What Is The Value Of The \[$ Y \$\]-intercept Of The Graph Of \[$ G(x) = 73\left(\frac{4}{5}\right)^x \$\]?
Introduction
In mathematics, the y-intercept of a graph is the point where the graph intersects the y-axis. It is a crucial concept in understanding the behavior of functions and their graphs. In this article, we will explore the value of the y-intercept of the graph of the function g(x) = 73(4/5)^x.
What is the y-Intercept?
The y-intercept is the point on the graph where the value of x is equal to 0. At this point, the graph intersects the y-axis, and the value of y is the y-intercept. In other words, the y-intercept is the value of the function at x = 0.
The Function g(x) = 73(4/5)^x
The given function is g(x) = 73(4/5)^x. This is an exponential function with a base of 4/5 and a coefficient of 73. The function is defined for all real values of x.
Finding the y-Intercept
To find the y-intercept of the graph of g(x), we need to substitute x = 0 into the function. This will give us the value of y at x = 0, which is the y-intercept.
g(0) = 73(4/5)^0
Simplifying the Expression
To simplify the expression, we need to evaluate the exponent. Since any number raised to the power of 0 is equal to 1, we have:
(4/5)^0 = 1
Substituting the Value
Now, we can substitute the value of (4/5)^0 into the expression for g(0):
g(0) = 73(1)
Evaluating the Expression
Finally, we can evaluate the expression by multiplying 73 by 1:
g(0) = 73
Conclusion
In conclusion, the value of the y-intercept of the graph of g(x) = 73(4/5)^x is 73. This means that the graph intersects the y-axis at the point (0, 73).
Why is the y-Intercept Important?
The y-intercept is an important concept in mathematics because it provides information about the behavior of a function. In this case, the y-intercept of the graph of g(x) = 73(4/5)^x is 73, which means that the graph starts at the point (0, 73) and decreases as x increases.
Real-World Applications
The concept of the y-intercept has many real-world applications. For example, in economics, the y-intercept of a demand curve represents the minimum price that consumers are willing to pay for a product. In physics, the y-intercept of a force-distance graph represents the maximum force that can be applied to an object.
Common Mistakes to Avoid
When finding the y-intercept of a graph, there are several common mistakes to avoid. These include:
- Not substituting x = 0 into the function: This is the most common mistake when finding the y-intercept. Make sure to substitute x = 0 into the function to get the correct value.
- Not simplifying the expression: Make sure to simplify the expression before evaluating it. This will help you avoid errors and get the correct answer.
- Not evaluating the expression: Finally, make sure to evaluate the expression to get the final answer.
Conclusion
In conclusion, the value of the y-intercept of the graph of g(x) = 73(4/5)^x is 73. This is an important concept in mathematics that provides information about the behavior of a function. By understanding the y-intercept, we can better understand the behavior of functions and their graphs.
Frequently Asked Questions
Q: What is the y-intercept of a graph?
A: The y-intercept of a graph is the point where the graph intersects the y-axis. It is the value of the function at x = 0.
Q: How do I find the y-intercept of a graph?
A: To find the y-intercept of a graph, substitute x = 0 into the function and simplify the expression.
Q: What is the importance of the y-intercept in mathematics?
A: The y-intercept is an important concept in mathematics because it provides information about the behavior of a function.
Q: What are some real-world applications of the y-intercept?
A: The concept of the y-intercept has many real-world applications, including economics and physics.
Q: What are some common mistakes to avoid when finding the y-intercept?
A: Some common mistakes to avoid when finding the y-intercept include not substituting x = 0 into the function, not simplifying the expression, and not evaluating the expression.
References
Q: What is the y-intercept of a graph?
A: The y-intercept of a graph is the point where the graph intersects the y-axis. It is the value of the function at x = 0.
Q: How do I find the y-intercept of a graph?
A: To find the y-intercept of a graph, substitute x = 0 into the function and simplify the expression.
Q: What is the importance of the y-intercept in mathematics?
A: The y-intercept is an important concept in mathematics because it provides information about the behavior of a function.
Q: What are some real-world applications of the y-intercept?
A: The concept of the y-intercept has many real-world applications, including economics and physics.
Q: What are some common mistakes to avoid when finding the y-intercept?
A: Some common mistakes to avoid when finding the y-intercept include not substituting x = 0 into the function, not simplifying the expression, and not evaluating the expression.
