Problem 58 - A Graphical Approach to College Algebra

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-x^{2}-x-6, x=3$$

Step-by-Step Solution

Answer

\( f(3) = -18 \)

Step 1: Understand the Problem

The task is to find the value of the function \( f(x) = -x^2 - x - 6 \) when \( x = 3 \). This means we substitute \( x = 3 \) into the function and compute the resulting value.

Step 2: Substitute the Value

Replace \( x \) in the function with 3: \( f(3) = -(3)^2 - (3) - 6 \). This substitution will allow us to calculate the exact value of the function at \( x = 3 \).

Step 3: Simplify the Expression

Calculate \( (3)^2 \) which equals 9. Then compute \( -9 - 3 - 6 \). This simplifies to \( f(3) = -9 - 3 - 6 \).

Step 4: Compute the Result

First compute \( -9 - 3 = -12 \), then \( -12 - 6 = -18 \). Therefore, \( f(3) = -18 \).

Key Concepts

Quadratic Function

A quadratic function is essentially any polynomial function where the highest degree of the variable is 2. Such functions take the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. Here, the variable \( x \) is raised to the power of 2, making it a 'quadratic' term.
Quadratic functions are easily recognizable by their unique U-shaped curve when graphed, known as a parabola. The direction the parabola opens (upward or downward) is determined by the sign of the coefficient \( a \). If \( a \) is positive, the parabola opens upward, and if negative, it opens downward. In our example, \( f(x) = -x^2 - x - 6 \), the coefficient of \( x^2 \) is negative, which means the parabola will open downwards. This particular function has a few key components:

• Leading coefficient: The \( x^2 \) term, which is \(-1\) in this case, defines the width and direction of the parabola.
• Linear term: The \( x \) term, which is \(-1\), affects the slope and offset of the parabola.
• Constant term: The number \(-6\) serves as the y-intercept, the point where the parabola crosses the y-axis.

Understanding these components is crucial when working with quadratic functions, especially when you're tasked with evaluating them at specific points.

Substitution

Substitution is a fundamental step in function evaluation, especially crucial in simplifying expressions and getting a numerical result. When substituting, you simply replace every instance of the variable in the function with a specified number.
In the problem we're exploring, we need to find the value of \( f(x) = -x^2 - x - 6 \) when \( x = 3 \). This is done by substituting \( x = 3 \) everywhere in the equation, transforming it as follows:

• Start with the original expression: \( f(x) = -x^2 - x - 6 \).
• Replace every \( x \) with 3: \( f(3) = -(3)^2 - 3 - 6 \).

This substitution allows us to transform an abstract algebraic expression into a concrete numerical one, making it much easier to evaluate and obtain a specific result.

Simplification Steps

Simplification is the process of converting expressions into their simplest possible form, often by performing arithmetic operations. This stage is crucial in determining the evaluated value of a function after substitution.
Once the substitution is done, as in \( f(3) = -(3)^2 - 3 - 6 \), the following simplification steps should be undertaken:

• Compute the square of 3: \( (3)^2 = 9 \).
• Substitute this back into the expression: \( f(3) = -9 - 3 - 6 \).
• Proceed with arithmetic operations from left to right: \(-9 - 3 = -12\).
• Continue simplifying to add the last term: \(-12 - 6 = -18\).

The final simplified value of \( f(3) \) is \(-18\). Following each simplification step methodically is important to accurately reach the correct answer. Simplification not only helps in finding the solution but also ensures that the function has been correctly evaluated.