Problem 57 - A Graphical Approach to College Algebra

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: Substitute the value of x

To find the value of the function at a specific point, substitute the given value of \(x = 4\) into the function \(f(x) = -x^2 + x + 2\). This becomes:\[f(4) = -(4)^2 + (4) + 2\]

Step 2: Calculate the square of x

Compute the square of 4, which is \(4^2 = 16\). Substitute this back into the expression:\[f(4) = -16 + 4 + 2\]

Step 3: Perform the arithmetic operations

Add and subtract the numbers in the expression:\[f(4) = -16 + 4 + 2 = -10\]

Step 4: State the final result

The value of the function at \(x = 4\) is \(-10\). Thus, \(f(4) = -10\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function characterized by its highest degree of two. It generally takes the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This function creates a parabolic graph, often resembling a U or an inverted U shape, known as a parabola.
The quadratic function in our exercise is \(f(x) = -x^2 + x + 2\). Here:

• \(a = -1\), which makes the parabola open downward due to the negative sign.
• \(b = 1\), influencing the parabola's horizontal position.
• \(c = 2\), shifting the parabola vertically.

Understanding the basic form and components of a quadratic function is crucial. It helps predict characteristics like the parabola's direction and its steepness.

Substitution Method

The substitution method is a straightforward algebraic technique used to evaluate functions or equations by replacing variables with specific values. This process allows you to find particular solutions quickly by turning a general function into a particular instance.
In the context of our problem, we are asked to find the value of the quadratic function \(f(x) = -x^2 + x + 2\) at \(x = 4\). To do this, we simply replace every instance of \(x\) in the function with the given value:

• Write down the original function: \(f(x) = -x^2 + x + 2\).
• Substitute \(4\) in for every \(x\): \(f(4) = -(4)^2 + 4 + 2\).

This step is crucial as it transforms the general function into a specific arithmetic expression that can be calculated directly.

Arithmetic Operations

Arithmetic operations are fundamental mathematical calculations, including addition, subtraction, multiplication, and division. When applied to algebraic expressions, these operations allow us to simplify or evaluate them to find specific numerical answers.
In our step-by-step solution, we used several arithmetic operations to find \(f(4)\):

• **Calculate the square root:** Begin by squaring 4: \((4)^2 = 16\).
• **Apply arithmetic operations:** Substitute the square value into the expression, resulting in \(-16 + 4 + 2\).
• **Combine like terms:** Perform the addition and subtraction sequentially: \(-16 + 4 = -12\), then \(-12 + 2 = -10\).

Each arithmetic step brings us closer to the solution, simplifying expressions until we arrive at a clear numerical answer: \(f(4) = -10\). Understanding these operations is essential for solving equations efficiently and accurately.