Math Snap

STEP 1

1. We are given the function $f(x) = 5\left(\frac{1}{4}\right)^{x}$.
2. We need to graph this function and its reflection about the $y$-axis.
3. We need to find the $y$-intercept of the function.

STEP 2

1. Graph the function $f(x) = 5\left(\frac{1}{4}\right)^{x}$.
2. Reflect the function about the $y$-axis.
3. Find the $y$-intercept of the function.

STEP 3

To graph the function $f(x) = 5\left(\frac{1}{4}\right)^{x}$, plot several points by choosing values for $x$ and calculating $f(x)$.
For example:
- When $x = 0$, $f(0) = 5\left(\frac{1}{4}\right)^{0} = 5$.
- When $x = 1$, $f(1) = 5\left(\frac{1}{4}\right)^{1} = \frac{5}{4}$.
- When $x = -1$, $f(-1) = 5\left(\frac{1}{4}\right)^{-1} = 20$.
Plot these points and draw a smooth curve through them.

STEP 4

To reflect the function about the $y$-axis, replace $x$ with $-x$ in the original function.
The reflected function is:
$$f(-x) = 5\left(\frac{1}{4}\right)^{-x} = 5\left(4\right)^{x}$$ Plot this reflected function by choosing values for $x$ and calculating $f(-x)$.
For example:
- When $x = 0$, $f(-0) = 5\left(4\right)^{0} = 5$.
- When $x = 1$, $f(-1) = 5\left(4\right)^{1} = 20$.
- When $x = -1$, $f(1) = 5\left(4\right)^{-1} = \frac{5}{4}$.
Plot these points and draw a smooth curve through them.

The $y$-intercept of a function is the point where the graph intersects the $y$-axis. This occurs when $x = 0$.
For the original function $f(x) = 5\left(\frac{1}{4}\right)^{x}$, when $x = 0$, $f(0) = 5$.
Therefore, the $y$-intercept is the ordered pair $(0, 5)$.
The $y$-intercept is:
$$\boxed{(0, 5)}$$