Analyzing Quadratic Function F(x) = 4(x^2 - 8x + 12)

Hey guys! Today, we're diving deep into analyzing the quadratic function F(x) = 4(x^2 - 8x + 12). We'll break down each statement, making sure we understand why it's true or false. Let's get started!

Graph Opens Upwards

The first statement we need to evaluate is whether the graph of the function F(x) = 4(x^2 - 8x + 12) opens upwards. To figure this out, we need to look at the coefficient of the x^2 term.

Quick refresher: A quadratic function is generally represented as f(x) = ax^2 + bx + c. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards.

In our function, F(x) = 4(x^2 - 8x + 12), we can distribute the 4 to get F(x) = 4x^2 - 32x + 48. Here, the coefficient of the x^2 term is 4, which is positive. Therefore, the graph of the function F(x) opens upwards. This makes the first statement TRUE.

When analyzing a quadratic function, the sign of the leading coefficient (the coefficient of the x^2 term) plays a crucial role in determining the parabola's orientation. A positive leading coefficient indicates that the parabola opens upwards, implying that the function has a minimum value. Conversely, a negative leading coefficient signifies that the parabola opens downwards, indicating a maximum value. This is because as x moves away from the vertex in either direction, the y-values increase if the parabola opens upwards and decrease if it opens downwards.

Moreover, the magnitude of the leading coefficient influences the "width" of the parabola. A larger absolute value of the leading coefficient results in a narrower parabola, while a smaller absolute value produces a wider parabola. Understanding these nuances helps in quickly visualizing and sketching the graph of a quadratic function. To further illustrate, consider two functions: f(x) = 2x^2 and g(x) = 0.5x^2. The graph of f(x) will be narrower compared to the graph of g(x), owing to the larger leading coefficient in f(x). This concept is vital in various applications, such as optimization problems, where determining the maximum or minimum value of a function is essential.

In addition to the leading coefficient, the other coefficients in the quadratic function also provide valuable information about its graph. The coefficient of the x term (b) and the constant term (c) influence the position of the vertex and the y-intercept, respectively. By analyzing these coefficients together, a comprehensive understanding of the quadratic function's behavior and its graphical representation can be obtained.

Graph Intersects the Line y = -18

Now, let's investigate whether the graph of F(x) intersects the line y = -18. To do this, we need to see if there's any real value of x for which F(x) = -18.

So, we set up the equation: 4(x^2 - 8x + 12) = -18.

First, divide both sides by 4: x^2 - 8x + 12 = -4.5

Next, add 4.5 to both sides: x^2 - 8x + 16.5 = 0

To determine if this quadratic equation has real solutions, we can use the discriminant (Δ), which is given by Δ = b^2 - 4ac.

In our equation, a = 1, b = -8, and c = 16.5. So, Δ = (-8)^2 - 4 * 1 * 16.5 = 64 - 66 = -2.

Since the discriminant (Δ) is negative, the quadratic equation has no real solutions. This means that the graph of F(x) does not intersect the line y = -18. Therefore, the second statement is FALSE.

To gain a deeper understanding, the discriminant (Δ) provides critical insights into the nature of the roots of a quadratic equation. A positive discriminant indicates that the equation has two distinct real roots, implying that the parabola intersects the x-axis at two different points. A discriminant of zero signifies that the equation has exactly one real root (a repeated root), meaning that the parabola touches the x-axis at its vertex. A negative discriminant, as in our case, implies that the equation has no real roots, which means that the parabola does not intersect the x-axis.

Understanding the discriminant is particularly useful in various practical applications. For instance, in physics, when analyzing the trajectory of a projectile, the discriminant can help determine whether the projectile will hit a target or not. Similarly, in engineering, when designing structures, the discriminant can be used to assess the stability of the structure under different loading conditions. Therefore, mastering the concept of the discriminant is essential for anyone working with quadratic equations in both theoretical and real-world contexts.

Furthermore, when the discriminant is negative, the roots of the quadratic equation are complex numbers. Complex numbers have both real and imaginary parts and are represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). Although complex roots do not correspond to points on the real number line (the x-axis), they provide valuable information about the behavior of the quadratic function in the complex plane. The study of complex roots is crucial in various advanced mathematical and engineering fields, such as signal processing, quantum mechanics, and control theory.

Graph Does Not Pass Through Quadrant 3

Finally, let's determine if the graph of F(x) does not pass through quadrant 3. Quadrant 3 is where both x and y values are negative. To check this, we need to analyze the behavior of the function for negative x values.

First, let's find the vertex of the parabola. The x-coordinate of the vertex is given by x = -b / 2a. In our function F(x) = 4x^2 - 32x + 48, a = 4 and b = -32.

So, x = -(-32) / (2 * 4) = 32 / 8 = 4.

Now, let's find the y-coordinate of the vertex by plugging x = 4 into the function: F(4) = 4(4^2 - 8*4 + 12) = 4(16 - 32 + 12) = 4(-4) = -16.

The vertex is at (4, -16), which is in quadrant 4.

Since the parabola opens upwards, and the vertex is in quadrant 4, the function will have positive y values for sufficiently large negative x values. To be sure, let's test a negative x value, say x = -1: F(-1) = 4((-1)^2 - 8*(-1) + 12) = 4(1 + 8 + 12) = 4(21) = 84.

Since F(-1) is positive, the graph will eventually enter quadrant 2 as we move towards negative infinity along the x-axis. Given that the vertex is in quadrant 4 and the parabola opens upwards, the graph will not pass through quadrant 3. Thus, the third statement is TRUE.

When assessing whether a graph passes through a particular quadrant, consider key features such as the vertex, intercepts, and end behavior. The vertex represents the minimum or maximum point of the parabola and provides valuable information about the function's range. The intercepts, particularly the y-intercept, indicate where the graph intersects the y-axis and can help determine the sign of the function for certain intervals. The end behavior describes how the function behaves as x approaches positive or negative infinity, and it helps identify which quadrants the graph will eventually enter. By analyzing these features, you can effectively determine the regions of the coordinate plane where the graph lies.

For example, if a parabola opens upwards and its vertex is located in the first quadrant, the graph will necessarily pass through the first and second quadrants. On the other hand, if the vertex is in the fourth quadrant, the graph will pass through the first and fourth quadrants. Similarly, if a parabola opens downwards and its vertex is in the second quadrant, the graph will pass through the second and third quadrants. By considering these relationships, you can quickly and accurately determine the quadrants that the graph traverses.

Also, remember that the absence of real roots (as indicated by a negative discriminant) means the graph does not intersect the x-axis. This further constrains the possible quadrants the graph can pass through, making the analysis more straightforward. Always consider the interplay between the vertex, intercepts, end behavior, and discriminant to paint a complete picture of the quadratic function's trajectory.

Summary

Alright, to recap:

• The graph of F(x) opens upwards: TRUE
• The graph of F(x) intersects the line y = -18: FALSE
• The graph of F(x) does not pass through quadrant 3: TRUE

Hope this breakdown was helpful, guys! Keep practicing, and you'll nail these quadratic function analyses in no time!