Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-22: (a) Use the quadratic formula to find the zeros of \(f\). (b) Find the maximum or minimum value of \(f(x)\). (c) Sketch the graph of \(f\). $$ f(x)=-12 x^{2}+11 x+15 $$

Step-by-Step Solution

Answer

Zeros: 1.6667, -0.5; Maximum value: 17.04 at x = 0.4583.

Step 1: Identify the coefficients

The given quadratic function is \( f(x) = -12x^2 + 11x + 15 \). Here, the coefficients are \( a = -12 \), \( b = 11 \), and \( c = 15 \).

Step 2: Apply the Quadratic Formula

To find the zeros of the function, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substituting \( a = -12 \), \( b = 11 \), and \( c = 15 \), calculate it.

Step 3: Calculate the Discriminant

Calculate the discriminant \( \Delta \): \( \Delta = b^2 - 4ac = 11^2 - 4(-12)(15) = 121 + 720 = 841 \). Since \( \Delta > 0 \), there are two distinct real roots.

Step 4: Solve for the Zeros

Substitute \( \Delta = 841 \) into the quadratic formula: \[ x = \frac{-11 \pm \sqrt{841}}{2(-12)} \]. Thus, \( x = \frac{-11 \pm 29}{-24} \). The roots are \( x_1 = 1.6667 \) and \( x_2 = -0.5 \).

Step 5: Determine Maximum or Minimum Value

Since the coefficient \( a = -12 \) is negative, the parabola opens downwards, hence there is a maximum value. The x-coordinate of the vertex can be found using \( x = -\frac{b}{2a} \), giving \( x = -\frac{11}{-24} = \frac{11}{24} \approx 0.4583 \).

Step 6: Calculate the Maximum Value

Substitute \( x = 0.4583 \) into the function to find the maximum value: \( f(0.4583) = -12(0.4583)^2 + 11(0.4583) + 15 = 17.04 \).

Step 7: Sketch the Graph

The parabola opens downward, has roots at \( x = 1.6667 \) and \( x = -0.5 \), and a maximum at \( x = 0.4583, y = 17.04 \). Plot these key points and sketch a symmetrical parabola through them.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to find the zeros, or roots, of a quadratic function, which are the values of \( x \) where the function equals zero.
The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

• The quadratic formula requires substitution of coefficients \( a \), \( b \), and \( c \) from the equation.
• "\( \pm \)" indicates that there are typically two solutions: one using addition, \( + \sqrt{...} \), and one using subtraction, \( - \sqrt{...} \).

By plugging in the given coefficients \( a = -12 \), \( b = 11 \), and \( c = 15 \), you can find the zeros of the function. This formula is not only systematic but works perfectly when manual guesswork won't suffice.
Applying this reveals the roots to be \( x_1 = 1.6667 \) and \( x_2 = -0.5 \) in this scenario. These are the points where the function crosses the x-axis.

Discriminant

The discriminant is a special component of the quadratic formula. It is crucial for determining the nature of the roots of a quadratic equation. It is the expression under the square root in the quadratic formula, calculated as:

• \( \Delta = b^2 - 4ac \)

• A positive discriminant (\( \Delta > 0 \)) indicates two distinct real roots.
• A discriminant of zero (\( \Delta = 0 \)) results in exactly one real root, meaning it's a perfect square.
• A negative discriminant (\( \Delta < 0 \)) implies there are no real roots, only complex ones.

In the given problem, the discriminant is calculated as 841, which is positive. This means there are two real roots, emphasizing that the parabola intersects the x-axis at two distinct points. The fact that \( \Delta = 841 \) gives us the indication that both roots (also called zeros) are real and different, assisting in forming a clear sketch of the quadratic graph.

Vertex of a Parabola

The vertex of a parabola is an essential landmark, especially when sketching its graph. It represents the peak (maximum or minimum) of the parabola, depending on the direction it opens. For the quadratic function \( ax^2 + bx + c = 0 \), the x-coordinate of the vertex is given by the formula:

• \( x = -\frac{b}{2a} \)

• If \( a > 0 \), the parabola opens upwards, and the vertex is the minimum point.
• If \( a < 0 \), as in our example (-12), the parabola opens downwards, and the vertex is at the maximum point.

By substituting \( b = 11 \) and \( a = -12 \), the x-coordinate of the vertex is found to be approximately 0.4583.
To find the y-coordinate, substitute \( x = 0.4583 \) back into the function, yielding the vertex at \( (0.4583, 17.04) \). The vertex represents the highest point of the parabola, useful for graphing and understanding the behavior of the function over its domain.

Zeros of a Function

Zeros of a function, often called roots or solutions, are the values of \( x \) where the function equals zero. Identifying these points is critical for understanding the intersections of the graph with the x-axis. For a quadratic function, zeros can be effectively found using the quadratic formula, as discussed earlier.
These zeros are solutions to the equation \( ax^2 + bx + c = 0 \). In our example, substituting into the quadratic formula gives the zeros at \( x_1 = 1.6667 \) and \( x_2 = -0.5 \). These values suggest where the graph of the function \( f(x) = -12x^2 + 11x + 15 \) touches or crosses the x-axis.
Understanding zeros is pivotal:

• They reveal where the graph intersects or touches the x-axis.
• This can also indicate the solutions to equations set by practical real-world models or problems.

Interpreting the zeros and their structure assists in constructing the quadratic graph effectively, offering a complete visual account of the function's behavior.