Problem 39 - College Algebra with Modeling and Visualization

Question

Find an equation of the parabola with vertex \((0,0)\) that satisfies the given conditions. Horizontal axis, passing through \((1,-2)\)

Step-by-Step Solution

Answer

The equation of the parabola is \( y^2 = 4x \).

Step 1: Identify the Parabola Form

Since the parabola has a horizontal axis, its general equation is of the form \( y^2 = 4px \). The parabola opens to the right or the left depending on the sign of \( p \). Since the vertex is at the origin \((0,0)\), the equation simplifies to \( y^2 = 4px \).

Step 2: Use Given Point to Find 'p'

We know the parabola passes through the point \((1, -2)\). Substitute \(x = 1\) and \(y = -2\) into the equation \( y^2 = 4px \):\( (-2)^2 = 4p(1) \).This gives us \( 4 = 4p \), so \( p = 1 \).

Step 3: Write the Final Equation

Substitute the value of \( p \) back into the parabola's equation, we get:\( y^2 = 4(1)x \) or \( y^2 = 4x \). This is the equation of the parabola with the vertex at \((0,0)\) and a horizontal axis passing through the point \((1, -2)\).

Key Concepts

Vertex Form

The vertex form of a parabola's equation is a special way to express it, focusing on the parabola's vertex—a point that's often key to understanding its shape and orientation. The vertex of a parabola is its "peak"; the highest or lowest point depending on the orientation of the parabola. For a vertical parabola, the vertex form is given by:

• Vertical Parabola: \( y = a(x-h)^2 + k \)

Here, \((h, k)\) represents the vertex. The parameter \(a\) affects the "width" and direction of the parabola. If \(a\) is positive, the parabola opens upwards, and if negative, downwards.
However, for horizontal parabolas, like in our exercise, the equation is structured differently:

• Horizontal Parabola: \( x = a(y-k)^2 + h \)

In this form, \((h, k)\) is again the vertex. Here, if \(a\) is positive, the parabola opens to the right, and if negative, to the left. In our problem, the vertex is at \((0, 0)\), simplifying the vertex form to either \( y^2 = 4px \) or \( x^2 = 4py \) depending on the axis direction.
Understanding the vertex form helps in quickly identifying the vertex, direction, and size of a parabola just by looking at its equation.

Horizontal Axis

When a parabola has a horizontal axis, it means it opens either left or right instead of the usual up or down. This impacts the orientation and shapes the form of its equation. In such cases, the standard form of the parabola equation shifts from the usual \( y = ax^2 + bx + c \) to:

• For horizontal open parabolas: \( y^2 = 4px \).

Here, \( p \) is a parameter that represents the distance from the vertex to the focus, and \( 4p \) dictates how "wide" or "narrow" the parabola will look. In our solved exercise, because the parabola passes through the point \((1, -2)\), plugging these coordinates into \( y^2 = 4px \) allows us to solve for \( p = 1 \) confirming the horizontal direction.
This equation makes it clear that, if we were to graph it, the parabola would start from the vertex at \((0, 0)\) and would "head" to the right, given that \( p \,= \,1 \) is a positive value.
Such understanding allows us to easily plot and understand the behavior of the curve on a graph without plotting multiple points.

Focus and Directrix

The concepts of focus and directrix are fundamental in understanding the properties of a parabola. A parabola is defined as the set of all points equidistant from a point known as the "focus" and a line called the "directrix." This geometric property not only defines the shape but also affects the analytic form of the equation.

• Focus: This point lies along the axis of symmetry of the parabola. For the equation \( y^2 = 4px \), the focus is located at \((p, 0)\).
• Directrix: It is a line that is perpendicular to the axis of symmetry of the parabola. For the equation \( y^2 = 4px \), the directrix is the line \( x = -p \).

In our example, since \( p = 1 \), the focus would be at \((1, 0)\) and the directrix would be the vertical line \( x = -1 \). These two components help define how the parabola opens and its orientation, painting the full picture beyond just the vertex's position.
Visualizing the focus and directrix helps learners understand why a parabola "bends" the way it does and provides a more comprehensive grasp of the curve's geometric nature.