Problem 82 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Through \((-3,3) ;\) opening to the left

Step-by-Step Solution

Answer

The equation is \( y^2 = -3x \).

Step 1: Understand the Standard Form

The equation of a parabola with the vertex at the origin \((0, 0)\) is given by \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus. For a parabola opening to the left, \(p\) should be negative. The focus will be located at \((-p, 0)\).

Step 2: Substitute the Given Point into the Equation

The point \((-3, 3)\) lies on the parabola. Substitute \(x = -3\) and \(y = 3\) into the equation \(y^2 = 4px\) to find the value of \(p\). Substitute:\[3^2 = 4p(-3)\] This simplifies to: \[9 = -12p\].

Step 3: Solve for p

To find the value of \(p\), solve the equation from the previous step: \[9 = -12p\]Divide both sides by \(-12\): \[p = -\frac{9}{12} = -\frac{3}{4}\].Thus, the value of \(p\) is \(-\frac{3}{4}\).

Step 4: Write the Final Equation

Now that we know \(p = -\frac{3}{4}\), substitute this value back into the standard form equation \(y^2 = 4px\): \[y^2 = 4\left(-\frac{3}{4}\right)x\] This simplifies to: \[y^2 = -3x\].

Key Concepts

Vertex Form

The vertex form of a parabola is a method of expressing the equation of a parabola that highlights its vertex. The general formula is \[ y = a(x-h)^2 + k \]
where

• \( (h, k) \) stands for the coordinates of the vertex.
• \( a \) determines the direction and size of the parabola's opening.

This form is particularly useful when dealing with the vertex, as it allows us to easily identify the vertex without much computation.
In situations where the parabola is opening horizontally, you might encounter the equation in the form \[ x = a(y-k)^2 + h \].The primary advantage of the vertex form is its ability to clearly show transformations like translations and reflections.Understanding the vertex form is key in simplifying the process of graphing parabolas. Smaller modifications to the parameters \( h \) and \( k \) in the equation can give you insight into how the entire graph shifts.

Focus of Parabola

The focus of a parabola is a special point that, together with the directrix, defines the shape and position of the parabola. The parabola is the locus of points equidistant from the focus and the directrix.
For a parabola with a vertex at the origin \((0,0)\) and opening horizontally (either left or right), the equation looks like \[ y^2 = 4px \].In this equation:

• \( p \) represents the directed distance from the vertex to the focus.
• A positive \( p \) means the parabola opens to the right, and a negative \( p \) indicates it opens to the left.

After calculating \( p \), the coordinates of the focus can be determined as \((-p, 0)\).The distance \( |p| \) not only tells us where the focus is, but it also gives information about the width and steepness of the parabola's curves.Knowing the focus and the directrix allows you to fully understand the geometric properties that define the parabola. This symmetry and relationship are crucial when solving problems that involve reflective properties of parabolas.

Standard Form of Parabola

The standard form of a parabola's equation is \[ y^2 = 4px \] when the parabola is horizontal, and \[ x^2 = 4py \] when vertical. In these forms:

• \( p \)is the distance from the vertex to the focus or directrix.
• This measurement is crucial as it helps determine the parabola's exact position and orientation.

In contexts such as our original exercise, understanding the standard form helps simplify the calculation of the parabola's properties.
To find the standard form equation, begin by identifying if the parabola opens horizontally or vertically. This starts the process of substitution and transformation needed to solve or sketch the parabola.For the problem with a given vertex at the origin, the equation is \[ y^2 = -3x \], meaning it opens to the left. Identifying these details quickly provides a sketch and understanding of the parabola's behavior and helps visualize its shape and orientation on a graph.