Problem 84 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Through \((3,2) ;\) symmetric with respect to the \(x\) -axis

Step-by-Step Solution

Answer

The equation is \(y^2 = \frac{4}{3}x\).

Step 1: Identify Parabola's Orientation

Since the parabola is symmetric with respect to the \(x\)-axis and has its vertex at the origin, the parabola must open left or right. This indicates that the equation of the parabola will be of the form \(y^2 = 4px\) or \(y^2 = -4px\).

Step 2: Determine the Formula for the Parabola

Given that the parabola passes through the point \((3,2)\), we will test the equation form \(y^2 = 4px\).

Step 3: Substitute Coordinates to Find 'p'

Substitute \((x, y) = (3, 2)\) into the equation \(y^2 = 4px\). This gives us: \(2^2 = 4p(3)\).

Step 4: Solve for 'p'

Simplify the equation from Step 3: \(4 = 12p\). Divide both sides by 12 to find \(p = \frac{1}{3}\).

Step 5: Write the Final Equation

Now that \(p = \frac{1}{3}\), substitute \(p\) back into the equation \(y^2 = 4px\) to write the final equation for the parabola: \(y^2 = \frac{4}{3}x\).

Key Concepts

Vertex

The vertex is a crucial point of a parabola, acting as its midpoint or the point where it turns. In "vertex form," the equation of a parabola is typically shown with

• A vertex at the origin, written as \[(h, k) = (0, 0),\]resulting in simple formulations.
• For the parabola opening left or right, the equation becomes \[y^2 = 4px.\]
• When a parabola's vertex is at the origin, solving becomes simpler as there are no horizontal or vertical shifts.

The vertex provides a starting point for analyzing and solving the parabola's characteristics and is essential for writing its equation.

Symmetry

Symmetry refers to how one side of the parabola mirrors the other. For a parabola symmetric with respect to the x-axis:

• This symmetry means that for every point \((x, y),\)there's a corresponding point \((x, -y)\).
• The equation \[y^2 = 4px\]indicates this symmetrical nature.

Understanding symmetry helps you predict the parabola's structure and simplifies solving. Symmetrical properties are key in graphing and understanding the relationships between points.

Coordinate Geometry

Coordinate geometry, or analytic geometry, enables us to study geometrical shapes in a numerical way. When dealing with parabolas:

• You need points like \((3,2)\)to find key characteristics such as the value 'p' in the parabola equation \[y^2 = 4px.\]
• Using these points in an equation showcases how coordinate geometry mixes algebra with geometry.

This fusion allows for solving problems by plotting shapes on graphs, providing a visual approach to understand and solve parabolas.

Solving Equations

Solving parabola equations involves substituting known values into the general form and solving for unknowns:

• Substitute the point \((3,2)\) into \[y^2 = 4px\] which helps in finding unknown constants like 'p'.
• Begin by plugging in \((3, 2)\). This results in solving for 'p' with calculations like \[4 = 12p,\]leading to \[p = \frac{1}{3}.\]
• Solve incrementally, focusing on each part of the equation step by step.
• Finally, rewrite the equation with the found values, \[y^2 = \frac{4}{3}x,\]representing the parabola.

Step-by-step solving makes the process manageable, ensuring clarity in finding the equation of a parabola.