Select All Functions That Have A { Y $}$-intercept Of { (0, 5)$}$.1. { F(x) = -3(b)^x - 5 $}$2. { F(x) = 7(b)^x - 2 $}$3. { F(x) = 5(b)^x - 1 $}$4. { F(x) = 2(b)^x + 5 $} 5. \[ 5. \[ 5. \[ F(x)
In mathematics, the y-intercept of a function is the point at which the graph of the function crosses the y-axis. It is the value of the function when the input (or x-value) is zero. In this article, we will explore how to select functions that have a specific y-intercept, specifically the point (0, 5).
Understanding the y-Intercept
The y-intercept of a function is a critical point that helps us understand the behavior of the function. It is the starting point of the function, and it can give us valuable information about the function's growth or decay. In this case, we are looking for functions that have a y-intercept of (0, 5), which means that when x = 0, the function's value is 5.
Analyzing the Functions
Let's analyze each of the given functions to determine which ones have a y-intercept of (0, 5).
Function 1: f(x) = -3(b)^x - 5
To determine if this function has a y-intercept of (0, 5), we need to substitute x = 0 into the function and see if the result is 5.
f(0) = -3(b)^0 - 5
Since b^0 = 1, we can simplify the equation:
f(0) = -3(1) - 5 f(0) = -3 - 5 f(0) = -8
This function does not have a y-intercept of (0, 5).
Function 2: f(x) = 7(b)^x - 2
To determine if this function has a y-intercept of (0, 5), we need to substitute x = 0 into the function and see if the result is 5.
f(0) = 7(b)^0 - 2
Since b^0 = 1, we can simplify the equation:
f(0) = 7(1) - 2 f(0) = 7 - 2 f(0) = 5
This function has a y-intercept of (0, 5).
Function 3: f(x) = 5(b)^x - 1
To determine if this function has a y-intercept of (0, 5), we need to substitute x = 0 into the function and see if the result is 5.
f(0) = 5(b)^0 - 1
Since b^0 = 1, we can simplify the equation:
f(0) = 5(1) - 1 f(0) = 5 - 1 f(0) = 4
This function does not have a y-intercept of (0, 5).
Function 4: f(x) = 2(b)^x + 5
To determine if this function has a y-intercept of (0, 5), we need to substitute x = 0 into the function and see if the result is 5.
f(0) = 2(b)^0 + 5
Since b^0 = 1, we can simplify the equation:
f(0) = 2(1) + 5 f(0) = 2 + 5 f(0) = 7
This function does not have a y-intercept of (0, 5).
Function 5: f(x) = 3(b)^x + 2
To determine if this function has a y-intercept of (0, 5), we need to substitute x = 0 into the function and see if the result is 5.
f(0) = 3(b)^0 + 2
Since b^0 = 1, we can simplify the equation:
f(0) = 3(1) + 2 f(0) = 3 + 2 f(0) = 5
This function has a y-intercept of (0, 5).
Conclusion
In conclusion, the functions that have a y-intercept of (0, 5) are:
- f(x) = 7(b)^x - 2
- f(x) = 3(b)^x + 2
In the previous article, we explored how to select functions that have a specific y-intercept, specifically the point (0, 5). In this article, we will answer some frequently asked questions (FAQs) about selecting functions with a specific y-intercept.
Q: What is a y-intercept?
A: The y-intercept of a function is the point at which the graph of the function crosses the y-axis. It is the value of the function when the input (or x-value) is zero.
Q: Why is the y-intercept important?
A: The y-intercept is an important point in a function's graph because it gives us valuable information about the function's behavior. It can help us understand the function's growth or decay, and it can also help us determine the function's domain and range.
Q: How do I find the y-intercept of a function?
A: To find the y-intercept of a function, you need to substitute x = 0 into the function and solve for the resulting value. This value is the y-intercept of the function.
Q: What if the function has a variable base?
A: If the function has a variable base, you need to use the properties of exponents to simplify the expression. For example, if the function is f(x) = 2(b)^x + 5, you can simplify it by using the property b^0 = 1.
Q: Can I have multiple y-intercepts for a function?
A: No, a function can only have one y-intercept. The y-intercept is a specific point on the graph of the function, and it is determined by the function's equation.
Q: How do I determine if a function has a y-intercept of (0, 5)?
A: To determine if a function has a y-intercept of (0, 5), you need to substitute x = 0 into the function and see if the resulting value is 5. If it is, then the function has a y-intercept of (0, 5).
Q: Can I use the y-intercept to determine the function's domain and range?
A: Yes, the y-intercept can give you information about the function's domain and range. For example, if the function has a y-intercept of (0, 5), it means that the function's value is 5 when x = 0. This can help you determine the function's domain and range.
Q: Are there any other ways to determine the y-intercept of a function?
A: Yes, there are other ways to determine the y-intercept of a function. For example, you can use the graph of the function to find the y-intercept. You can also use the function's equation to find the y-intercept by substituting x = 0 into the equation.
Conclusion
In conclusion, selecting functions with a specific y-intercept is an important concept in mathematics. By understanding the y-intercept and how to find it, you can gain valuable insights into a function's behavior and use it to determine the function's domain and range.