Graphing F(x) = -x² + 6x - 9 A Step-by-Step Analysis

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Hey guys! Ever stumbled upon a quadratic function and felt a bit lost trying to visualize its graph? Well, you're not alone! Quadratic functions, with their signature parabolic curves, might seem intimidating at first, but trust me, once you grasp the key concepts, they become super interesting and predictable. Today, we're diving deep into the function f(x) = -x² + 6x - 9 to explore its graph and understand why it looks the way it does. We'll break down each component of the function, analyze its impact on the graph's shape and orientation, and ultimately determine the correct graphical representation from the given options. So, buckle up and let's embark on this mathematical journey together!

Understanding Quadratic Functions

Before we zoom in on our specific function, let's take a step back and refresh our understanding of quadratic functions in general. A quadratic function is essentially a polynomial function of degree two, which means the highest power of the variable x is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero (otherwise, it wouldn't be a quadratic function!). The graph of a quadratic function is always a parabola, a U-shaped curve that can open either upwards or downwards.

The coefficient a plays a crucial role in determining the parabola's orientation. If a is positive, the parabola opens upwards, resembling a smiling face. This is because as x moves away from the vertex (the turning point of the parabola), the x² term dominates, and the positive a ensures that f(x) increases. On the other hand, if a is negative, the parabola opens downwards, resembling a frowning face. In this case, the negative a causes f(x) to decrease as x moves away from the vertex. This is a fundamental concept to remember, guys, as it's the first clue in visualizing the graph of any quadratic function. The magnitude of a also affects the "width" of the parabola; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider one. Understanding this interplay between a and the parabola's shape is key to quickly sketching a quadratic function's graph.

The other coefficients, b and c, also influence the parabola's position and shape. The coefficient b is related to the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. The x-coordinate of the vertex lies on this axis of symmetry, and its value is given by -b / 2a. So, b helps determine the horizontal position of the parabola. The constant term c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. This is because when x = 0, f(x) = a(0)² + b(0) + c = c. The y-intercept gives us a direct visual anchor point on the graph. Together, a, b, and c orchestrate the parabola's dance on the coordinate plane, dictating its orientation, position, and overall shape. Mastering the roles of these coefficients empowers us to predict and interpret quadratic graphs with confidence. Remember, guys, practice makes perfect, so keep exploring different quadratic functions and observing how their coefficients influence their graphs!

Analyzing f(x) = -x² + 6x - 9

Now that we've recapped the general characteristics of quadratic functions, let's focus on our specific function: f(x) = -x² + 6x - 9. The first thing we should do, guys, is identify the coefficients a, b, and c. By comparing this function to the general form f(x) = ax² + bx + c, we can easily see that a = -1, b = 6, and c = -9. This simple identification is the key to unlocking a wealth of information about the graph.

Let's start with a. Since a = -1, which is negative, we immediately know that the parabola opens downwards. This eliminates option A from the choices right away, because option A describes a parabola opening upwards. Remember, guys, a negative a always means a downward-facing parabola, a fundamental rule in quadratic function graphing. This seemingly small piece of information significantly narrows down our options and gets us closer to the correct answer. Now, let's dig deeper and see what else we can uncover.

Next, let's determine the vertex of the parabola. The vertex is the turning point of the parabola, either the highest point (if it opens downwards) or the lowest point (if it opens upwards). The x-coordinate of the vertex is given by the formula -b / 2a. In our case, this is -(6) / (2 * -1) = -6 / -2 = 3. So, the x-coordinate of the vertex is 3. To find the y-coordinate, we substitute this value back into the function: f(3) = -(3)² + 6(3) - 9 = -9 + 18 - 9 = 0. Therefore, the vertex of our parabola is at the point (3, 0). This is a crucial piece of information, guys, as it pinpoints the exact location of the parabola's peak. Knowing the vertex helps us visualize the graph more accurately and discard options that don't align with this key point.

The fact that the y-coordinate of the vertex is 0 also tells us something important: the vertex lies on the x-axis. This means the parabola touches the x-axis at exactly one point. Furthermore, since the parabola opens downwards, it will be entirely below the x-axis, except for the vertex itself. This gives us a clear picture of the parabola's position and orientation in the coordinate plane. We can also find the y-intercept by setting x = 0 in the function. This gives us f(0) = -(0)² + 6(0) - 9 = -9. So, the y-intercept is at the point (0, -9). This point, along with the vertex, provides us with two key reference points for sketching the graph. By carefully analyzing the coefficients and calculating these key features, we're building a comprehensive understanding of the parabola's behavior and appearance. Remember, guys, the more we dissect the function, the clearer its graph becomes!

Determining the Correct Option

Alright, guys, we've done some serious analysis of the function f(x) = -x² + 6x - 9. We know the parabola opens downwards because a = -1. We've calculated the vertex to be (3, 0), which lies on the x-axis. And we've found the y-intercept to be (0, -9). With all this information in hand, we can now confidently determine the correct option from the given choices.

Let's revisit the options:

  • A. Uma parábola com abertura voltada para cima. (A parabola opening upwards)
  • B. Uma parábola com abertura voltada para baixo. (A parabola opening downwards)
  • C. Uma reta crescente (An increasing line)
  • D. Uma hipérbole com uma parte no primeiro quadrante e outra no (A hyperbola with one part in the first quadrant and another in)

We already eliminated option A because we know the parabola opens downwards. Option C can be ruled out because the graph of a quadratic function is a parabola, not a straight line. Option D describes a hyperbola, which is a different type of conic section altogether. So, by the process of elimination, we're left with option B. But let's confirm that option B indeed matches our analysis.

Option B states: Uma parábola com abertura voltada para baixo. (A parabola opening downwards). This aligns perfectly with our finding that a = -1, which indicates a downward-opening parabola. Furthermore, the vertex at (3, 0) and the y-intercept at (0, -9) confirm that the parabola is positioned below the x-axis, touching it only at the vertex. Therefore, option B accurately describes the graph of f(x) = -x² + 6x - 9. Woohoo! We nailed it, guys!

Final Answer and Key Takeaways

So, the correct answer is B. Uma parábola com abertura voltada para baixo. (A parabola opening downwards). We arrived at this conclusion by systematically analyzing the function, identifying the coefficients, determining the orientation, and calculating the vertex and y-intercept. This step-by-step approach is crucial for tackling any quadratic function graphing problem.

Let's recap the key takeaways from this exploration, guys:

  • The coefficient a determines the parabola's orientation: positive a means it opens upwards, and negative a means it opens downwards.
  • The vertex is the turning point of the parabola, and its x-coordinate is given by -b / 2a.
  • The y-intercept is the point where the parabola intersects the y-axis, and it's found by setting x = 0 in the function.
  • By analyzing these key features, we can accurately visualize and graph quadratic functions.

Remember, guys, practice is the key to mastering quadratic functions. The more you work with them, the more intuitive they become. So, keep exploring, keep analyzing, and keep graphing! You've got this!

Further Exploration

To solidify your understanding, try graphing other quadratic functions with different coefficients. See how changing the values of a, b, and c affects the parabola's shape, position, and orientation. You can also explore the concept of the discriminant (b² - 4ac), which tells us about the number of real roots (x-intercepts) of the quadratic equation. Happy graphing, guys! And remember, math can be fun when we break it down and explore it together!