Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-x^{2}+x+2 ; x=4$$

Step-by-Step Solution

Answer

\( f(4) = -10 \)

Step 1: Identify the Function and Value

The function provided is \( f(x) = -x^2 + x + 2 \). We are given \( x = 4 \) and need to evaluate the function at this specific point.

Step 2: Substitute x with 4

Replace \( x \) with 4 in the function. This gives us \( f(4) = -(4)^2 + 4 + 2 \).

Step 3: Simplify the Squared Term

Calculate \( (4)^2 \), which equals 16. Substitute this back into the equation: \( f(4) = -16 + 4 + 2 \).

Step 4: Perform the Addition and Subtraction

First, add 4 and 2 together to get 6. Then subtract 16 from this result: \( f(4) = 6 - 16 \).

Step 5: Calculate the Final Result

Finally, perform the subtraction: \( 6 - 16 = -10 \). This means \( f(4) = -10 \).

Key Concepts

Substitution

Substitution is a fundamental process in mathematics where we replace a variable with a specific value. In this exercise, we start with the quadratic function \(f(x) = -x^2 + x + 2\). Here, our goal is to find the value of \(f(x)\) when \(x = 4\).
To do this, we take every occurrence of \(x\) in the function and substitute it with 4.

• First, identify the function: \(f(x) = -x^2 + x + 2\).
• Then substitute: \(f(4) = -(4)^2 + 4 + 2\).
• Remember: Substitution is simply about plugging in the number given for \(x\) into the function.

This step sets the stage for further simplification, which allows us to find the specific value of the function at \(x = 4\). Consistent application of substitution skills is crucial in various math problems, particularly in algebra.

Simplification

Simplification is a method we use to make mathematical expressions easier to work with. After substituting \(x = 4\) into our function, we have \(f(4) = -(4)^2 + 4 + 2\). The next step involves simplification to get a final result. Here's a closer look:

• First, calculate \((4)^2\), which equals 16. Then substitute this back: \(f(4) = -16 + 4 + 2\).
• Now, it's about performing operations in the correct order:
  • First, add the numbers: 4 + 2 = 6.
  • Then, perform the subtraction: 6 - 16 = -10.

This structured approach of breaking down each mathematical operation helps simplify even complex equations, making it easier to arrive at the correct answer efficiently. Mastering simplification is key to solving algebraic expressions and functions.

Quadratic Function

A quadratic function is an important algebraic concept characterized by the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Our example function \(f(x) = -x^2 + x + 2\) fits this form well:

• The term \(-x^2\) is known as the quadratic term and it determines the "U" shaped curve of the parabola. The negative sign in front indicates the parabola opens downwards.
• The linear term, \(x\), affects the slope or tilt of the parabola.
• Finally, the constant term, \(+2\), shifts the parabola vertically.

When evaluating quadratic functions, such as when finding \(f(x)\) at a given point, the substitution and simplification steps allow the calculation of precise values of the function. Understanding the components and the behavior of quadratic functions is essential for graphical interpretations and mathematical problem-solving.