Which Quadratic Inequality Does The Graph Below Represent? Y = 2X^2 - 3
Quadratic inequalities are a fundamental concept in mathematics. They are used to represent the relationship between variables in an equation, and they have many applications in real-life situations. In this article, we will explore which quadratic inequality does the graph below represent, which is y = 2x^2 - 3. We will break down the problem step by step, so you can understand the process and apply it to other quadratic inequalities.
Understanding Quadratic Inequalities
Before we dive into the problem, let's review what quadratic inequalities are. A quadratic inequality is an inequality that involves a quadratic function, which is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants. Quadratic inequalities are represented on a graph as a parabola, which is a U-shaped curve. The direction of the curve depends on the sign of the coefficient a. If a is positive, the parabola opens upward, and if a is negative, it opens downward.
Graphing the Quadratic Function
To determine which quadratic inequality the graph represents, we must first graph the function y = 2x^2 - 3. We can do this by plotting points or by using the vertex form of the quadratic function, which is f(x) = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.
To find the vertex of the parabola, we can use the formula h = -b/2a and k = f(h). In this case, a = 2, b = 0, and c = -3, so h = 0 and k = -3. Therefore, the vertex of the parabola is (0,-3).
Next, we can plot a few points to sketch the graph of the quadratic function. We can choose any values of x and substitute them into the function to find the corresponding values of y. For example, when x = -1, y = -1, and when x = 1, y = -1. Therefore, we can plot the points (-1,-1) and (1,-1) on the graph.
Using these points and the vertex, we can sketch the graph of the quadratic function as a U-shaped curve that opens upward. The graph intersects the x-axis at two points, which are approximately x = ±1.22. These points are called the roots or zeros of the quadratic function.
Identifying the Quadratic Inequality
Now that we have graphed the quadratic function y = 2x^2 - 3, we can use the graph to identify the quadratic inequality that it represents. To do this, we need to look at the shape and position of the parabola.
First, we notice that the coefficient a is positive, which means that the parabola opens upward. This tells us that the quadratic inequality is greater than or equal to, which is represented by the symbol ≥.
Second, we notice that the vertex of the parabola is at the point (0,-3), which is below the x-axis. This tells us that the quadratic inequality is less than or equal to -3, which is represented by the symbol ≤ -3.
Therefore, the quadratic inequality that the graph represents is y ≥ 2x^2 - 3, or equivalently, 2x^2 - 3 ≤ y ≤ ∞.
Applications of Quadratic Inequalities
Quadratic inequalities have many applications in real-life situations, such as optimization problems, physics, and engineering. For example, a company may use quadratic inequalities to optimize their production process by minimizing costs and maximizing profits. A physicist may use quadratic inequalities to model the trajectory of a projectile, such as a baseball or a rocket. An engineer may use quadratic inequalities to design a bridge or a building that can withstand certain loads and forces.
Solving Quadratic Inequalities
In addition to graphing quadratic inequalities, we can also solve them algebraically. To do this, we need to find the roots of the quadratic function and determine the sign of the function in each interval between the roots.
For example, if we have the quadratic inequality y < 2x^2 - 3, we can rewrite it as 2x^2 - 3 > y, or equivalently, 2x^2 - y - 3 > 0. Then, we can find the roots of the quadratic function 2x^2 - y - 3 = 0 by using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a). This gives us x = (√(y + 13) ± 1)/2.
Next, we can test the sign of the function 2x^2 - y - 3 in each interval between the roots. We can choose any value of x in each interval and substitute it into the function to find the corresponding sign of the function. For example, if we choose x = 0, we get 2(0)^2 - y - 3 = -y - 3, which is negative if y > -3 and positive if y < -3. Therefore, the solution to the quadratic inequality is y < -3, or equivalently, -∞ < y < 2x^2 - 3.
Conclusion
Quadratic inequalities are an important concept in mathematics that have many applications in real-life situations. In this article, we explored which quadratic inequality does the graph y = 2x^2 - 3 represent. We graphed the quadratic function, identified the shape and position of the parabola, and determined the quadratic inequality. We also discussed some applications of quadratic inequalities and how to solve them algebraically. By understanding quadratic inequalities, you can develop your problem-solving skills and apply them to a wide range of fields.
Remember to practice graphing and solving quadratic inequalities to improve your understanding and proficiency in mathematics.
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