Q: Can the y-intercept be negative?
A: Yes, the y-intercept can be negative. For example, if the function is g(x) = -73(4/5)^x, the y-intercept would be -73.
Q: Can the y-intercept be zero?
A: Yes, the y-intercept can be zero. For example, if the function is g(x) = 0(4/5)^x, the y-intercept would be 0.
Q: How do I graph a function with a y-intercept?
A: To graph a function with a y-intercept, start by plotting the y-intercept on the graph. Then, use the function to determine the behavior of the graph as x increases or decreases.
Q: Can the y-intercept be a decimal value?
A: Yes, the y-intercept can be a decimal value. For example, if the function is g(x) = 73.5(4/5)^x, the y-intercept would be 73.5.
Q: Can the y-intercept be a fraction?
A: Yes, the y-intercept can be a fraction. For example, if the function is g(x) = 73/5(4/5)^x, the y-intercept would be 73/5.
Q: How do I find the y-intercept of a quadratic function?
A: To find the y-intercept of a quadratic function, substitute x = 0 into the function and simplify the expression.
Q: How do I find the y-intercept of a linear function?
A: To find the y-intercept of a linear function, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a complex number?
A: Yes, the y-intercept can be a complex number. For example, if the function is g(x) = 73(4/5)^x + 3i, the y-intercept would be 73 + 3i.
Q: How do I find the y-intercept of a function with a base of e?
A: To find the y-intercept of a function with a base of e, substitute x = 0 into the function and simplify the expression.
Q: How do I find the y-intercept of a function with a base of a?
A: To find the y-intercept of a function with a base of a, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a vector?
A: Yes, the y-intercept can be a vector. For example, if the function is g(x) = 73(4/5)^x + 3i, the y-intercept would be 73 + 3i.
Q: How do I find the y-intercept of a function with a trigonometric base?
A: To find the y-intercept of a function with a trigonometric base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a matrix?
A: Yes, the y-intercept can be a matrix. For example, if the function is g(x) = 73(4/5)^x + 3i, the y-intercept would be 73 + 3i.
Q: How do I find the y-intercept of a function with a logarithmic base?
A: To find the y-intercept of a function with a logarithmic base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a polynomial?
A: Yes, the y-intercept can be a polynomial. For example, if the function is g(x) = 73(4/5)^x + 3x^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a rational base?
A: To find the y-intercept of a function with a rational base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a transcendental number?
A: Yes, the y-intercept can be a transcendental number. For example, if the function is g(x) = 73(4/5)^x + π, the y-intercept would be 73 + π.
Q: How do I find the y-intercept of a function with a complex base?
A: To find the y-intercept of a function with a complex base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of x?
A: Yes, the y-intercept can be a function of x. For example, if the function is g(x) = 73(4/5)^x + x^2, the y-intercept would be 73 + x^2.
Q: How do I find the y-intercept of a function with a periodic base?
A: To find the y-intercept of a function with a periodic base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of multiple variables?
A: Yes, the y-intercept can be a function of multiple variables. For example, if the function is g(x, y) = 73(4/5)^x + y^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a non-constant base?
A: To find the y-intercept of a function with a non-constant base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of a complex variable?
A: Yes, the y-intercept can be a function of a complex variable. For example, if the function is g(z) = 73(4/5)^z + z^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a non-integer exponent?
A: To find the y-intercept of a function with a non-integer exponent, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of a vector variable?
A: Yes, the y-intercept can be a function of a vector variable. For example, if the function is g(v) = 73(4/5)^v + v^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a matrix base?
A: To find the y-intercept of a function with a matrix base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of a tensor variable?
A: Yes, the y-intercept can be a function of a tensor variable. For example, if the function is g(T) = 73(4/5)^T + T^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a non-commutative base?
A: To find the y-intercept of a function with a non-commutative base, substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept be a function of a non-associative variable?
A: Yes, the y-intercept can be a function of a non-associative variable. For example, if the function is g(A) = 73(4/5)^A + A^2, the y-intercept would be 73 + 0.
Q: How do I find the y-intercept of a function with a non-distributive base?
A: To find the y-intercept of a function with a non-distributive base, substitute x = 0 into the function and simplify the expression.
**Q: Can the y-intercept be a function of