Problem 17 - Algebra and Trigonometry

Question

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=2 x^{2}+3 x-4} \\ {f(0), f(2), f(-2), f(\sqrt{2}), f(x+1), f(-x)}\end{array} $$

Step-by-Step Solution

Answer

\( f(0) = -4, f(2) = 10, f(-2) = -2, f(\sqrt{2}) = 3\sqrt{2}, f(x+1) = 2x^2 + 7x + 1, f(-x) = 2x^2 - 3x - 4 \).

Step 1: Evaluate f(0)

To find \( f(0) \), substitute \( x = 0 \) into the function \( f(x) = 2x^2 + 3x - 4 \).Calculate:\[ f(0) = 2(0)^2 + 3(0) - 4 = -4 \]So, \( f(0) = -4 \).

Step 2: Evaluate f(2)

Substitute \( x = 2 \) into the function:\[ f(2) = 2(2)^2 + 3(2) - 4 \]Calculate:\[ f(2) = 2(4) + 6 - 4 = 8 + 6 - 4 = 10 \]So, \( f(2) = 10 \).

Step 3: Evaluate f(-2)

Substitute \( x = -2 \) into the function:\[ f(-2) = 2(-2)^2 + 3(-2) - 4 \]Calculate:\[ f(-2) = 2(4) - 6 - 4 = 8 - 6 - 4 = -2 \]So, \( f(-2) = -2 \).

Step 4: Evaluate f(\sqrt{2})

Substitute \( x = \sqrt{2} \) into the function:\[ f(\sqrt{2}) = 2(\sqrt{2})^2 + 3(\sqrt{2}) - 4 \]Calculate:\[ f(\sqrt{2}) = 2(2) + 3\sqrt{2} - 4 = 4 + 3\sqrt{2} - 4 = 3\sqrt{2} \]Thus, \( f(\sqrt{2}) = 3\sqrt{2} \).

Step 5: Evaluate f(x+1)

Substitute \( x = x+1 \) into the function:\[ f(x+1) = 2(x+1)^2 + 3(x+1) - 4 \]Expand and simplify:\[ (x+1)^2 = x^2 + 2x + 1 \]\[ f(x+1) = 2(x^2 + 2x + 1) + 3(x+1) - 4 \]\[ = 2x^2 + 4x + 2 + 3x + 3 - 4 \]Combine like terms:\[ f(x+1) = 2x^2 + 7x + 1 \].

Step 6: Evaluate f(-x)

Substitute \( x = -x \) into the function:\[ f(-x) = 2(-x)^2 + 3(-x) - 4 \]Calculate:\[ f(-x) = 2x^2 - 3x - 4 \]Thus, \( f(-x) = 2x^2 - 3x - 4 \).

Key Concepts

Polynomial Functions

Polynomial functions are a central part of algebra and appear frequently in math exercises and real-world applications. Polynomials are mathematical expressions that include variables raised to whole-number powers, and they are summed together with coefficients. The general form of a polynomial function looks like this:

• For any integer value of n, where n is the degree of the polynomial, the general polynomial is: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]
• Each of the terms consists of the coefficients \(a_i\) and the variable \(x\) raised to a power.
• In the function evaluated in the original exercise, \(f(x) = 2x^2 + 3x - 4\) is a quadratic polynomial because the highest power of \(x\) is 2.

Understanding polynomial functions is fundamental for performing operations such as addition, subtraction, multiplication, and division, and for evaluating functions, like substituting values and analyzing their behavior.

Substitution Method

The substitution method is a technique used to evaluate functions by replacing the variable with given values. It allows us to compute the output for any specific input, helping us understand how the function behaves. Following this method involves a simple and systematic approach:

• Replace the variable \(x\) in the function with the given number or expression.
• Carefully perform the arithmetic or algebraic operations following the order of operations (PEMDAS/BODMAS).
• Simplify to find the value of the function.

In the original exercise, the function \(f(x)\) was evaluated at several values like \(0, 2, -2, \sqrt{2}, x+1,\) and \(-x\). By substituting each into the function, we found corresponding outputs of the function \(f(x)\). This method is especially valuable in solving math problems and understanding function graphs.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest power of the variable is 2. They have some distinct characteristics that make them interesting and useful in various contexts:

• The general form is: \[ f(x) = ax^2 + bx + c \]where \(a, b,\) and \(c\) are constants, and \(a eq 0\).
• Quadratic functions graph as parabolas. The direction of the opening depends on the sign of \(a\): they open upwards if \(a\) is positive and downwards if \(a\) is negative.
• Their symmetry is around a vertical line called the axis of symmetry.
• They possess a vertex, which is the minimum or maximum point, depending on the direction the parabola opens.

In the exercise, \(f(x) = 2x^2 + 3x - 4\) is a quadratic function, showcasing a parabola opening upwards as \(a = 2\) is positive. Understanding the features of quadratic functions aids in graphing and solving quadratic equations.