Analyzing The Quadratic Function F(x) = -3x² + 12x - 4
In the realm of mathematics, quadratic functions hold a significant position, often serving as the foundation for more complex mathematical concepts. These functions, characterized by their parabolic graphs, possess unique properties that make them indispensable tools in various fields, from physics to economics. In this comprehensive exploration, we will delve into the intricacies of a specific quadratic function, f(x) = -3x² + 12x - 4, unraveling its characteristics and uncovering its hidden potential.
a. Determining the Existence of Minimum or Maximum Value Without Graphing
The very first question that arises when encountering a quadratic function is whether it possesses a minimum or maximum value. This crucial aspect determines the overall shape and behavior of the function's graph. To answer this question without resorting to graphing, we can leverage the power of the quadratic function's coefficient of the x² term.
In our case, the quadratic function is f(x) = -3x² + 12x - 4. Notice that the coefficient of the x² term is -3, a negative value. This negative coefficient holds the key to determining the existence of a maximum value. When the coefficient of the x² term is negative, the parabola opens downwards, resembling an upside-down U. This downward-opening nature implies that the function reaches a peak, a maximum point, beyond which the function values decrease. Conversely, if the coefficient of the x² term were positive, the parabola would open upwards, indicating the presence of a minimum value.
Therefore, without even sketching the graph, we can confidently conclude that the function f(x) = -3x² + 12x - 4 has a maximum value. This conclusion stems directly from the negative coefficient of the x² term, which dictates the downward-opening nature of the parabola.
b. Finding the Minimum or Maximum Value and Its Location
Having established the existence of a maximum value for the function f(x) = -3x² + 12x - 4, our next objective is to pinpoint this maximum value and determine the x-coordinate at which it occurs. This quest involves employing the powerful technique of completing the square, a method that transforms the quadratic function into a more revealing form.
The process of completing the square begins by factoring out the coefficient of the x² term, which is -3 in our case, from the first two terms of the function. This yields:
f(x) = -3(x² - 4x) - 4
Next, we focus on the expression within the parentheses, x² - 4x. To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is -4, so half of it is -2, and the square of -2 is 4. Adding and subtracting 4 within the parentheses, we get:
f(x) = -3(x² - 4x + 4 - 4) - 4
Now, we can rewrite the expression within the parentheses as a perfect square:
f(x) = -3((x - 2)² - 4) - 4
Distributing the -3, we obtain:
f(x) = -3(x - 2)² + 12 - 4
Simplifying, we arrive at the vertex form of the quadratic function:
f(x) = -3(x - 2)² + 8
The vertex form, f(x) = a(x - h)² + k, provides invaluable information about the quadratic function. The vertex of the parabola, which represents the maximum or minimum point, is located at the coordinates (h, k). In our case, the vertex is at (2, 8). Since the coefficient a is negative (-3), the parabola opens downwards, confirming that the vertex represents the maximum point. Therefore, the maximum value of the function is 8, and it occurs at x = 2.
In essence, completing the square has allowed us to unveil the function's maximum value and its corresponding x-coordinate. The vertex form of the quadratic function serves as a powerful tool for extracting this information, providing a clear understanding of the function's behavior.
c. Unveiling the Function's Identity: Intercepts and Axis of Symmetry
Having determined the maximum value and its location, our exploration of the function f(x) = -3x² + 12x - 4 continues as we delve into its intercepts and axis of symmetry. These elements further illuminate the function's characteristics and graphical representation.
Unearthing the Intercepts: Where the Parabola Meets the Axes
Intercepts are the points where the graph of the function intersects the x-axis and the y-axis. The x-intercepts, also known as roots or zeros, are the solutions to the equation f(x) = 0. To find the x-intercepts, we set the function equal to zero and solve for x:
-3x² + 12x - 4 = 0
This quadratic equation can be solved using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a = -3, b = 12, and c = -4. Plugging these values into the quadratic formula, we get:
x = (-12 ± √(12² - 4(-3)(-4))) / 2(-3)
x = (-12 ± √(144 - 48)) / -6
x = (-12 ± √96) / -6
x = (-12 ± 4√6) / -6
Simplifying, we obtain two x-intercepts:
x₁ = (6 - 2√6) / 3 ≈ 0.38 x₂ = (6 + 2√6) / 3 ≈ 3.63
These x-intercepts represent the points where the parabola crosses the x-axis. To find the y-intercept, we set x = 0 in the function:
f(0) = -3(0)² + 12(0) - 4
f(0) = -4
Thus, the y-intercept is -4, indicating the point where the parabola intersects the y-axis.
The Axis of Symmetry: A Mirror for the Parabola
The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex of the parabola. The equation of the axis of symmetry is given by:
x = -b / 2a
In our case, a = -3 and b = 12, so the axis of symmetry is:
x = -12 / 2(-3)
x = 2
This vertical line, x = 2, acts as a mirror, reflecting one half of the parabola onto the other. The axis of symmetry provides a crucial reference point for understanding the parabola's symmetry and overall shape.
Synthesis: A Comprehensive Understanding
By meticulously examining the function f(x) = -3x² + 12x - 4, we have unveiled its key characteristics. We determined the existence of a maximum value, pinpointed its location, and calculated the intercepts and axis of symmetry. This comprehensive analysis has provided us with a deep understanding of the function's behavior and graphical representation. The negative coefficient of the x² term dictated the downward-opening nature of the parabola, leading to a maximum value. Completing the square allowed us to express the function in vertex form, revealing the maximum value and its corresponding x-coordinate. The quadratic formula enabled us to find the x-intercepts, while setting x = 0 yielded the y-intercept. Finally, the formula x = -b / 2a provided the equation of the axis of symmetry, highlighting the parabola's symmetrical nature.
In conclusion, this exploration has demonstrated the power of mathematical tools and techniques in unraveling the secrets of quadratic functions. By understanding the interplay between coefficients, intercepts, and the axis of symmetry, we can gain a profound appreciation for these fundamental mathematical entities.