Problem 2 - Algebra for College Students

Question

If \(f(x)=x^{2}-3 x-4\), find \(f(2), f(4)\), and \(f(-3)\). \(f(2)=-6 ; f(4)=0 ; f(-3)=14\)

Step-by-Step Solution

Answer

f(2) = -6, f(4) = 0, f(-3) = 14.

Step 1: Substitute and Find f(2)

To find \( f(2) \), substitute \( x = 2 \) into the function \( f(x) = x^2 - 3x - 4 \). This gives:\[ f(2) = (2)^2 - 3(2) - 4 \]Calculate this:\[ f(2) = 4 - 6 - 4 \]\[ f(2) = -6 \]

Step 2: Substitute and Find f(4)

To find \( f(4) \), substitute \( x = 4 \) into the function \( f(x) = x^2 - 3x - 4 \). This gives:\[ f(4) = (4)^2 - 3(4) - 4 \]Calculate this:\[ f(4) = 16 - 12 - 4 \]\[ f(4) = 0 \]

Step 3: Substitute and Find f(-3)

To find \( f(-3) \), substitute \( x = -3 \) into the function \( f(x) = x^2 - 3x - 4 \). This gives:\[ f(-3) = (-3)^2 - 3(-3) - 4 \]Calculate this:\[ f(-3) = 9 + 9 - 4 \]\[ f(-3) = 14 \]

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two. It's commonly written in the form \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a \) is not zero. The graph of a quadratic function is a curve called a parabola. Parabolas have a distinctive U-shape and can open upwards or downwards depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Quadratic functions are commonly used in problems involving area, projectile motion, and other real-world applications. The function \( f(x)=x^{2}-3x-4 \) is an example of a quadratic function with \( a = 1 \), \( b = -3 \), and \( c = -4 \). Understanding how to manipulate and evaluate these functions is an important mathematical skill.

Substitution Method

The substitution method is a useful technique for evaluating functions at specific values. In the context of this problem, substitution involves replacing the variable \( x \) in the function \( f(x) \) with a given number.
Here’s how to use substitution step by step:

• Identify the function and the value you wish to substitute.
• Replace every occurrence of \( x \) in the function with the specified value.
• Perform the arithmetic operations to simplify the expression and find the result.

For instance, to find \( f(2) \), substitute \( x = 2 \) into the function \( f(x) = x^2 - 3x - 4 \):
1. Replace \( x \) with 2:\[ f(2) = (2)^2 - 3(2) - 4 \]2. Simplify the expression:\[ f(2) = 4 - 6 - 4 = -6 \]
Through substitution, solving and evaluating for \( f(4) \) or \( f(-3) \) becomes straightforward once these steps are applied consistently.

Polynomial Evaluation

Polynomial evaluation is the process of calculating the output of a polynomial function for given input values. In this process, the entire polynomial expression is simplified by computing the operations for a specific input.
When evaluating a polynomial such as \( f(x) = x^2 - 3x - 4 \), the following steps provide a clear pathway:

• Substitute the chosen value into every instance of the variable in the expression.
• Follow the order of operations – first calculate exponents, then perform multiplication and division, followed by addition and subtraction.
• Simplify to find the result.

Let’s break down the evaluation using the example \( f(-3) = 9 + 9 - 4 \):

• Compute the square first: \[ (-3)^2 = 9 \]
• Then the product: \[ 3(-3) = -9 \]
• Simplify: Add \( 9 \), add \( 9 \) from multiplication, and subtract \( 4 \).
• The result: 9 + 9 - 4 = 14.

This repetition of the process for different values helps reinforce the understanding of polynomial behavior and functions.