Find X-Intercepts F(x) = (1/2)x^2 + X - 9 On The Negative Axis
Finding where a graph crosses the x-axis is a fundamental concept in algebra, particularly when dealing with quadratic functions. In this article, we will explore how to determine the points where the graph of the quadratic function f(x) = (1/2)x^2 + x - 9 intersects the negative x-axis. This involves understanding the significance of x-intercepts, utilizing the quadratic formula, and interpreting the results in the context of ordered pairs.
Understanding X-Intercepts and Quadratic Functions
X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function intersects the x-axis. At these points, the value of the function, f(x), is equal to zero. For a quadratic function of the form f(x) = ax^2 + bx + c, the x-intercepts represent the solutions to the quadratic equation ax^2 + bx + c = 0. These solutions can be found using various methods, including factoring, completing the square, or the quadratic formula. Understanding the x-intercepts is vital for analyzing the behavior and characteristics of the quadratic function, such as its concavity, vertex, and symmetry.
The quadratic function f(x) = (1/2)x^2 + x - 9 represents a parabola, which is a U-shaped curve. The coefficient of the x^2 term, which is 1/2 in this case, determines the parabola's concavity. Since the coefficient is positive, the parabola opens upwards. The x-intercepts of this parabola are the points where the curve crosses the x-axis, and they provide valuable information about the function's roots and behavior. Determining the intervals between which these x-intercepts lie is crucial for understanding the function's graph and its relationship to the x-axis. By finding the x-intercepts, we can pinpoint the exact locations where the function's value changes from negative to positive or vice versa, which is essential for various applications in mathematics and real-world scenarios.
Applying the Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions (roots) of any quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the quadratic function is f(x) = (1/2)x^2 + x - 9. To find the x-intercepts, we need to solve the equation (1/2)x^2 + x - 9 = 0. By identifying the coefficients, we have a = 1/2, b = 1, and c = -9. Substituting these values into the quadratic formula, we get:
x = (-1 ± √(1^2 - 4(1/2)(-9))) / (2(1/2))
Simplifying the expression under the square root:
x = (-1 ± √(1 + 18)) / 1
x = (-1 ± √19) / 1
Thus, the two solutions for x are:
x₁ = -1 + √19
x₂ = -1 - √19
These two values represent the x-coordinates of the points where the graph of the function intersects the x-axis. The quadratic formula provides a systematic way to find these x-intercepts, regardless of whether the quadratic equation can be easily factored or not. By calculating these x-intercepts, we gain critical information about the function's behavior, such as where it crosses the x-axis and the intervals where the function's value is positive or negative. The quadratic formula's ability to handle any quadratic equation makes it an indispensable tool in algebra and calculus.
Calculating the X-Intercepts and Identifying the Interval
Now that we have the solutions in terms of the square root of 19, we need to approximate these values to determine the intervals between which the graph crosses the negative x-axis. We know that √19 is between √16 (which is 4) and √25 (which is 5). A closer approximation is √19 ≈ 4.36.
Using this approximation, we can find the approximate values of the x-intercepts:
x₁ = -1 + √19 ≈ -1 + 4.36 ≈ 3.36
x₂ = -1 - √19 ≈ -1 - 4.36 ≈ -5.36
We are interested in the negative x-axis crossing, so we focus on x₂ ≈ -5.36. This x-intercept lies between the ordered pairs (-6, 0) and (-5, 0). The value of -5.36 indicates that the graph of the function crosses the x-axis at a point between -6 and -5 on the x-axis. This means that the function changes its sign from negative to positive or vice versa within this interval. Understanding the approximate values of the x-intercepts helps us visualize the graph's position relative to the x-axis and identify the specific regions where the function's behavior changes. By pinpointing the interval containing the negative x-intercept, we gain valuable insights into the function's properties and its interactions with the x-axis.
Determining the Correct Ordered Pairs
The question asks between which two ordered pairs the graph of the function crosses the negative x-axis. We have found that the negative x-intercept is approximately -5.36. This value lies between -6 and -5.
Therefore, the graph crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0). These ordered pairs represent points on the x-axis where the y-coordinate is zero. The x-intercept of -5.36 falls within the range defined by these two points, indicating that the parabola intersects the x-axis somewhere between these two locations. Identifying the correct ordered pairs is crucial for accurately describing the function's behavior and its relationship to the coordinate plane. By understanding the significance of x-intercepts and their corresponding ordered pairs, we can effectively analyze and interpret the graph of the quadratic function.
Conclusion
In summary, by applying the quadratic formula to the function f(x) = (1/2)x^2 + x - 9, we found the x-intercepts to be approximately 3.36 and -5.36. The negative x-intercept, -5.36, indicates that the graph crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0). This process demonstrates the importance of the quadratic formula in finding the roots of quadratic equations and understanding the behavior of quadratic functions. The ability to determine x-intercepts is essential for analyzing graphs, solving equations, and applying these concepts in various mathematical and real-world contexts.
By mastering the techniques involved in finding x-intercepts and interpreting their significance, we can gain a deeper understanding of quadratic functions and their applications. The quadratic formula serves as a fundamental tool in this process, allowing us to solve for the roots of any quadratic equation and analyze the behavior of the corresponding parabolic graph. Understanding the relationship between the equation, its solutions, and the graph is crucial for success in algebra and calculus.
Repair Input Keyword: Between which two ordered pairs does the graph of f(x) = (1/2)x^2 + x - 9 cross the negative x-axis?
Title: Find X-Intercepts f(x) = (1/2)x^2 + x - 9 on the Negative Axis
