Problem 26 - Precalculus

Question

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 \(y = -1/3x^2\).

Step 1: Setup the general equation of a parabola.

Based on the problem, it is known that the vertex is at the origin and the parabola has a vertical axis. Therefore, the equation of the parabola can be written as \(y = ax^2\). Now we substitute the given point (-3, -3) into this equation to derive the value of 'a'.

Step 2: Substitute the point (-3, -3) into the equation

The point (-3,-3) tells us that when \(x = -3\), \(y = -3\). By replacing x and y coordinates with this information, we get \(-3 = a(-3)^2\). This simplifies to \(-3 = 9a\). Now, solving this equation next will help to find 'a'.

Step 3: Solve for 'a'

To find the value of 'a', we simply divide both sides of the equation by 9. That would yield: \(a = -3/9\), which simplifies to \(a = -1/3\).

Step 4: Write the standard form equation of the parabola

Now that 'a' is known, we can finalize the parabolic equation by substituting 'a' back into the general form. This gives us the final equation \(y = -1/3x^2\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a significant feature, marking the turning point or apex of the curve. In the context of a parabola with its vertex at the origin, like in our exercise, the coordinates of the vertex are (0,0). This specific location simplifies the equation and helps in understanding the parabola's shape and direction.

For a parabola opening upwards or downwards, the vertex will be the minimum or maximum point respectively. When the standard equation of a parabola is given by \(y = ax^2 + bx + c\), if the coefficient \(a\) is positive, the parabola opens upwards, and if it's negative, the parabola opens downwards. Identifying the vertex is crucial in graphing the curve and solving related optimization problems.

Parabolic Equations

Parabolic equations, such as \(y = ax^2 + bx + c\), define the U-shaped curve that we know as a parabola. The standard form is particularly valuable because it makes certain properties like the vertex and direction of opening immediately apparent. In the standard form, \(a\), \(b\), and \(c\) are constants, where \(a\) is not zero, as it dictates the curvature of the parabola.

When you have additional information, like a point through which the parabola passes—as in the given exercise—you can further refine the equation by solving for the coefficients. Remember that substituting the coordinates of any point on the parabola into the standard form equation allows you to solve for these unknowns, just as we found the value of \(a\) by substituting \((-3, -3)\) and using the fact that the vertex is at the origin.

Vertical Axis of Symmetry

Every parabola has an axis of symmetry, a straight line that divides it into two mirror images. For parabolas that open upwards or downwards, this axis of symmetry is vertical. The vertical axis of symmetry passes through the vertex, meaning for the parabola described in our exercise, it lies along the y-axis.

This axis is crucial when graphing as it shows that for every point on one side of the parabola, there is an identical point mirrored across the axis. Mathematically, for a parabola in standard form \(y = ax^2 + bx + c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\). However, when the vertex is at the origin, like with our equation \(y = -\frac{1}{3}x^2\), the axis of symmetry is simply the y-axis, or x = 0.