Problem 11 - Algebra for College Students

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: Determine f(-a)

To find \( f(-a) \), substitute \( x = -a \) into the function \( f(x) = x^2 - 4x + 10 \). This gives us:\[f(-a) = (-a)^2 - 4(-a) + 10\]Simplify the expression:\[= a^2 + 4a + 10\]

Step 2: Determine f(a-4)

To find \( f(a-4) \), substitute \( x = a-4 \) into the function \( f(x) = x^2 - 4x + 10 \). We have:\[f(a-4) = (a-4)^2 - 4(a-4) + 10\]First, expand \((a-4)^2\):\[= a^2 - 8a + 16\]Then, simplify \(-4(a-4)\):\[= -4a + 16\]Putting it all together:\[f(a-4) = (a^2 - 8a + 16) + (-4a + 16) + 10\]Combine like terms:\[= a^2 - 12a + 42\]

Step 3: Determine f(a+h)

To find \( f(a+h) \), substitute \( x = a+h \) into the function \( f(x) = x^2 - 4x + 10 \). We have:\[f(a+h) = (a+h)^2 - 4(a+h) + 10\]First, expand \((a+h)^2\):\[= a^2 + 2ah + h^2\]Then simplify \(-4(a+h)\):\[= -4a - 4h\]Putting it all together:\[f(a+h) = (a^2 + 2ah + h^2) + (-4a - 4h) + 10\]Combine like terms:\[= a^2 + 2ah + h^2 - 4a - 4h + 10\]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of one or more variables. These functions have various coefficients and exponents. A typical polynomial function can be written as:

• \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)

In this equation, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(n\) represents the highest power of the variable \(x\), called the degree of the polynomial. Evaluation of polynomial functions involves substituting specific values for \(x\) in order to find the corresponding \(y\) value. For example, the function \(f(x) = x^2 - 4x + 10\) is of degree 2 because the highest power of \(x\) is 2. Each term of the polynomial (such as \(x^2\), \(-4x\), \(+10\)) plays a role in shaping the overall graph or curvature of the function. Understanding polynomial functions helps in analyzing the behavior of equations in algebra, calculus, and beyond.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. In essence, they are the building blocks of polynomials and more complex algebraic structures.

• An example of a simple algebraic expression is \(3x + 4\).

The current polynomial function \(f(x) = x^2 - 4x + 10\) is an algebraic expression that contains three terms:

• The term \(x^2\) (a square term indicating non-linear behavior).
• The term \(-4x\) (a linear term impacting the slope).
• The constant 10 (shifting the graph vertically).

In algebra, combining different expressions involves manipulating these terms, such as expanding or factoring them, to simplify or evaluate functions more effectively. Mastering how to work with algebraic expressions can enhance skills in solving equations, graphing, and analyzing mathematical relationships.

Substitution Method

The substitution method is a useful algebraic process where values are replaced in a function or equation to find the result. It involves plugging in a given variable value into an expression and simplifying the outcome to evaluate the expression or solve an equation.

• For example, in the function \(f(x) = x^2 - 4x + 10\), to find \(f(-a)\), substitute \(-a\) for \(x\):
• \(f(-a) = (-a)^2 - 4(-a) + 10\), which simplifies to \(a^2 + 4a + 10\).

Using substitution, different values of \(x\) are evaluated:

• To find \(f(a-4)\), replace \(x\) with \(a-4\), leading to \(f(a-4) = a^2 - 12a + 42\).
• For \(f(a+h)\), replace \(x\) with \(a+h\), resulting in \(f(a+h) = a^2 + 2ah + h^2 - 4a - 4h + 10\).

The substitution method streamlines solving problems by making complex expressions simpler as step-by-step substitutions lead to easily calculable results. It is a fundamental technique in algebra with applications in calculus, physics, engineering, and beyond.