91Ó°ÊÓ

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is about finding the unknown or placing real-life variables into equations and then solving them. In this exercise, algebra is used to manipulate the expression for \( f(a+h) \). For instance, expanding \( (a+h)^2 \) follows the algebraic rule:

• \((a + b)^2 = a^2 + 2ab + b^2\)

To break it down, you multiply everything inside the parenthesis:

• First, \(a \times a = a^2\)
• Then, \(a \times h = ah \)
• Next, \(h \times a = ah \)
• Lastly, \(h \times h = h^2 \).

Combine all these and simplify to yield \(a^2 + 2ah + h^2\). Understanding these rules makes algebra easier as it builds up the foundational structure in solving equations.

• Tip: Memorize key multiplication formulas like \( (a+b)^2 \) and practice them often.

Linear equations describe a relationship between two variables that appears as a straight line when graphed. A basic form is \( y = mx + b \), where \(m \) is the slope and \(b \) is the y-intercept.

In this exercise, we derive the slope formula by analyzing the change in the y-values over the change in x-values for two points given by the function \(f(x) = x^2 + 5 \), which is a quadratic function. While this is not a linear equation itself, the concept of finding the slope between two points utilizes linear equation principles.

The formula for the slope \( m \) between any two points \((x_1, y_1) \) and \((x_2, y_2) \) is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula conceptually represents how a linear equation's slope is calculated by the rise over run method.

Function evaluation involves replacing the variable in the function with a given number and then performing the operations. For the function given by \(f(x) = x^2 + 5\), evaluating at specific points is crucial to finding the slope. Let's see this step-by-step:

• Evaluate the function at point \(a:\)
  \( f(a) = a^2 + 5 \)


• Evaluate at point \(a + h:\)
  \(f(a + h) = (a+h)^2 + 5 \)

Steps: Substitute \(a + h \) into \( x \) operating within the function:

• First calculate the square: \( (a+h)^2 = a^2 + 2ah + h^2 \)
• Then add 5 to the result: \(a^2 + 2ah + h^2 + 5 \)

This updated formula becomes integral to finding the difference in y-values, which is used in the slope calculation.

Simplifying expressions means condensing them to their simplest form, making them easier to understand and work with. During the solution, several steps involve simplifying the equation:

For example, in the slope calculation step:

• Begin by substituting values into the slope formula:
  \[ m = \frac{(a^2 + 2ah + h^2 + 5) - (a^2+5)}{(a+h) - a} \]
• Next, cancel out common terms in the numerator:
  \[ a^2 + 5 \], which leaves us with \[2ah + h^2\]
• Divide each term by \(h \) in the remaining expression:
  \[ m = \frac{2ah + h^2}{h} \]
• Lastly, factor out \(h\) from the numerator and cancel it with the denominator:
  \[ m = 2a + h \]

In this way, each step becomes more manageable and leads to the simplest form.