Problem 21 - Algebra 2 and Trigonometry

Question

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=5 x^{2}+4 x+1 \text { and } a=-1 $$

Step-by-Step Solution

Answer

\(\mathrm{f}(-1) = 2\); \(a = -1\) is not a root.

Step 1: Substitute the Value of 'a' into the Function

To find \( \mathrm{f}(a) \) for the given function \( \mathrm{f}(x) = 5x^2 + 4x + 1 \) with \( a = -1 \), we substitute \( -1 \) for \( x \). This gives us \( \mathrm{f}(-1) = 5(-1)^2 + 4(-1) + 1 \).

Step 2: Calculate \( \mathrm{f}(-1) \)

First, compute \( (-1)^2 \), which is \( 1 \). Then, substitute to get \( \mathrm{f}(-1) = 5 \times 1 + 4 \times (-1) + 1 \). This simplifies to \( 5 - 4 + 1 \).

Step 3: Simplify the Expression

Now, simplify the expression: \( 5 - 4 + 1 = 2 \). Therefore, \( \mathrm{f}(-1) = 2 \).

Step 4: Determine if 'a' is a Root of the Function

A number \( a \) is a root of a function \( \mathrm{f}(x) \) if \( \mathrm{f}(a) = 0 \). Since \( \mathrm{f}(-1) = 2 \), and not 0, \( a = -1 \) is not a root of the function.

Key Concepts

Evaluating Functions

When you encounter a function like \( f(x) = 5x^2 + 4x + 1 \), and you are asked to find \( f(a) \) for some number \( a \), this is called evaluating the function. It's like checking what the function equals when you plug a specific value for \( x \).
To evaluate, you follow these steps:

• Start by substituting the given value of \( a \) into the function wherever you see \( x \).
• Perform the arithmetic operations according to the order of operations: first exponents, then multiplication and division, and finally addition and subtraction.

In our example, we substituted \(-1\) for \( x \) in \( 5x^2 + 4x + 1 \), leading to the computation and simplification of \( 5(-1)^2 + 4(-1) + 1 = 2 \). Now you know the function's value at that specific point, which is indispensable in many areas of math and its applications.

Roots of Functions

The concept of roots of functions is about identifying when a function equals zero. A root is a value of \( x \) such that \( f(x) = 0 \). Determining roots is crucial because they tell us where the function crosses or touches the x-axis on a graph.
To check if a value is a root:

• Substitute the value into the function.
• Simplify and see if the result is 0.

For instance, if we substitute \(-1\) into \( f(x) = 5x^2 + 4x + 1 \) and get \( f(-1) = 2 \), it confirms that \(-1\) is not a root because the result isn't zero. Understanding this helps you graph functions and solve equations effectively.

Polynomial Functions

Polynomial functions like \( f(x) = 5x^2 + 4x + 1 \) are expressions involving terms composed of constants and variables raised to whole number powers. These are some of the most common types of functions you'll encounter, and they appear across different fields of math.
Key characteristics of polynomial functions:

• The degree of the polynomial is the highest power of \( x \). In our example, it's 2, which makes it a quadratic function.
• Polynomial functions are smooth, continuous, and defined for all numbers.
• The coefficients and constants (like 5, 4, and 1 in our function) determine the function's shape and position on a graph.

Understanding polynomials supports various problem-solving strategies, including finding roots, analyzing end behavior, and optimizing solutions for real-world problems.