Problem 151 - Algebra and Trigonometry

Question

Consider the quadratic function $$f(x)=-4 x^{2}-16 x+3$$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. (Section 3.1, { Example } 4)

Step-by-Step Solution

Answer

a. The function has a maximum value. b. The maximum value is -27 and it occurs at \(x = 2\). c. The domain of the function is all real numbers \(-\infty, \infty\) and the range is \(-\infty, -27]\).

Step 1: Determining the nature of extreme value

Analyze the coefficient of the \(x^2\) term in the function \(f(x) = -4x^{2} - 16x + 3 \). If the coefficient is positive, the function opens upwards and has a minimum. If the coefficient is negative, it opens downwards and has a maximum. In this case, the coefficient is -4, which is negative. Therefore, this function has a maximum value.

Step 2: Find the maximum value

The maximum or minimum value of a quadratic function is found at its vertex. The x-coordinate of the vertex can be found using the formula \(-b / (2a)\) where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\). In this case, \(a = -4\) and \(b = -16\), so applying the formula gives \(x = -(-16) / (2*-4) = 2\). Substituting \(x = 2\) back into the function: \(f(2) = -4(2)^{2} -16(2) +3 = -27\). Therefore, the maximum value of the function is -27 and it occurs at \(x = 2\).

Step 3: Identify the function's domain and range

The domain of a quadratic function is all real numbers since it is defined for all x. Hence the domain is all real numbers or in interval notation, \(-\infty, \infty\).\nThe range of a quadratic function that opens downwards is all y values up to and including the maximum value. The maximum value as calculated in step 2 is -27. Hence, the range by interval notation is \(-\infty, -27] \).

Key Concepts

Vertex of Quadratic Function

The vertex of a quadratic function is a pivotal concept in understanding its graph. This point is the "turning point" where the function switches direction. Quadratic functions are typically written in the form \(f(x) = ax^2 + bx + c\). To find the vertex, we use the formula for the x-coordinate: \(-\frac{b}{2a}\). This formula derives from completing the square or calculus-based derivations, but it's handy to memorize due to its frequent use.

In our specific problem, the given function is \(f(x) = -4x^2 - 16x + 3\). Here, \(a = -4\) and \(b = -16\). Plugging into the formula, we find that the x-coordinate of the vertex is \(2\). To find the y-coordinate, substitute \(x = 2\) back into the function:

• \(f(2) = -4(2)^2 - 16(2) + 3 = -27\)

Thus, the vertex is at \((2, -27)\), which indicates not only the "turning point" but also the maximum or minimum value of the function.

Domain of Quadratic Function

The domain of a quadratic function is pleasantly straightforward. Regardless of the coefficients or constants, the domain is all real numbers. This universal property arises because there is no restriction on x-values for which quadratic functions can be evaluated.

In interval notation, the domain is expressed as \((-\infty, \infty)\). This means that for any x value you choose, there is a corresponding y value, creating a smooth, continuous parabola with no breaks or gaps.

• In practical terms, it means the function's graph extends infinitely to the left and right on the x-axis.

Range of Quadratic Function

The range of a quadratic function describes all the possible y-values that the function can output. Unlike the domain, it's not always all real numbers; it depends on whether the parabola opens upwards or downwards. If it opens downwards, as in the original problem where the coefficient of \(x^2\) is negative, the parabola has a maximum point.

For our function \(f(x) = -4x^2 - 16x + 3\), we found that the maximum value is -27, occurring at the vertex. Thus, the range includes all y-values less than or equal to -27. In interval notation, this range is expressed as \((-\infty, -27]\).

• This range means the function's y-values stretch downwards infinitely but will never surpass -27.

Maximum Value of Quadratic Function

Determining whether a quadratic function has a maximum or minimum value is steered by the sign of the \(x^2\) coefficient. In our scenario, \(f(x) = -4x^2 - 16x + 3\) has a coefficient \(a = -4\), which is negative. Thus, the function opens downwards and possesses a maximum value, not a minimum.

Finding the maximum entails identifying the vertex because the vertex's y-coordinate represents this extremum. Using our vertex calculations, we found this maximum value occurs at \(x=2\), and \(f(2) = -27\). Therefore, the maximum value that \(f(x)\) can reach is -27.

• This value is a peak point, above which the function doesn't venture.

Understanding these concepts allows mathematicians and students alike to sketch a rough profile of the quadratic function without plotting numerous points.