Problem 27 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Horizontal axis and passes through the point (-2,5)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola given is \(y=1.25x^2\).

Step 1 - Identify the Vertex

The vertex of the given parabola is at the origin, given by coordinates (0,0). Therefore, the values of h and k in the standard equation \(y=a(x-h)^2 + k\) is 0. Thus the equation simplifies to \(y=ax^2\).

Step 2 - Substitute the given point

We know the parabola goes through the point (-2, 5). This gives us an x and y that we can substitute into the equation \(y=ax^2\), which will allow us to solve for \(a\). Substituting the values, we get: \(5=a(-2)^2\).

Step 3 - Solve for a

We now need to solve the equation for \(a\). The (-2) squared is 4. So the equation is now: \(5=4a\). To solve for \(a\), we divide both sides by 4 to give: \(a=5/4=1.25\).

Step 4 - Write out the standard form equation of the parabola

Now that we have solved for \(a\), we substitute the value back into the equation to get: \(y=1.25x^2\). This is the standard form equation of the given parabola with the vertex at the origin and passing through the point (-2, 5).

Key Concepts

Standard Form

In mathematics, especially when dealing with parabolas, the term "standard form" is used to describe a specific way of writing equations. The standard form for a quadratic equation representing a parabola with a vertical or horizontal axis is \( y = a(x-h)^2 + k \), where:

• \(a\) determines the opening direction and the width of the parabola.
• \((h, k)\) is the vertex of the parabola.

When the vertex is at the origin, meaning \((h, k) = (0, 0)\), the equation simplifies significantly. It reduces to \( y = ax^2 \) for a vertical parabola or \( x = ay^2 \) for a horizontal parabola.
This simplified form makes it easier to understand the shape and direction of the parabola with a few parameters.

Vertex

The vertex of a parabola is a crucial point that reveals key characteristics about the parabola's graph. It can be considered the "tip" or "peak" of the parabola. In the standard form equation \( y = a(x-h)^2 + k \), the vertex is expressed as the point \((h, k)\).

• A parabola that opens upwards or downwards has its vertex as the lowest or highest point, respectively.
• When the vertex is at the origin, as in our exercise, the parabola takes a symmetric and often simpler form.

Understanding where the vertex is positioned helps in sketching the parabola and predicting how it behaves. In many problems, identifying the vertex can simplify the equation significantly.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in one variable, typically in the form \( ax^2 + bx + c = 0 \). For parabolas, however, the quadratic function is more commonly associated with equations like \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \).

• The term including \(x^2\) or \(y^2\) signifies the quadratic nature, making the graph a parabola.
• The coefficient \(a\) affects how "steep" or "shallow" the parabola becomes.

In situations where the vertex form is used, such as \( y = a(x-h)^2 + k \), recognizing the equation's quadratic nature is important for solving and understanding its properties.
This foundational understanding can guide further exploration into properties like the axis of symmetry, direction of opening, and maximum or minimum values.