Problem 74 - Algebra 2

Question

Each point lies on a parabola with vertex \((0,2) .\) Write the equation of the parabola. $$ (1,3) $$

Step-by-Step Solution

Answer

The equation of the parabola is \(y = x^2 + 2\).

Step 1: Substitute the vertex into the parabola equation

Given a vertex of (0,2), the equation of the parabola can be written as \(y = a(x-0)^2 + 2\), which simplifies to \(y = ax^2 + 2\).

Step 2: Substitute the given point into the equation

Substitute the given point (1,3) into the equation so as to solve for 'a'. This gives \(3 = a(1)^2 + 2\), which simplifies to \(3 = a + 2\).

Step 3: Solve for 'a'

Solving the above equation for 'a' gives \(a = 3 - 2\), which simplifies to \(a = 1\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola’s equation is particularly useful because it gives direct information about the vertex. The general equation in this form is \( y = a(x-h)^2 + k \), where \( (h,k) \) is the vertex of the parabola. This form is quite appealing as it tells us straight away:

• The coordinates \( (h, k) \) are where the peak or the lowest point of the parabola is.
• The value of \( a \) affects the "width" and the direction in which the parabola opens. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Using this form, students can quickly sketch the parabola by plotting its vertex and determining whether it opens upwards or downwards. This simplicity makes it incredibly valuable for solving problems involving parabolas. Indeed in our example, starting from the vertex \( (0, 2) \), we could initially express the parabola as \( y = a(x-0)^2 + 2 \).

Equation of a Parabola

The equation of a parabola serves as a description of the curve itself. In its simplest form, the standard equation is \( y = ax^2 + bx + c \). However, this general form can also be adapted into the vertex form \( y = a(x-h)^2 + k \) by completing the square or other methods.

An important point in deriving the vertex form from the given vertex is knowing the additional constraints like a point through which the parabola passes. In our example, given the vertex \( (0, 2) \) and a point \( (1, 3) \) on the parabola, we could use these to ascertain the specific value of \( a \). Substituting these into our vertex form \( y = a(x-0)^2 + 2 \), we find that \( a \) equates to \( 1 \). Thus, the complete equation becomes \( y = x^2 + 2 \).

Parabola Calculation

Calculating the specific details of a parabola, such as determining its equation, is a process that relies on understanding its key properties. In many exercises:

• First, identify the known values: often the vertex and at least one other point on the parabola.
• Substitute these known values into the vertex equation \( y = a(x-h)^2 + k \).
• Solve for \( a \) to find the specific equation of the parabola. This often involves basic algebraic manipulation.

In our scenario, beginning with the vertex \( (0, 2) \) simplifies our starting form to \( y = ax^2 + 2 \). Then, using the point \( (1, 3) \) plugged into this equation, it evolves into \( 3 = a \times 1^2 + 2 \). From there, solving gives \( a = 1 \), finalizing the equation to \( y = x^2 + 2 \). This step-by-step identification and substitution approach helps in visualizing the path from general concept to specific expression.