Problem 81 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{10},-5) ;\) opening downward

Step-by-Step Solution

Answer

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

Step 1: Understand the Parabola's Equation

Parabolas with the vertex at the origin have equations in the form \(y = ax^2\) or \(x = ay^2\), depending on whether they open vertically or horizontally. Since the parabola is opening downward, we use the form \(y = ax^2\) with \(a < 0\).

Step 2: Substitute the Given Point

The parabola passes through the point \((\sqrt{10}, -5)\). Substitute this point into the equation \(y = ax^2\): \(-5 = a(\sqrt{10})^2\).

Step 3: Solve for the Coefficient 'a'

Simplify the equation: \(-5 = a \times 10\). Divide both sides by 10 to solve for \(a\): \(a = -\frac{1}{2}\).

Step 4: Write the Final Equation

Substitute \(a = -\frac{1}{2}\) back into the equation of the parabola: \(y = -\frac{1}{2}x^2\). This is the equation for the parabola that satisfies all the given conditions.

Key Concepts

Vertex Form

Understanding the vertex form of a parabola's equation is crucial in graphing and analyzing its shape. For parabolas that open vertically, the vertex form is structured as:

• \(y = a(x - h)^2 + k\)

Here, \(h\) and \(k\) are the coordinates of the vertex. This form intuitively shows how the parabola shifts around the graph.
When the vertex is at the origin, like in our exercise, the vertex coordinates are \((0, 0)\). This simplifies the equation to:

• \(y = ax^2\)

No horizontal or vertical shifting occurs in this simplified version.

Coefficient Determination

Determining the coefficient \(a\) is a key part of creating a parabola's equation. This value affects the direction and 'width' of the parabola.

• It determines whether it opens upwards or downwards.
• Provides information about its steepness.

A more negative \(a\) means a steeper parabola.
To find \(a\), we use a known point on the parabola; in our case, \(\((\sqrt{10}, -5)\)\).
Substitute this point into the simplified vertex form:

• \(-5 = a(\sqrt{10})^2\)

and solve for \(a\).
This will yield \(a = -\frac{1}{2}\), determining the specific characteristics of our parabola.

Downward Opening

A parabola can open either upward or downward. This direction is controlled by the sign of the coefficient \(a\).
In our instance, the parabola opens downward because \(a < 0\).

• If \(a > 0\), it opens upward.
• If \(a < 0\), it opens downward.

This impacts not only the visual orientation but also the way solutions behave.
A downward-opening parabola indicates maximum values, rather than minimum.
This characteristic is crucial for problems where maximum values need to be determined.

Origin as Vertex

When the vertex is situated at the origin, it significantly simplifies the parabola's equation

• \((0, 0)\) makes transformations around this point easier to calculate

The simplified form, \(y = ax^2\), results with no additional constants beyond \(a\).
This also makes these parabolas simpler to work with algebraically.
Placement at the origin means there's a symmetrical shape around the y-axis.
Understanding this symmetry can assist with visualizing and solving graphical parabola problems efficiently.