Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(y=a(x-h)^{2}+k\) for \(a \quad\) (Mathematics)

Step-by-Step Solution

Answer

\(a = \frac{y - k}{(x-h)^2}\)

Step 1: Isolate the Squared Term

To solve for \(a\) in the equation \(y = a(x-h)^2 + k\), we must first isolate the term containing \(a\). Start by subtracting \(k\) from both sides of the equation:\[y - k = a(x-h)^2\]

Step 2: Solve for 'a'

Now that we have \(y - k = a(x-h)^2\), divide both sides by \((x-h)^2\) to solve for \(a\):\[a = \frac{y - k}{(x-h)^2}\]

Key Concepts

Isolating Variables

When solving equations, especially when dealing with multiple variables, isolating a specific variable is often the first important step. This means you want to get the variable you're solving for alone on one side of the equation.
This step simplifies the equation and makes it easier to see what needs to be done next.Here's how you can approach isolating variables:

• Identify the variable you need to isolate. Look at the terms in the equation and find where your target variable is located.
• Perform the opposite operation to move other terms across the equal sign. For example, if a term is added, subtract it from both sides; if it's multiplied, divide both sides by it.
• Keep the equation balanced by doing the same operation on both sides.

In our exercise, to solve for \(a\) in the formula \(y = a(x-h)^2 + k\), we started by isolating the term \(a(x-h)^2\). We did this by subtracting \(k\) from both sides of the equation. This left us with \(y - k = a(x-h)^2\), which is a simpler form to work with.

Factoring

Factoring is a common strategy used in solving equations, especially when working with polynomials. It involves rewriting an expression as a product of its factors, which can simplify the problem and make it easier to solve.Basic steps to factor an expression include:

• Look for common factors in each term. If present, factor them out.
• Apply special factoring formulas such as difference of squares, or perfect square trinomials, if applicable.
• Re-write the expression as a product of its factors.

Though our exercise involves a quadratic component \((x-h)^2\), factoring wasn't needed directly in solving for \(a\) due to the structure of the equation. Instead, operations were performed to isolate and solve for \(a\). Factoring is more commonly used in solving quadratic equations or simplifying expressions before solving.

Quadratic Equations

Quadratic equations are a type of polynomial equation that include a squared term as their highest degree term. These equations take the general form \(ax^2 + bx + c = 0\). They appear frequently in algebra, and can be solved by different methods such as factoring, completing the square, or using the quadratic formula.In our exercise, the term \((x-h)^2\) suggests the presence of a quadratic form, but since the problem asked for isolation of a variable \(a\), rather than solving the equation, traditional methods of solving quadratic equations didn't apply.
To solve quadratic equations generally, you might:

• Factor the quadratic if possible, and then set each factor equal to zero.
• Use completing the square to rewrite the equation in a solvable form.
• Apply the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), if the equation cannot be factored easily.

Understanding these methods is crucial for tackling more complex algebraic problems involving quadratic equations.