Problem 45 - Calculus with Concepts in Calculus

Question

Approximate all zeros of the function to the nearest hundredth. $$ f(x)=-4.9 x^{2}+5.1 x+1.2 $$

Step-by-Step Solution

Answer

The zeros are approximately -0.20 and 1.24.

Step 1: Identify Method to Use

To approximate the zeros of a quadratic function like \( f(x) = -4.9x^2 + 5.1x + 1.2 \), we can use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -4.9 \), \( b = 5.1 \), and \( c = 1.2 \).

Step 2: Calculate the Discriminant

Compute the discriminant \( \Delta = b^2 - 4ac \).Substitute the values in: \( \Delta = 5.1^2 - 4(-4.9)(1.2) = 26.01 + 23.52 = 49.53 \).

Step 3: Compute the Square Root of the Discriminant

Evaluate \( \sqrt{\Delta} \):\( \sqrt{49.53} \approx 7.039 \).

Step 4: Calculate the Zeros Using the Quadratic Formula

Substitute \( a, b, \) and \( \sqrt{\Delta} \) into the quadratic formula to find the zeros.1. \( x_1 = \frac{-5.1 + 7.039}{2(-4.9)} = \frac{1.939}{-9.8} \approx -0.198 \).2. \( x_2 = \frac{-5.1 - 7.039}{2(-4.9)} = \frac{-12.139}{-9.8} \approx 1.238 \).

Step 5: Round the Solutions to the Nearest Hundredth

Round the results to the nearest hundredth:\( x_1 \approx -0.20 \) and \( x_2 \approx 1.24 \).

Key Concepts

Quadratic Formula

When dealing with quadratic equations, the quadratic formula is often our best friend. This formula allows us to find the roots or zeros of a quadratic equation, which is critical for understanding the behavior of the function. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this formula, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation in the form \( ax^2 + bx + c = 0 \). The symbol "±" indicates that there are usually two solutions: one for the plus and another for the minus. This method is powerful because it doesn't assume that the quadratic can be easily factored, making it suitable for all forms of quadratic equations.
By using this formula, we can systematically find the zeros of functions like \( f(x)=-4.9x^2+5.1x+1.2 \).

Discriminant

The discriminant is a pivotal component of the quadratic formula. You find it inside the square root: \( b^2 - 4ac \). It helps determine the nature of the roots of a quadratic equation.

• If the discriminant is positive, like in our example where \( \Delta = 49.53 \), this means there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, which means the parabola touches the x-axis at a single point.
• If the discriminant is negative, there are no real roots, indicating that the graph does not intersect the x-axis at all.

By calculating the discriminant, we can anticipate the type of solutions we’re about to calculate. This is why it's often the first step after identifying the coefficients when solving using the quadratic formula.

Zeros of Functions

The zeros of a function are the x-values that make the function equal to zero. For quadratic functions, finding the zeros is akin to finding the x-intercepts of the graph of the function.In the exercise at hand, we used the quadratic formula to approximate these values for \( f(x)=-4.9x^2+5.1x+1.2 \) where we found two zeros:

• \(x_1 \approx -0.20\)
• \(x_2 \approx 1.24\)

These zeros tell us at which points the parabola represented by our quadratic equation crosses the x-axis. Understanding this concept allows you to grasp more about the nature of the function and its graph.

Approximation Methods

In real-life applications, sometimes solving exactly isn't feasible or necessary, and approximation becomes important. For quadratic functions, once we have applied the quadratic formula and derived solutions, we often need to round these solutions to make them more manageable.In our example, we approximated:

• \(x_1 \approx -0.20\)
• \(x_2 \approx 1.24\)

We rounded these results to the nearest hundredth. This process is a simple yet crucial step, ensuring that our results are both precise and practical. When solving problems using approximations, it's essential to maintain consistency in your method of rounding to maintain accuracy and reliability.