Problem 56 - Algebra 2

Question

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(a+1) $$

Step-by-Step Solution

Answer

The value of \(f(a+1)\) is \(a^2 - 3a + 2\).

Step 1: Understand the Expression

The function given is \(f(x) = x^2 - 5x + 6\). We need to find the value of \(f(a+1)\). This means we substitute \(x\) with \(a+1\) in the function.

Step 2: Substitute \(a+1\) into \(f(x)\)

Replace each occurrence of \(x\) in the function \(f(x) = x^2 - 5x + 6\) with \(a+1\). This gives us \(f(a+1) = (a+1)^2 - 5(a+1) + 6\).

Step 3: Expand \((a+1)^2\)

Expand the squared term \((a+1)^2\). It becomes \(a^2 + 2a + 1\). Thus, the expression is \(a^2 + 2a + 1 - 5(a+1) + 6\).

Step 4: Distribute \(-5(a+1)\)

Distribute the \(-5\) across the terms \(a+1\). This gives us \(-5a - 5\). Substitute back into the expression to get \(a^2 + 2a + 1 - 5a - 5 + 6\).

Step 5: Simplify the Expression

Combine like terms: \(a^2\) terms remain as is, \(2a - 5a = -3a\), and the constant terms \(1 - 5 + 6 = 2\). So, \(f(a+1) = a^2 - 3a + 2\).

Key Concepts

Polynomial Function

A polynomial function is like a special recipe involving powers of variables with coefficients. In our exercise, the polynomial function is given by the equation \(f(x) = x^2 - 5x + 6\). This is a quadratic polynomial, which means the highest power of \(x\) in it is 2.

Quadratic polynomials are written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Here, our polynomial has \(a = 1\), \(b = -5\), and \(c = 6\).

Understanding the parts of a polynomial is important because it helps us understand how changes in \(x\) affect the value of the function.

• The term \(x^2\) affects the curve of the graph significantly because it has the highest degree.
• The term \(-5x\) influences the slope of the graph.
• The constant term \(6\) shifts the graph up or down without changing its shape.

By studying polynomials, we learn valuable lessons about algebra and the functions we use to model the world around us.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with. In this exercise, one key manipulation was expanding \((a+1)^2\).

Here's how it breaks down:

• Start with: \((a+1)^2 = (a+1) \times (a+1)\).
• Expand: Use the distributive property (FOIL method for binomials), you get: \((a \times a) + (a \times 1) + (1 \times a) + (1 \times 1) = a^2 + 2a + 1\).

In addition to expansion, we also distributed \(-5\) across \((a+1)\), yielding

• Distribute: \(-5(a+1) = -5a - 5\).

Algebraic manipulation is about clearing away the clutter so that both calculations and understanding become clearer.

Once these terms are expanded and combined, simplifying involves combining like terms, which reduces the expression to its simplest form: \(a^2 - 3a + 2\). Simplifying helps us understand both specific numerical solutions and the general behavior of functions.

Function Substitution

Function substitution is a powerful tool that allows us to evaluate functions using alternative input values. In this case, to find \(f(a+1)\), we replaced every \(x\) in the original function \(f(x) = x^2 - 5x + 6\) with \(a+1\).

Substitution changes the independent variable \(x\) into a new expression \(a+1\). Here's how to handle it:

• **Identify the function:** Start with \(f(x) = x^2 - 5x + 6\).
• **Substitute:** Replace \(x\) with \(a+1\): \(f(a+1) = (a+1)^2 - 5(a+1) + 6\).

Substitution is essential for evaluating functions for specific values or expressions in more complex cases. It transforms a problem that might seem daunting into a straightforward substitution step, followed by necessary algebraic manipulations.

This technique allows us to explore how functions respond when input values change, which is fundamental for analyzing and interpreting mathematical and real-world situations.