- Substitute the x-value of each ordered pair into the function \(f(x) = -9^{x+1}\).
- Compare the result \(f(x)\) to the given y-value.
- The point \((0, -9)\) is on the graph because \(f(0) = -9\).
- The point \((1, 81)\) is not on the graph because \(f(1) = -81
eq 81\).
- The point \((-1, -1)\) is on the graph because \(f(-1) = -1\).
- Final results: \((0, -9) \rightarrow \boxed{\text{Yes}}\), \((1, 81) \rightarrow \boxed{\text{No}}\), \((-1, -1) \rightarrow \boxed{\text{Yes}}\).
Explanation
- Problem Analysis
We are given the function \(f(x) = -9^{x+1}\) and three points: \((0, -9)\), \((1, 81)\), and \((-1, -1)\). To determine if these points lie on the graph of the function, we must check if the \(y\)-coordinate of each point is equal to the value of the function evaluated at the given \(x\)-coordinate, i.e., \(f(x) = y\).
- Checking the point (0, -9)
For the ordered pair \((0, -9)\), we substitute \(x = 0\) into the function:
\[f(0) = -9^{0+1} = -9^1 = -9\]
Since the calculated value is \(-9\), which matches the given \(y\)-coordinate, the point \((0, -9)\) lies on the graph. Therefore, the answer is "Yes".
- Checking the point (1, 81)
For the ordered pair \((1, 81)\), we substitute \(x = 1\) into the function:
\[f(1) = -9^{1+1} = -9^2 = -81\]
Since the calculated value is \(-81\), which does not match the given \(y\)-coordinate of \(81\), the point \((1, 81)\) does not lie on the graph. Therefore, the answer is "No".
- Checking the point (-1, -1)
For the ordered pair \((-1, -1)\), we substitute \(x = -1\) into the function:
\[f(-1) = -9^{-1+1} = -9^0 = -1\]
Since the calculated value is \(-1\), which matches the given \(y\)-coordinate, the point \((-1, -1)\) lies on the graph. Therefore, the answer is "Yes".
Examples
Functions are used in real life to model relationships, such as population growth or financial compounding. For instance, if you invest money in an account where the balance is determined by a specific function of time, you can verify if a statement about your account balance is accurate by evaluating the function at a given time point. Checking if coordinates lie on a graph is a fundamental skill for interpreting data models and ensuring calculations are correct.