Problem 36 - Calculus Early Transcendentals

Question

Find an equation for the conic that satisfies the given conditions. Parabola, horizontal axis, passing through \((-1,0),(1,-1),\) and \((3,1)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem requires finding the equation of a parabola with a horizontal axis passing through three given points: \((-1,0)\), \((1,-1)\), and \((3,1)\). A horizontal parabola typically has the form \(x = a(y-k)^2 + h\). Our goal is to determine the constants \(a\), \(h\), and \(k\).

Step 2: State the Equation

We assume the equation of the parabola as \(x = a(y-k)^2 + h\). We will substitute each of the points \((-1,0)\), \((1,-1)\), and \((3,1)\) into this equation to create three equations with unknowns \(a\), \(h\), and \(k\).

Step 3: Substituting the First Point

Substitute \((-1,0)\) into the equation: \(-1 = a(0-k)^2 + h\). This simplifies to \(-1 = ak^2 + h\).

Step 4: Substituting the Second Point

Substitute \((1,-1)\) into the equation: \(1 = a(-1-k)^2 + h\). This provides the second equation: \(1 = a(k+1)^2 + h\).

Step 5: Substituting the Third Point

Substitute \((3,1)\) into the equation: \(3 = a(1-k)^2 + h\). This gives the third equation: \(3 = a(k-1)^2 + h\).

Step 6: Solve the System of Equations

We solve the system of three equations: \(-1 = ak^2 + h\), \(1 = a(k+1)^2 + h\), \(3 = a(k-1)^2 + h\). Start by solving for \(h\) using any two equations and then find \(a\) and \(k\).

Step 7: Simplify and Compute 'a', 'h', and 'k'

From the first two equations: \[ak^2 + h = -1,\]\[a(k+1)^2 + h = 1\]Subtract to eliminate \(h\):\[a(k+1)^2 - ak^2 = 2\]\[a(k^2 + 2k + 1 - k^2) = 2\]\[2ak + a = 2\]Assuming \(k = 0\), solve for \(a\):\[a = 2\]And using \(k = 0\) in \(-1 = ak^2 + h\):\[h = -1\]Thus, \(x = 2y^2 - 1\).

Step 8: Validate the Solution

Substitute back the points to ensure they satisfy the equation: - For \((-1,0)\): \(-1 = 2(0)^2 - 1\). - For \((1,-1)\): \(1 = 2(-1)^2 - 1\). - For \((3,1)\): \(3 = 2(1)^2 + 1\).All points satisfy the equation, validating our solution is correct.

Key Concepts

parabola

A parabola is a U-shaped curve that is defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed straight line, called the directrix. Parabolas can open upwards, downwards, or horizontally. In basic geometry, when the parabola opens upwards or downwards, it is commonly expressed in the standard form of a vertical parabola:

• \(y = ax^2 + bx + c\)

However, for parabolas that open horizontally, the equation looks a bit different, as it's based on a similar principle but pivoted to the side. Parabolas are one of the four conic sections, alongside ellipses, circles, and hyperbolas. These are important shapes in mathematics because they model many real-world structures and paths, like satellite dishes and projectile paths.
Understanding the orientation of a parabola helps in graphing it and solving problems that involve finding the intersection of points or the vertex of the parabola.

horizontal axis

A horizontal axis in the context of parabolas refers to the symmetry axis of the curve when the parabola opens to the left or right. This is in contrast to the vertical axis in vertically oriented parabolas. The horizontal axis is crucial when determining the direction in which the parabola opens. Often, the equation of such a parabola is given as:

• \(x = a(y-k)^2 + h\)

Here, the "horizontal axis" refers to a line parallel to the x-axis, and the shape of the parabola is determined by the vertex

• (h, k),

which is also the point of maximum or minimum value.
A horizontal axis helps visualize how the parabola stretches across the plane, impacting how the bases of equations like quadratic systems are built. In real-world terms, this was used to determine objects that might have a wider horizontal span, like bridges or road signs, which mirror the parabola’s signature curve.

equation of a parabola

The equation of a parabola describes how it is positioned on the Cartesian plane. In a typical scenario, the equation takes different forms depending on the direction in which it opens. For a parabola with a horizontal axis, the equation is usually:

• \(x = a(y-k)^2 + h\)

Where \(a\) affects the width of the parabola (with larger absolute values of \(a\) leading to narrower parabolas),

• \(h\) and \(k\) designate the vertex (horizontally and vertically),

respectively.
This quadratic form allows one to use specific points through which the parabola passes to determine the unknown constants. For example, given points

• \((-1,0), (1,-1),\) and \((3,1)\),

one can substitute these into the equation to derive values for \(a\), \(h\), and \(k\). This process helps confirm the parabola not only fits the given points but also adheres to its natural geometric properties. Knowing the specific equation of a parabola becomes especially useful in fields like physics and engineering, where precise modeling is needed.