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Chapter 2: Problem 55

Writing the Equation of a Parabola In Exercises \(47-56\) , write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: \(\left(-\frac{5}{2}, 0\right) ;\) point: \(\left(-\frac{7}{2},-\frac{16}{3}\right)\)

Short Answer

Expert verified
The equation of the parabola in standard form is \(y = -16/3 (x + 5/2)^2\).

Step by step solution

01

Identify the given points

From the given exercise, identify the vertex (h, k) as \(-\frac{5}{2}, 0\) and another point on the parabola as \(-\frac{7}{2},-\frac{16}{3}\).
02

Substitute the given points into the standard equation

Substitute (h, k) and the given point into the standard form equation of the parabola, \(y = a(x-h)^2 + k\), to solve for 'a'. Substitute h = -5/2, k = 0, x = -7/2, y = -16/3 into the equation to get -16/3 = a(-7/2+5/2)^2 + 0.
03

Solve the equation to find the value of 'a'

Solve -16/3 = a(-7/2+5/2)^2 + 0 to find the value of 'a'. This simplifies to -16/3 = a(-1)^2, which then simplifies to -16/3 = a. Therefore, a = -16/3.
04

Substitute the values of 'a', 'h', and 'k' into the standard form

Substitute a = -16/3, h = -5/2, and k = 0 into the standard form, which is \(y = a(x-h)^2 + k\). This will give the equation of the parabola as \(y = -16/3 (x + 5/2)^2\). This is the equation of the parabola in standard form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Parabola
The standard form of a parabola's equation provides a clear picture of its graph and important characteristics. It is typically expressed as
\(y = a(x-h)^2 + k\)
where \(h\) and \(k\) represent the coordinates of the vertex鈥攖he highest or lowest point on the parabola鈥攁nd \(a\) dictates the direction and width of the parabolic opening.
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), the parabola opens downwards.
  • The greater the absolute value of \(a\), the 'steeper' or 'narrower' the parabola.
  • The smaller the absolute value of \(a\), the 'wider' the parabola.
Understanding the standard form enables students to sketch the graph easily and identify the vertex without completing a full square.
Vertex of a Parabola
The vertex of a parabola is a cornerstone concept in quadratic functions. It's the point where the parabola changes direction, and it's located at the coordinates \( (h, k) \) when the parabola's equation is in standard form. The vertex can serve as a starting point when graphing a parabola and is of vital importance for various applications in physics, engineering, and economics.
In the exercise, the vertex is given as \(\left(-\frac{5}{2}, 0\right)\), making it straightforward to plot. By comparison, when the vertex is not provided, one would need to complete the square on a general quadratic equation to find the vertex, which adds a few extra steps.

Finding the Optimal Path

For instance, many real-life optimization problems, like finding the optimal path of a projectile or maximizing revenue in business, are related to analyzing the vertex of a parabola.

Solving Parabolas
Solving for the specifics of a parabola requires identifying the vertex and the value of \(a\), which affects the open direction and width of the parabola. In our exercise, the steps to solve for the parabola's equation begin by substituting the vertex and another point on the parabola into the standard form equation.
Once you have substituted the points, the next step is to solve for \(a\). This is done by rearranging the equation and isolating \(a\). The value of \(a\) is crucial because it tells us how the parabola behaves. In our exercise, we found that \(a = -16/3\), which means our parabola opens downward and is relatively steep.

Interpreting the Solution

Interpreting the solution involves analyzing the sign and magnitude of \(a\) alongside the vertex. This process encapsulates understanding the shape and positioning of the parabola on the coordinate grid鈥攁 fundamental step before sketching its graph or applying the equation to problem-solving scenarios.

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Most popular questions from this chapter

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