Problem 56 - A Graphical Approach to College Algebra

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x^{2}+2 x-5, x=2$$

Step-by-Step Solution

Answer

When \(x = 2\), \(f(x) = 11\).

Step 1: Understand the Problem

We are given a quadratic function \(f(x) = 3x^2 + 2x - 5\) and need to find the value of \(f(x)\) when \(x = 2\).

Step 2: Substitute the Value of x

Substitute \(x = 2\) into the function \(f(x)\). This means you replace every \(x\) in the equation with \(2\). Thus, the expression becomes \(f(2) = 3(2)^2 + 2(2) - 5\).

Step 3: Simplify the Expression

Now, simplify the expression by calculating each term:1. First, calculate \(3(2)^2\). - \(2^2 = 4\) - \(3 \times 4 = 12\)2. Then calculate \(2 \times 2 = 4\).3. Finally, simplify \(12 + 4 - 5\).

Step 4: Perform the Addition and Subtraction

Add the values from the simplified terms: - \(12 + 4 = 16\)- Subtract \(5\) from \(16\), which gives \(16 - 5 = 11\).

Step 5: Conclude the Calculation

The result of \(f(2)\) is \(11\). This means when \(x = 2\), \(f(x)\) becomes \(11\).

Key Concepts

Quadratic Functions

Quadratic functions are a vital part of algebra and appear frequently in math problems and real-world applications. A standard quadratic function is expressed as \[ f(x) = ax^2 + bx + c \] where:

• \(a\), \(b\), and \(c\) are constants,
• \(x\) is the variable,
• and \(ax^2\) is the quadratic term.

The quadratic term \(ax^2\) is what makes this function different from linear functions. It introduces a curve to the graph, usually forming a shape known as a parabola.
This parabola can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards, resembling a 'U' shape. If \(a < 0\), it opens downwards, similar to an upside-down 'U'. Quadratic functions are often used to model situations such as projectile motion and area optimization problems. Understanding how to manipulate and evaluate quadratic functions is essential in developing algebraic skills.

Function Substitution

Function substitution is a straightforward yet essential process in algebra. It involves replacing a variable within a function with a specific value or another expression. In our exercise, we are tasked with evaluating the function \[ f(x) = 3x^2 + 2x - 5 \] at \( x = 2 \).
To perform the substitution, follow these steps:

• Identify the variable to be replaced, which is \( x \) in our case.
• Plug the given value (2) into the function wherever \( x \) appears.
• Ensure that the replacement is consistent across the entire expression, converting it to: \[ f(2) = 3(2)^2 + 2(2) - 5 \]

Function substitution is a fundamental skill that aids in evaluating functions at specific points, helping to understand the behavior and properties of various mathematical expressions.

Algebraic Simplification

Algebraic simplification is the process of reducing complex expressions into simpler forms without changing their values. This makes them easier to interpret and calculate. When simplifying, follow these general steps:

• Perform any operations within parentheses or brackets first.
• Evaluate exponentials or powers.
• Continue with any multiplications or divisions.
• Lastly, carry out additions or subtractions.

In our specific exercise example, you would simplify the expression \[ 3(2)^2 + 2(2) - 5 \] by:

• Calculating the square: \( 2^2 = 4 \).
• Multiplying: \( 3 imes 4 = 12 \) and \( 2 imes 2 = 4 \).
• Adding these intermediate results: \( 12 + 4 = 16 \).
• Performing the final subtraction: \( 16 - 5 = 11 \).

These steps demonstrate how careful simplification ensures accurate solutions and helps develop a deeper understanding of quadratic functions and other algebraic forms.