Analyzing Rate Of Change For Quadratic Function Q
Let's dive into analyzing the rate of change for a quadratic function, specifically function q. This function has some interesting properties: it has zeros at -3 and 8, and it intersects the y-axis at the point (0, 12). Understanding how the rate of change behaves for such a function is a fundamental concept in mathematics, especially in calculus and pre-calculus. So, guys, let’s break it down and see what we can learn about function q!
Understanding the Basics of Quadratic Functions
To really grasp the rate of change of function q, we first need to understand the basics of quadratic functions. A quadratic function is a polynomial function of degree 2, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. This directionality is crucial because it directly impacts how the function's rate of change behaves.
The zeros of a quadratic function, also known as the roots or x-intercepts, are the points where the function intersects the x-axis. These are the values of x for which f(x) = 0. In our case, function q has zeros at -3 and 8. This means that when x is -3 or 8, the function q(x) equals zero. The y-intercept, on the other hand, is the point where the function intersects the y-axis. This occurs when x = 0. For function q, the y-intercept is (0, 12), indicating that q(0) = 12. These intercepts give us vital clues about the function's shape and position on the coordinate plane.
The vertex of a parabola is the point where the function reaches its minimum (if the parabola opens upwards) or maximum (if the parabola opens downwards) value. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Knowing the vertex helps us understand the symmetry of the parabola, as the parabola is symmetrical around the vertical line passing through the vertex. Understanding these fundamental aspects of quadratic functions sets the stage for analyzing the rate of change.
Determining the Quadratic Function q(x)
Before we can delve into the rate of change, we need to determine the equation for function q(x). We know that q(x) is a quadratic function, and we have three key pieces of information: the zeros at -3 and 8, and the y-intercept at (0, 12). Since we know the zeros, we can express q(x) in the factored form:
q(x) = a(x - r₁)(x - r₂)
Where r₁ and r₂ are the zeros of the function, and a is a constant that determines the parabola's direction and stretch. Plugging in the zeros -3 and 8, we get:
q(x) = a(x + 3)(x - 8)
Now, we need to find the value of a. This is where the y-intercept comes into play. We know that q(0) = 12, so we can substitute x = 0 into our equation:
12 = a(0 + 3)(0 - 8) 12 = a(3)(-8) 12 = -24a
Solving for a, we get:
a = -12 / 24 = -1 / 2
Therefore, the quadratic function q(x) is:
q(x) = -1 / 2(x + 3)(x - 8)
To make it easier to analyze the rate of change, we can expand this equation into the standard form q(x) = ax² + bx + c:
q(x) = -1 / 2(x² - 8x + 3x - 24) q(x) = -1 / 2(x² - 5x - 24) q(x) = -1 / 2 x² + 5 / 2 x + 12
Now that we have the equation for q(x) in both factored and standard forms, we are well-equipped to analyze its rate of change.
Analyzing the Rate of Change
The rate of change of a function describes how the function's output changes with respect to its input. For a quadratic function, the rate of change is not constant; it varies across the function's domain. This is because the parabola is a curve, not a straight line. The average rate of change over an interval [a, b] is given by:
(q(b) - q(a)) / (b - a)
This represents the slope of the secant line connecting the points (a, q(a)) and (b, q(b)) on the parabola. However, to understand the instantaneous rate of change at a specific point, we need to consider the derivative of the function.
The derivative of q(x), denoted as q'(x), gives us the instantaneous rate of change at any point x. For a quadratic function in the form q(x) = ax² + bx + c, the derivative is:
q'(x) = 2ax + b
In our case, q(x) = -1 / 2 x² + 5 / 2 x + 12, so the derivative is:
q'(x) = 2(-1 / 2)x + 5 / 2 q'(x) = -x + 5 / 2
This derivative q'(x) is a linear function, which means the rate of change of q(x) changes linearly. To analyze the rate of change, we can consider different intervals and points along the parabola.
Intervals of Increase and Decrease
The rate of change is positive when the function is increasing and negative when the function is decreasing. To find the intervals where q(x) is increasing or decreasing, we need to analyze the sign of q'(x). We set q'(x) = 0 to find the critical point:
0 = -x + 5 / 2 x = 5 / 2 = 2.5
This critical point, x = 2.5, is the x-coordinate of the vertex of the parabola. Now we can analyze the sign of q'(x) in the intervals determined by this critical point:
- For x < 2.5, q'(x) = -x + 5 / 2 > 0, so q(x) is increasing.
- For x > 2.5, q'(x) = -x + 5 / 2 < 0, so q(x) is decreasing.
This means that q(x) increases until it reaches its vertex at x = 2.5, and then it decreases. The vertex represents the maximum point of the parabola since a is negative.
Behavior Around the Zeros
Let's consider the behavior of the rate of change around the zeros of q(x), which are x = -3 and x = 8:
- At x = -3, q'(-3) = -(-3) + 5 / 2 = 3 + 2.5 = 5.5. The rate of change is positive, indicating that q(x) is increasing as it approaches the zero from the left.
- At x = 8, q'(8) = -8 + 5 / 2 = -8 + 2.5 = -5.5. The rate of change is negative, indicating that q(x) is decreasing as it passes the zero.
These observations align with our understanding of the parabola opening downwards. As x increases from a large negative value towards -3, the function is increasing. After passing the vertex at x = 2.5, the function starts decreasing, eventually passing through the zero at x = 8.
Rate of Change at the Y-intercept
Finally, let's consider the rate of change at the y-intercept, x = 0:
q'(0) = -0 + 5 / 2 = 5 / 2 = 2.5
The rate of change at the y-intercept is 2.5, which is positive. This means that the function is increasing as it crosses the y-axis at the point (0, 12). This further confirms the shape of the parabola and its behavior around different points.
Conclusion
In conclusion, analyzing the rate of change of a quadratic function like q(x) involves understanding its zeros, y-intercept, and vertex, as well as calculating its derivative. By determining the derivative q'(x) = -x + 5 / 2, we were able to identify intervals of increase and decrease, and the rate of change at specific points, such as the zeros and the y-intercept. The function q(x) = -1 / 2 x² + 5 / 2 x + 12 increases until it reaches its vertex at x = 2.5, and then decreases. The rate of change is positive for x < 2.5 and negative for x > 2.5. This comprehensive analysis provides a clear picture of how function q behaves across its domain. Guys, I hope this breakdown helps you understand how to tackle similar problems! Remember, math can be fun when you break it down step by step!