Determine The { Y $}$-intercept Of The Graph Of The Function { F(x) = X^2 + 4x - 5 $}$.A. { (4, -5) $}$B. { (1, 0) $}$C. { (0, -5) $}$D. { (-5, 0) $}$
Understanding the y-intercept
The y-intercept of a function is the point at which the graph of the function intersects the y-axis. In other words, it is the value of the function when the input (or x-value) is equal to zero. To determine the y-intercept of a quadratic function, we need to substitute x = 0 into the function and solve for y.
The Given Function
The given function is f(x) = x^2 + 4x - 5. This is a quadratic function in the form of f(x) = ax^2 + bx + c, where a = 1, b = 4, and c = -5.
Substituting x = 0 into the Function
To find the y-intercept, we need to substitute x = 0 into the function. This means we replace every instance of x with 0.
f(0) = (0)^2 + 4(0) - 5
Simplifying the Expression
Now, we simplify the expression by evaluating the exponent and the multiplication.
f(0) = 0 + 0 - 5
Finding the y-intercept
Finally, we find the value of the function when x = 0, which is the y-intercept.
f(0) = -5
Conclusion
The y-intercept of the graph of the function f(x) = x^2 + 4x - 5 is (0, -5). This means that when x = 0, the value of the function is -5.
Answer
The correct answer is C. (0, -5).
Why is this important?
Understanding the y-intercept of a function is crucial in various mathematical and real-world applications. For example, in physics, the y-intercept of a function can represent the initial velocity of an object. In economics, the y-intercept of a function can represent the initial cost of a product.
Real-world Applications
The concept of y-intercept is used in various real-world applications, such as:
- Physics: The y-intercept of a function can represent the initial velocity of an object.
- Economics: The y-intercept of a function can represent the initial cost of a product.
- Engineering: The y-intercept of a function can represent the initial value of a system.
Conclusion
Q: What is the y-intercept of a function?
A: The y-intercept of a function is the point at which the graph of the function intersects the y-axis. In other words, it is the value of the function when the input (or x-value) is equal to zero.
Q: How do I determine the y-intercept of a quadratic function?
A: To determine the y-intercept of a quadratic function, you need to substitute x = 0 into the function and solve for y.
Q: What is the formula for finding the y-intercept of a quadratic function?
A: The formula for finding the y-intercept of a quadratic function is f(0) = a(0)^2 + b(0) + c, where a, b, and c are the coefficients of the quadratic function.
Q: How do I simplify the expression when finding the y-intercept of a quadratic function?
A: When simplifying the expression, you need to evaluate the exponent and the multiplication. For example, if the expression is f(0) = 0^2 + 4(0) - 5, you would simplify it to f(0) = 0 + 0 - 5.
Q: What is the significance of the y-intercept of a quadratic function?
A: The y-intercept of a quadratic function is significant because it represents the initial value of the function. In physics, the y-intercept of a function can represent the initial velocity of an object. In economics, the y-intercept of a function can represent the initial cost of a product.
Q: Can the y-intercept of a quadratic function be negative?
A: Yes, the y-intercept of a quadratic function can be negative. For example, if the function is f(x) = x^2 + 4x - 5, the y-intercept would be -5.
Q: Can the y-intercept of a quadratic function be zero?
A: Yes, the y-intercept of a quadratic function can be zero. For example, if the function is f(x) = x^2 - 4x, the y-intercept would be 0.
Q: How do I determine the y-intercept of a quadratic function with a negative coefficient?
A: To determine the y-intercept of a quadratic function with a negative coefficient, you need to follow the same steps as before. For example, if the function is f(x) = -x^2 + 4x - 5, you would substitute x = 0 into the function and simplify the expression.
Q: Can the y-intercept of a quadratic function be a fraction?
A: Yes, the y-intercept of a quadratic function can be a fraction. For example, if the function is f(x) = x^2 + 4x - 1/2, the y-intercept would be -1/2.
Q: How do I determine the y-intercept of a quadratic function with a fractional coefficient?
A: To determine the y-intercept of a quadratic function with a fractional coefficient, you need to follow the same steps as before. For example, if the function is f(x) = x^2 + 4x - 1/2, you would substitute x = 0 into the function and simplify the expression.
Conclusion
In conclusion, determining the y-intercept of a quadratic function is a crucial concept in mathematics. By following the steps outlined in this article, you can determine the y-intercept of any quadratic function, regardless of the coefficients or the complexity of the function.