Problem 30 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point \((-3, -3)\)

Step-by-Step Solution

Answer

The standard form equation of the parabola is \(y=-1/3*x^2\)

Step 1: Identify the Form of Equation

Since the parabola has a vertical axis and the vertex at the origin, the equation is in the form \(y=ax^2\).

Step 2: Substitute the given Point

We are given that the parabola passes through the point (-3,-3). Substitute x=-3 and y=-3 into the equation to solve for 'a'. \(-3=a*(-3)^2\). Solve for a.

Step 3: Solve for 'a'

Solving \(-3=a*(-3)^2\) or \(-3=9a\) for 'a' gives \( a=-3/9 \) or \( a=-1/3 \).

Step 4: Write the Final Equation

Substitute \( a=-1/3 \) into the original equation \(y=ax^2\) to get the finalized form of equation which is \(y=-1/3*x^2\).

Key Concepts

Vertex Form

The vertex form of a parabolic equation is essential in understanding the parabola's properties. For a parabola with a vertical axis and the vertex at the origin, it takes a simplified form. Normally, the vertex form is given as \[ y = a(x-h)^2 + k \] where \( (h, k) \) represents the vertex of the parabola. However, when the vertex is at the origin \((0,0)\), this simplifies to \[ y = ax^2 \]. This equation highlights two main points:

• **The term \( 'a' \)** – This coefficient determines the parabola's direction and width. A positive \(a\) opens upwards, while a negative \(a\) opens downwards. Larger values of \(a\) lead to a narrower parabola, while smaller values spread it wider.
• **Simplicity with origin as the vertex** – The form simplifies calculations since the origin \((0,0)\) means there's no need to adjust for the vertex position.

Recognizing this form helps in identifying and manipulating the parabola within graphing problems.

Vertical Axis

Understanding the vertical axis of a parabola is crucial for graphing and solving problems. A vertical axis means the parabola opens either upwards or downwards. In this exercise, the axis is explicitly noted as vertical, leading us to use the formula \( y = ax^2 \).A few key points about the vertical axis:

• **Line of symmetry** – The linea divides the parabola into two mirror-image halves. This is helpful for predicting behavior and solving for points on either side of the axis.
• **Focus on simplicity** – A vertical axis significantly simplifies the process, especially when combined with a vertex at the origin. There's no need to adjust formulas for tilting or horizontal shifts.

This understanding aids in quickly establishing the equation you need for problem-solving.

Substituting Points

To find the exact form of the parabola's equation, inserting known points is essential. In this problem, given the parabola passes through the point \((-3, -3)\), we use these values to solve for the unknown coefficient \(a\). Here's how you substitute:

• **Identify the coordinates** – Here, \(x = -3\) and \(y = -3\).
• **Plug these into the equation** – Substitute into the formula \(y = ax^2\) to form the equation \(-3 = a(-3)^2 \).

The process of substituting makes the initially abstract equation yield specific values that define the parabola.

Solving for a

Once the values are substituted into the equation, the next critical step is solving for the coefficient \(a\). It dictates the parabola's direction and width. For the equation \[-3 = a(-3)^2\]Solving means isolating \(a\):

• Evaluate \((-3)^2 = 9\).
• Rewrite the equation as \(-3 = 9a\).
• Divide both sides by 9, solving gives \( a = -\frac{3}{9} = -\frac{1}{3} \).

With \(a\) found, it can be substituted back into the vertex form equation. Completing the calculation provides the precise equation of the parabola, in this case, \[ y = -\frac{1}{3}x^2 \]. This value directly impacts how you sketch or analyze the parabola in further problems.