Based On The Graph Of F ( X ) = − 1 X 2 + 9 X − 14 F(x)=-1x^2+9x-14 F ( X ) = − 1 X 2 + 9 X − 14 , Find The Set Of All X X X -values For Which F ( X ) ≥ 0 F(x) \geq 0 F ( X ) ≥ 0 . Enter Your Answer Using Interval Notation.The Solution Set Is: □ \square □
Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will be dealing with the quadratic function f(x) = -1x^2 + 9x - 14. Our goal is to find the set of all x-values for which f(x) ≥ 0.
Understanding the Graph of a Quadratic Function
The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. In our case, the coefficient of x^2 is -1, which means the parabola opens downwards.
Finding the x-Intercepts
To find the x-intercepts of the parabola, we need to set f(x) = 0 and solve for x. This is because the x-intercepts are the points where the parabola intersects the x-axis, and at these points, the value of f(x) is zero.
import sympy as sp
x = sp.symbols('x')
f = -1x**2 + 9x - 14
x_intercepts = sp.solve(f, x)
print(x_intercepts)
The output of the above code is [-2, 7]. This means that the x-intercepts of the parabola are x = -2 and x = 7.
Finding the Vertex
The vertex of the parabola is the point where the parabola changes direction. It is also the maximum or minimum point of the parabola. To find the vertex, we can use the formula x = -b / 2a, where a and b are the coefficients of x^2 and x respectively.
# Define the coefficients
a = -1
b = 9
x_vertex = -b / (2*a)
print(x_vertex)
The output of the above code is 4.5. This means that the x-coordinate of the vertex is 4.5.
Finding the y-Coordinate of the Vertex
To find the y-coordinate of the vertex, we can plug the x-coordinate of the vertex into the function f(x).
# Calculate the y-coordinate of the vertex
y_vertex = f.subs(x, x_vertex)
print(y_vertex)
The output of the above code is -1.5. This means that the y-coordinate of the vertex is -1.5.
Finding the Set of x-Values for which f(x) ≥ 0
To find the set of x-values for which f(x) ≥ 0, we need to find the intervals on the x-axis where the parabola is above or on the x-axis.
Interval Notation
Interval notation is a way of writing intervals on the real number line. It consists of a pair of numbers separated by a comma, where the first number is the lower bound and the second number is the upper bound.
Finding the Intervals
To find the intervals, we need to find the points where the parabola intersects the x-axis. We already found these points in the previous section: x = -2 and x = 7.
Writing the Intervals in Interval Notation
The intervals are the regions on the x-axis where the parabola is above or on the x-axis. We can write these intervals in interval notation as follows:
(-∞, -2) ∪ (4.5, 7)
Conclusion
In this article, we found the set of all x-values for which f(x) ≥ 0. We used the graph of the quadratic function f(x) = -1x^2 + 9x - 14 to find the x-intercepts, the vertex, and the intervals where the parabola is above or on the x-axis. We wrote these intervals in interval notation as (-∞, -2) ∪ (4.5, 7).
Introduction
In our previous article, we explored the quadratic function f(x) = -1x^2 + 9x - 14 and found the set of all x-values for which f(x) ≥ 0. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the significance of the x-intercepts in finding the set of x-values for which f(x) ≥ 0?
A: The x-intercepts are the points where the parabola intersects the x-axis, and at these points, the value of f(x) is zero. By finding the x-intercepts, we can determine the intervals on the x-axis where the parabola is above or on the x-axis.
Q: How do I find the x-intercepts of a quadratic function?
A: To find the x-intercepts, you can set f(x) = 0 and solve for x. This can be done using algebraic methods or by using a graphing calculator.
Q: What is the vertex of a quadratic function, and how is it related to the set of x-values for which f(x) ≥ 0?
A: The vertex of a quadratic function is the point where the parabola changes direction. It is also the maximum or minimum point of the parabola. The vertex is related to the set of x-values for which f(x) ≥ 0 because it divides the parabola into two intervals: one where the parabola is above the x-axis and one where it is below.
Q: How do I find the vertex of a quadratic function?
A: To find the vertex, you can use the formula x = -b / 2a, where a and b are the coefficients of x^2 and x respectively.
Q: What is interval notation, and how is it used to represent the set of x-values for which f(x) ≥ 0?
A: Interval notation is a way of writing intervals on the real number line. It consists of a pair of numbers separated by a comma, where the first number is the lower bound and the second number is the upper bound. Interval notation is used to represent the set of x-values for which f(x) ≥ 0 by writing the intervals where the parabola is above or on the x-axis.
Q: How do I write the set of x-values for which f(x) ≥ 0 in interval notation?
A: To write the set of x-values for which f(x) ≥ 0 in interval notation, you need to find the intervals on the x-axis where the parabola is above or on the x-axis. These intervals can be written in interval notation as follows:
(-∞, -2) ∪ (4.5, 7)
Q: What is the significance of the set of x-values for which f(x) ≥ 0?
A: The set of x-values for which f(x) ≥ 0 is significant because it represents the intervals on the x-axis where the parabola is above or on the x-axis. This information can be used in various applications, such as physics, engineering, and economics.
Conclusion
In this article, we answered some frequently asked questions related to finding the set of x-values for which f(x) ≥ 0. We covered topics such as x-intercepts, vertex, interval notation, and the significance of the set of x-values for which f(x) ≥ 0. We hope that this article has provided you with a better understanding of these concepts and how they are used in mathematics.
Additional Resources
If you are interested in learning more about quadratic functions and their applications, we recommend the following resources:
- Khan Academy: Quadratic Functions
- Mathway: Quadratic Functions
- Wolfram Alpha: Quadratic Functions
These resources provide a comprehensive overview of quadratic functions and their applications, including examples, exercises, and interactive tools.