Problem 80 - Algebra 2

Question

Each point lies on a parabola with vertex \((0,2) .\) Write the equation of the parabola. $$ (2,0) $$

Step-by-Step Solution

Answer

The equation of the parabola is \(y = -\frac{1}{2}x^2 + 2\).

Step 1: Understand the Vertex Form of a Parabola

The vertex form of a parabola's equation is given by \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Since the vertex is given as \((0, 2)\), the equation will have the form \(y = a(x-0)^2 + 2\) or simply \(y = ax^2 + 2\).

Step 2: Use the Given Point to Determine the Value of 'a'

A point \((x, y)=(2, 0)\) lies on the parabola. Substitute this point into the vertex form to find the value of \(a\). By substitution: \(0 = a(2)^2 + 2\).

Step 3: Solve for the Coefficient 'a'

Solve for \(a\) using the equation from Step 2. \[0 = a(2)^2 + 2\] \[0 = 4a + 2\] Subtracting 2 from both sides gives \[-2 = 4a\] Dividing by 4 gives \[a = -\frac{1}{2}\]

Step 4: Write the Final Equation of the Parabola

Now that we have the value of \(a\), substitute it into the vertex form to get the final equation of the parabola: \[y = -\frac{1}{2}x^2 + 2\]

Key Concepts

Vertex Form of a Parabola

Understanding the vertex form of a parabola is vital when you're dealing with quadratic functions. The standard vertex form equation is represented as \(y = a(x-h)^2 + k\), where the vertex is the point \(h, k\), which is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.

In this problem, the given vertex is \(0, 2\), which indicates that the parabola is either at its minimum or maximum when \(x = 0\). The fact that a point with a y-value of 0 (\(2, 0\)) is also on the graph suggests that the parabola opens downwards. Otherwise, the y-values would increase as the x-values move away from the vertex. This insight will help you predict the sign of the coefficient \(a\), which dictates the direction your parabola opens.

Parabola Coefficient Calculation

The coefficient \(a\) in the vertex form of a parabola determines not only the direction the parabola opens but also the breadth or 'width' of the parabola. A larger absolute value of \(a\) means a narrower parabola, while a smaller absolute value of \(a\) indicates a wider one.

To calculate \(a\), use a known point on the parabola to substitute for \(x\) and \(y\) in the vertex form equation. In our example, \(y = -\frac{1}{2}x^2 + 2\) was obtained by substituting the point \(2, 0\) and using algebraic techniques to isolate \(a\). This calculation is crucial as it solidifies the exact shape of your parabola, ensuring that it fits all given points and the vertex accurately.

Substitution Method in Algebra

The substitution method is a foundational algebraic technique where you replace variables with their equivalent values. In the context of finding a parabola’s equation, you substituted the given point's coordinates into the vertex form equation.

By placing \(2\) for \(x\) and \(0\) for \(y\), you formed an equation that could be manipulated to solve for \(a\). This involved standard algebraic operations such as distributing, combining like terms, and isolating variables. Remember, the substitution method is not unique to parabolas; it's a versatile technique useful across various branches of mathematics.