Solving Quadratic Inequalities: Vertex Form Explained

Hey guys! Let's dive into the world of quadratic inequalities. We're going to break down how to tackle these problems, especially when they involve a quadratic function's vertex form. This is super helpful, and I promise, we'll make it as easy as possible. Today, we're focusing on a specific scenario: a quadratic inequality that includes all the values less than or equal to a quadratic function. This function has a vertex at the point $(-9, -27)$ and magically passes through another point on its boundary, specifically at $\left(1, 39 \frac{2}{3}\right)$. Our mission? To figure out which inequality, written in vertex form, perfectly represents this situation.

Understanding Quadratic Inequalities

Alright, first things first: what exactly is a quadratic inequality? Well, it's pretty much an inequality that includes a quadratic function. Remember those parabolas we loved (or maybe didn't love!) in algebra class? That's what we're dealing with here. A quadratic function is essentially a function that can be written in the form of $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not zero. Graphically, these functions produce those lovely U-shaped curves (parabolas). Now, an inequality comes into play when we compare this quadratic function to something else, often a constant or another function, using symbols like <, >, ≤, or ≥. For example, $x^2 + 2x - 3 < 0$ is a quadratic inequality. The solutions to these inequalities aren't just single points, like with equations, but rather entire ranges of x-values. These ranges represent the parts of the x-axis where the function satisfies the inequality. This could be where the parabola dips below the x-axis, stays above it, or includes the points on the curve itself, depending on the inequality symbol.

So, what does it mean when we're dealing with 'less than or equal to'? In our scenario, we're looking for the x-values where the parabola's values are either below or at the same level as a certain boundary. The 'equal to' part is key because it tells us that any point on the curve is included in the solution. This is a super important concept because it changes the way we interpret the graph. If we had a 'less than' inequality, we'd exclude the points on the parabola. But, since we have 'less than or equal to', we're going to keep those points in our solution set. Think of it like a shaded region on the graph that includes the actual curve itself, not just the area below it. Understanding these subtleties is crucial when working with quadratic inequalities because they dictate the types of solutions we'll be dealing with and the visual representation of our answer on a number line or coordinate plane. This visual representation is fundamental for quickly grasping and communicating the solution set. It allows us to readily identify the intervals that satisfy the inequality, providing a clear and concise understanding of the problem's solution.

The Power of Vertex Form

Now, let's talk about the vertex form of a quadratic equation. This is where things get a whole lot easier, especially when you're given the vertex of the parabola. The vertex form is expressed as $f(x) = a(x - h)^2 + k$, where (h, k) are the coordinates of the vertex, and 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The vertex form is incredibly convenient because it directly provides you with the vertex's coordinates and makes it super simple to rewrite the equation. Using the vertex form is like having a secret weapon when tackling these problems. Given our scenario, we already know the vertex is at $(-9, -27)$. This immediately gives us $h = -9$ and $k = -27$. Our equation looks like this: $f(x) = a(x - (-9))^2 - 27$, which simplifies to $f(x) = a(x + 9)^2 - 27$. Notice how knowing the vertex just makes it plug-and-play? We only have to solve for 'a' to completely define the function. The 'a' value is also critically important because it defines the shape of the parabola, whether it opens upward (a positive value) or downward (a negative value). It also determines how wide or narrow the parabola is. Knowing the value of 'a' gives you a complete picture of the parabola’s behavior. Furthermore, when working with inequalities, we need to know the 'a' value to accurately determine the solution set. The sign of 'a' tells us the direction of the inequality’s solution and is crucial in understanding whether the solution lies inside or outside the parabola's curve. With our knowledge of the vertex form and the given vertex coordinates, we can accurately and efficiently define the quadratic inequality and find the correct solution.

Finding the Value of 'a'

We know the vertex and we have the general form, but we still have one missing piece: the value of 'a'. This is where the additional point $\left(1, 39 \frac{2}{3}\right)$ comes in handy. Remember, this point lies on the boundary of our inequality. This means that if we plug in x = 1 into our equation, we should get the corresponding y-value, which is $39 \frac{2}{3}$. So, let's substitute the coordinates of this point into the equation: $39 \frac{2}{3} = a(1 + 9)^2 - 27$. Convert $39 \frac{2}{3}$ to an improper fraction, $\frac{119}{3} = a(10)^2 - 27$. Simplify further: $\frac{119}{3} = 100a - 27$. Now, solve for 'a'. First, add 27 to both sides of the equation: $\frac{119}{3} + 27 = 100a$. To add the fraction and the whole number, convert 27 into a fraction with a denominator of 3: $27 = \frac{81}{3}$. So, our equation becomes $\frac{119}{3} + \frac{81}{3} = 100a$. Combining the fractions gives us $\frac{200}{3} = 100a$. Finally, divide both sides by 100 to isolate 'a': $a = \frac{200}{3} / 100$. This simplifies to $a = \frac{2}{3}$. With the value of 'a' in hand, we have the complete quadratic function: $f(x) = \frac{2}{3}(x + 9)^2 - 27$. This value is crucial because it helps us define how the inequality behaves across the entire graph. The sign of 'a' indicates the direction of the parabola (upward or downward), and its value defines the parabola’s width. Accurately determining 'a' is essential for writing the correct quadratic inequality and correctly finding its solution. Understanding and correctly calculating 'a' is a cornerstone for solving and understanding any quadratic inequality problems.

Constructing the Inequality

Okay, we're in the home stretch now. We have the vertex form of the quadratic function: $f(x) = \frac{2}{3}(x + 9)^2 - 27$. We are looking for the values where the function is less than or equal to this function. This translates directly into our inequality. Since we want values less than or equal to the function, we use the ≤ symbol. Thus, the correct inequality is: $y ≤ \frac{2}{3}(x + 9)^2 - 27$. Notice how this inequality reflects our problem's constraints: all the y-values that are either below the parabola or exactly on the parabola itself are included in the solution set. To visualize this, imagine the entire area under the parabola, including the line itself, is shaded. When plotting this inequality on a graph, the solution set consists of all points (x, y) that satisfy this condition. This forms a region of the Cartesian plane which directly represents the range of values that make the inequality true. The use of the '≤' symbol is a critical element, it signifies that the solutions must include the values along the parabola's boundary. This shows that the solution to our quadratic inequality includes both the area below the curve and the parabola itself. Understanding and constructing the inequality is vital for interpreting the solution in different contexts.

Conclusion

So there you have it! The quadratic inequality that represents the situation is $y ≤ \frac{2}{3}(x + 9)^2 - 27$. We've taken a journey from understanding quadratic inequalities, using vertex form, finding the crucial 'a' value, and finally, constructing the inequality itself. Knowing the vertex form is a game-changer because it gives you a direct link to the vertex's location and lets you quickly write the equation. Remember, always double-check whether you need to include the boundary (as in our 'less than or equal to' scenario) because this determines the solution set. Keep practicing, and you'll be acing these problems in no time! Keep in mind how important each piece of information is: the vertex, the point on the boundary, and the inequality symbol. Putting these elements together allows you to solve a wide variety of quadratic inequality problems confidently.