Problem 70 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Isolate the Quadratic Expression

Given the function \(y=a(x-h)^{2}+k\), first isolate the term involving \(a\). Subtract \(k\) from both sides to begin moving terms that don't involve \(a\): \[ y - k = a(x-h)^{2} \] This rearranges the equation to focus on the term with \(a\).

Step 2: Solve for the Variable \(a\)

With the equation \(y-k = a(x-h)^{2}\), divide both sides by \((x-h)^{2}\) to solve for \(a\):\[ a = \frac{y-k}{(x-h)^{2}} \]This isolates \(a\) as the subject of the formula.

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it involves a variable raised to the power of two. In this exercise, the quadratic expression is in the form \(a(x-h)^{2}\), which is part of the function \(y=a(x-h)^{2}+k\). This specific form is known as the vertex form of a quadratic equation, and it is beneficial in identifying the properties of a quadratic, such as its vertex.

• The quadratic term is \(x-h\), which indicates a horizontal shift.
• The square \( (x-h)^{2}\) means the parabola opens either upwards or downwards, depending on the sign and value of \(a\).
• The constant \(k\) shows the vertical shift, moving the entire graph up or down.

Understanding this structure makes it easier to manipulate and solve equations involving quadratics.

Formula Manipulation

Formula manipulation involves rearranging the components of an equation to solve for a specific variable or to simplify the equation. In our exercise, we started with the equation \(y=a(x-h)^{2}+k\) and manipulated it to solve for \(a\). Manipulating formulas is a critical skill in algebra that enables students to:

• Understand relationships between variables by rearranging terms.
• Isolate specific variables to make them the subject of the equation.
• Simplify complex equations for better interpretation or computation.

Effective manipulation starts by recognizing the structure of the formula and knowing which mathematical operations can be applied to both sides without changing the equality. For instance, subtracting \(k\) from both sides began the process of isolating the term \(a(x-h)^{2}\).

Variable Isolation

Variable isolation is the process of rearranging an equation to make one variable the subject. This typically requires reverse operations like addition or subtraction first, followed by division or multiplication. In this problem, we were tasked with isolating \(a\) in the function \(y=a(x-h)^{2}+k\).The isolated form of the equation provides direct information about how \(a\) varies with changes in other variables. Specifically, we:

• First, subtracted \(k\) from both sides to bring \(a(x-h)^{2}\) away from other terms.
• Next, divided both sides by \((x-h)^{2}\) to completely isolate \(a\).
• The final isolated expression, \ a = \frac{y-k}{(x-h)^{2}} \, clearly indicates \(a\)'s dependence on \(y\), \(x\), \(h\), and \(k\).

Mastering variable isolation allows for solving equations in various forms, giving us a clearer understanding of mathematical relationships.