The '2' multiplies the $arctan(x)$ values, stretching the graph vertically.
- asymptote Asymptotes: The horizontal asymptotes change from $y = \pm \frac{\pi}{2}$ to $y = \pm \pi$.
๐ Worked Example 3: $y = arctan(2x)$
This graph represents a horizontal compression of the basic $arctan(x)$ by a factor of 2.
- ๐ Horizontal Compression: The '2x' compresses the graph horizontally toward the y-axis.
๐งญ Worked Example 4: $y = -arctan(x)$
This graph represents a reflection of the basic $arctan(x)$ across the x-axis.
- ๐ Reflection: The negative sign reflects the graph over the x-axis.
๐ก Worked Example 5: $y = arctan(-x)$
This graph represents a reflection of the basic $arctan(x)$ across the y-axis.
- ้ๅ Reflection: The negative sign inside the arctangent function reflects the graph over the y-axis. Since $arctan(x)$ is an odd function, $arctan(-x) = -arctan(x)$, so this is equivalent to a reflection over the x-axis as well.
โ๏ธ Worked Example 6: $y = 3 \cdot arctan(x + 1) - 2$
This combines multiple transformations: a horizontal shift, a vertical stretch, and a vertical shift.
- โฌ
๏ธ Horizontal Shift: The term $(x + 1)$ shifts the graph 1 unit to the left.
- ๐ Vertical Stretch: The '3' stretches the graph vertically by a factor of 3.
- โฌ๏ธ Vertical Shift: The '- 2' shifts the graph 2 units down.
- asymptote Asymptotes: The horizontal asymptotes change from $y = \pm \frac{\pi}{2}$ to $y = 3(\frac{\pi}{2}) - 2$ and $y = 3(-\frac{\pi}{2}) - 2$.
๐งฉ Worked Example 7: $y = -2 \cdot arctan(\frac{1}{2}x) + 1$
This combines reflection, horizontal stretch, vertical stretch and a vertical shift.
- ๐ Reflection: The '-2' reflects the graph over the x-axis.
- โ๏ธ Horizontal Stretch: The $\frac{1}{2}x$ stretches the graph horizontally by a factor of 2.
- ๐ Vertical Stretch: The '2' (from -2) stretches the graph vertically by a factor of 2.
- โฌ๏ธ Vertical Shift: The '+ 1' shifts the graph 1 unit up.
- asymptote Asymptotes: The horizontal asymptotes change from $y = \pm \frac{\pi}{2}$ to $y = -2(\frac{\pi}{2}) + 1$ and $y = -2(-\frac{\pi}{2}) + 1$, i.e., $y=- \pi + 1$ and $y = \pi + 1$.