Transforming a function $f(x)$ into another function $g(x)$ involves several techniques that change the appearance of the graph of $f(x)$. These transformations include shifting, scaling, reflecting, and composing functions. Let’s explore these methods with examples.
Shifting
Vertical Shifts
A vertical shift moves the graph up or down. If you add a constant $c$ to $f(x)$, the graph shifts upward by $c$ units:
$g(x) = f(x) + c$
If you subtract $c$, the graph shifts downward by $c$ units:
$g(x) = f(x) – c$
For example, if $f(x) = x^2$, then $g(x) = x^2 + 3$ shifts the graph up by 3 units.
Horizontal Shifts
A horizontal shift moves the graph left or right. If you replace $x$ with $(x – c)$, the graph shifts to the right by $c$ units:
$g(x) = f(x – c)$
If you replace $x$ with $(x + c)$, the graph shifts to the left by $c$ units:
$g(x) = f(x + c)$
For instance, if $f(x) = x^2$, then $g(x) = (x – 2)^2$ shifts the graph right by 2 units.
Scaling
Vertical Scaling
Vertical scaling stretches or compresses the graph vertically. If you multiply $f(x)$ by a constant $a$, the graph is scaled vertically:
$g(x) = a times f(x)$
If $|a| > 1$, the graph stretches; if $0 < |a| < 1$, it compresses. For example, if $f(x) = x^2$ and $a = 2$, then $g(x) = 2x^2$ stretches the graph vertically.
Horizontal Scaling
Horizontal scaling stretches or compresses the graph horizontally. If you replace $x$ with $x/b$, the graph is scaled horizontally:
$g(x) = fbigg(frac{x}{b}bigg)$
If $|b| > 1$, the graph compresses; if $0 < |b| < 1$, it stretches. For example, if $f(x) = x^2$ and $b = 2$, then $g(x) = (x/2)^2$ compresses the graph horizontally.
Reflecting
Vertical Reflection
Reflecting the graph across the x-axis involves multiplying $f(x)$ by -1:
$g(x) = -f(x)$
For example, if $f(x) = x^2$, then $g(x) = -x^2$ reflects the graph across the x-axis.
Horizontal Reflection
Reflecting the graph across the y-axis involves replacing $x$ with $-x$:
$g(x) = f(-x)$
For instance, if $f(x) = x^3$, then $g(x) = (-x)^3$ reflects the graph across the y-axis.
Composing Functions
Composing functions involves combining two or more functions. If $h(x)$ is another function, then $g(x)$ can be formed by composing $f(x)$ and $h(x)$:
$g(x) = f(h(x))$
For example, if $f(x) = x^2$ and $h(x) = text{sin}(x)$, then $g(x) = ( text{sin}(x) )^2$
Conclusion
Understanding how to transform $f(x)$ to $g(x)$ through shifting, scaling, reflecting, and composing functions allows you to manipulate and analyze graphs effectively. These techniques are fundamental in algebra and calculus, providing a foundation for more complex mathematical concepts.