Problem 6 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=-4(2 x-3)+8(2 x+1)$$

Step-by-Step Solution

Answer

The zero of the function is \( x = -\frac{5}{2} \).

Step 1: Expand the Expression

First, expand the expression for the function. Begin with distributing the coefficients across the terms inside the parentheses: \[ f(x) = -4(2x - 3) + 8(2x + 1) \]Distribute -4 and 8:\[ -4 \times 2x = -8x, \quad -4 \times -3 = 12 \]\[ 8 \times 2x = 16x, \quad 8 \times 1 = 8 \]Thus, the expression becomes:\[ f(x) = -8x + 12 + 16x + 8 \]

Step 2: Combine Like Terms

Combine the like terms in the expanded expression from Step 1:First, combine the terms with \(x\):\[ -8x + 16x = 8x \]Then, combine the constant terms:\[ 12 + 8 = 20 \]So, the simplified expression is:\[ f(x) = 8x + 20 \]

Step 3: Find the Zero of the Function

To find the zero of the function, set \( f(x) \) equal to zero and solve for \( x \):\[ 8x + 20 = 0 \]Subtract 20 from both sides:\[ 8x = -20 \]Divide both sides by 8:\[ x = \frac{-20}{8} = -\frac{5}{2} \]

Key Concepts

Function expansion

Function expansion is a crucial step in simplifying expressions to solve equations. It involves distributing and multiplying each term inside the parentheses by the coefficient outside. This helps get rid of any parentheses in the expression, allowing for simpler calculations. In the given problem, we expanded the function by distributing -4 across \((2x-3)\) and 8 across \((2x+1)\). This can be broken down as follows:
\[\begin{align*}-4 \times 2x &= -8x, \-4 \times (-3) &= 12, \8 \times 2x &= 16x, \8 \times 1 &= 8.\end{align*} \]
After expanding all parts, combine them to form one expression without parentheses: \(-8x + 12 + 16x + 8\). Then, you move to the next step, which is simplification.

Combining like terms

After the expression is expanded, we simplify it by combining like terms. Like terms are terms in an expression that have identical variable parts with the same exponents. For example, in our expanded function \(f(x) = -8x + 12 + 16x + 8\), we have like terms that can be grouped together:
- Terms with \(x\): -8x and 16x
- Constant numbers: 12 and 8

To combine these terms, you perform the following calculations:

• Combine the terms with \(x\): \(-8x + 16x = 8x\)
• Combine the constants: \(12 + 8 = 20\)

By merging these, we turn the function into a much simpler one: \(f(x) = 8x + 20\). This makes it easier to take the next steps, such as finding zeros.

Zero of a function

Finding the zero of a function involves determining the point at which the function's output is zero. This means we want to find a value for \(x\) that makes \(f(x) = 0\). For the function \(f(x) = 8x + 20\), we start by setting the function equal to zero:
\[8x + 20 = 0\]
To isolate \(x\), subtract 20 from both sides:
\[8x = -20\]
Finally, divide by 8 to solve for \(x\):
\[x = \frac{-20}{8} = -\frac{5}{2}\]
This means that the zero of the function is \(-\frac{5}{2}\), indicating that if the function \(f(x)\) equals zero, then \(x\) must be \(-\frac{5}{2}\). Finding the zeros is essential, as it provides insights into the function's behavior and the points where it crosses the x-axis.