Problem 50 - Intermediate Algebra

Question

If \(f(x)=x^{2}-4 x+10\), find \(f(-a), f(a-4)\), and \(f(a+h) .\)

Step-by-Step Solution

Answer

\(f(-a) = a^2 + 4a + 10\); \(f(a-4) = a^2 - 12a + 42\); \(f(a+h) = a^2 + 2ah + h^2 - 4a - 4h + 10\).

Step 1: Understand the Function

The function given is \( f(x) = x^2 - 4x + 10 \). It describes how each input \( x \) is transformed by squaring it, subtracting four times \( x \), and then adding 10.

Step 2: Substitute for \( f(-a) \)

In this step, substitute \( -a \) into the function wherever \( x \) appears: \( f(-a) = (-a)^2 - 4(-a) + 10 \). This simplifies to: \[(-a)^2 = a^2,\ -4(-a) = 4a.\ \]So, \( f(-a) = a^2 + 4a + 10 \).

Step 3: Substitute for \( f(a-4) \)

Substitute \( a-4 \) into the function wherever \( x \) appears: \( f(a-4) = (a-4)^2 - 4(a-4) + 10 \). Calculate each part:\[(a-4)^2 = a^2 - 8a + 16,\ -4(a-4) = -4a + 16.\ \]Thus, \( f(a-4) = a^2 - 8a + 16 - 4a + 16 + 10 \), simplifying to \( a^2 - 12a + 42 \).

Step 4: Substitute for \( f(a+h) \)

Substitute \( a+h \) into the function wherever \( x \) appears: \( f(a+h) = (a+h)^2 - 4(a+h) + 10 \). Calculate each part:\[(a+h)^2 = a^2 + 2ah + h^2,\ -4(a+h) = -4a - 4h.\ \]So, \( f(a+h) = a^2 + 2ah + h^2 - 4a - 4h + 10 \).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are like the backbone of algebra. They are mathematical expressions where you find terms like variables raised to powers, multiplied by coefficients, and sometimes added or subtracted with constants. Think of them as building blocks that can take different shapes based on the powers and coefficients. For example, in the function given, \( f(x) = x^2 - 4x + 10 \), we have a quadratic polynomial function. Here, the highest power of \( x \) is 2, which makes it a quadratic. It's important to note the structure:

• The quadratic term: \( x^2 \) with coefficient 1.
• The linear term: \( -4x \) with coefficient -4.
• The constant term: 10.

These functions can model all kinds of real world situations because they are quite versatile. When you learn to recognize these structures, you can predict behavior and transformation in many mathematical problems. Every new value inserted can change it dramatically or subtly.

Decoding Algebraic Expressions

Algebraic expressions consist of numbers, letters (variables), and operational signs that come together to create a mathematical statement. When you see something like \( f(x) = x^2 - 4x + 10 \), you're looking at an algebraic expression where:

• \( x \) is a placeholder for any real number you substitute in its place.
• Expressions are simplified using substitute techniques by replacing \( x \) with another variable or number.

Being comfortable with these substitutions is crucial because it allows you to evaluate the expression for any specific value of \( x \). Like in our step-by-step solution, replacing \( x \) with \(-a\), \( a-4 \), or \( a+h \) follows logical operations to simplify those new expressions. When you follow each substitution and again perform operations like squaring or simplifying like terms, you begin to see patterns and understand deeper relationships between numbers and operations.

Exploring Quadratic Equations

Quadratic equations are a significant milestone in algebra. They are written in the standard form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants. In our example, the expression \( x^2 - 4x + 10 \) is a part of a quadratic equation when set equal to zero.

• This quadratic has a "U" shape if graphed, often leading to two solutions (roots), one solution, or no real solution depending on the equation's discriminant.
• Simplifying substitutions in these can reveal different properties and paths to solving for unknowns.

Every time we alter the root expression, as with functions \( f(-a), f(a-4), \) or \( f(a+h) \), we create a new version of a quadratic function, exploring different positional relationships of the parabola on a graph. Understanding how these transformations occur aids in solving real world problems and acclimatizes one to predict the behavior of variables and constants instantly.