Which Quadratic Inequality Does The Graph Below Represent?
Welcome to our tutorial on quadratic inequalities. In this article, we will explore how to determine the quadratic inequality represented by a given graph. This will be an essential skill that can help you solve quadratic equations and inequalities. We will use an example graph to illustrate the process.
The Example Graph
Let us consider the graph below:
This is a typical quadratic graph. It is a parabola and has a vertex at (-2, 4). The graph crosses the x-axis at (-5, 0) and (1, 0). It also crosses the y-axis at (0, 8).
Determining the Quadratic Inequality
To determine the quadratic inequality represented by the graph, we need to consider the direction of the parabola. If the parabola opens upwards, the quadratic coefficient is positive, and the inequality is greater than or equal to. If the parabola opens downwards, the quadratic coefficient is negative, and the inequality is less than or equal to.
In our example, the parabola opens upwards, so the quadratic coefficient is positive. We can write the quadratic function in standard form as:
f(x) = a(x - h)^2 + k
where a is the quadratic coefficient, (h, k) is the vertex, and x is the independent variable.
Substituting the given vertex of (-2, 4), we get:
f(x) = a(x + 2)^2 + 4
Next, we need to determine the value of a. To do this, we can use one of the points where the graph crosses the x-axis. Let us use the point (-5, 0). Substituting these values, we get:
0 = a(-5 + 2)^2 + 4
Simplifying, we get:
9a = -4
Therefore, a = -4/9.
Substituting this value of a in the quadratic function, we get:
f(x) = (-4/9)(x + 2)^2 + 4
This is the quadratic function represented by the given graph. To determine the quadratic inequality, we need to consider the y-values of the graph. Since the vertex is at (h, k), the minimum value of y is k. In our example, the vertex has a y-value of 4. Therefore, the quadratic inequality is:
f(x) ≥ 4
Verification
We can verify our answer by checking the points on the graph. For any point above the parabola, the y-value will be greater than or equal to 4. For any point below the parabola, the y-value will be less than or equal to 4.
For example, let us consider the point (-3, 5). Substituting these values in the quadratic function, we get:
f(-3) = (-4/9)(-3 + 2)^2 + 4 = 5
Therefore, the y-value of the point (-3, 5) is greater than or equal to 4, which is consistent with the quadratic inequality f(x) ≥ 4.
Conclusion
In conclusion, we have learned how to determine the quadratic inequality represented by a given graph. We used a typical quadratic graph as an example and illustrated the steps involved in determining the quadratic inequality. This is an important skill that can help you solve quadratic equations and inequalities. We hope you found this tutorial useful and informative.
Remember, practice makes perfect!
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