To solve this problem and determine the next point Tanisha should plot on the graph of the function \( f(x) = 25 \left( \frac{3}{5} \right)^x \), we will follow these steps:
1. Understand the function: The function given is \( f(x) = 25 \left( \frac{3}{5} \right)^x \).
2. Identify the given point: The point provided in the problem is \( (1, 15) \). This means that when \( x = 1 \), the function evaluates to 15. So, checking:
[tex]\[
f(1) = 25 \left( \frac{3}{5} \right)^1
\][/tex]
Simplifying this, we get:
[tex]\[
f(1) = 25 \left( \frac{3}{5} \right) = 25 \times 0.6 = 15
\][/tex]
This confirms that \( (1, 15) \) correctly lies on the function \( f(x) \).
3. Find the y-coordinate for the next x-value: Now, we need to determine the y-coordinate when \( x = 2 \). Plugging \( x = 2 \) into the function:
[tex]\[
f(2) = 25 \left( \frac{3}{5} \right)^2
\][/tex]
Simplifying this, we get:
[tex]\[
\left( \frac{3}{5} \right)^2 = \left( \frac{3}{5} \times \frac{3}{5} \right) = \frac{9}{25}
\][/tex]
Then multiply by 25:
[tex]\[
f(2) = 25 \times \frac{9}{25} = 9
\][/tex]
Therefore, the y-coordinate when \( x = 2 \) is \( 9 \), giving us the point \( (2, 9) \).
4. Choose the correct option: Among the options provided:
- \((2, 9)\)
- \((2, -10)\)
- \(\left(2, 14 \frac{2}{5}\right)\)
- \((2, 5)\)
The correct point that Tanisha should plot is \( (2, 9) \).
Thus, the next point she should plot on the graph is [tex]\((2, 9)\)[/tex].
