Problem 77 - Algebra 2

Question

Write an equation of a parabola opening left with vertex \((0,0)\) and focus \((-3,0)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x=-\sqrt{12y^2}\)

Step 1: Determine the Vertex and Focus

The vertex, \(V\), and the focus, \(F\), are given as \(V=(0,0)\) and \(F=(-3,0)\). They both lie on the x-axis.

Step 2: Determine the Value of p

The coordinate of the focus is \((-3,0)\), it can be proved by the fact that the absolute number of x-coordinate is the value of \(p\). Since the parabola opens to the left, we get \(p = -3\).

Step 3: Substitute the Values into the Equation

Given that \(h=0\), \(k=0\) and \(p=-3\), plug these values into the standard equation of a parabola \(4p(y-k)^2=(x-h)^2\). It becomes \(4(-3)y^2=x^2\), which simplifies to \(12y^2=x^2\). Thus, \(x^2=12y^2\). Because the parabola opens to the left, \(x\) should be negative, hence the equation should be written as \(x=-\sqrt{12y^2}\)

Key Concepts

Vertex Form of a Parabola

When dealing with parabolas, the **vertex form** is an essential concept. This form of the equation allows you to easily identify the vertex of the parabola, which is a point where the curve turns. The vertex of a parabola in vertex form is represented by the formula \[ y = a(x-h)^2 + k \] where

• \((h, k)\) are the coordinates of the vertex
• \(a\) determines the "width" or "narrowness" of the parabola's arms and its direction (upwards or downwards)

For a parabola where the vertex is at the origin \(0,0\), it simplifies the equation considerably because

• \(h = 0\) and \(k = 0\)
• the equation becomes \(y = ax^2\)

In our exercise example, since the parabola opens horizontally (either left or right), the concept is inverted, and the formula is used in the form of \[ x = a(y-k)^2 + h \] giving insights into how x and y interact in this horizontal orientation.

Focus of a Parabola

Understanding the focus of a parabola is key to mastering parabolic equations. The **focus** is a special point that lies along the axis of symmetry of a parabola. It dictates how "steep" or "wide" the parabola is. When a parabolic equation is represented as \[ y = ax^2 \] or \[ x = ay^2 \], its focus has a crucial relationship with the parameter \(p\), defined as the distance from the vertex to the focus.

• In our scenario, where the vertex is at \(0,0\) and the focus is \((-3,0)\), the distance \(p\) is -3 because the focus is along the negative x-direction.
• This tells us the parabola opens horizontally, specifically towards the left.

The distance \(p\), in combination with the vertex, helps to determine the precise equation of the parabola, by setting how far the parabolic arms "extend" in relation to this focus point.

Parabola Opening Direction

The **opening direction** of a parabola tells us which direction the arms of the parabola extend, and it is crucial for identifying correct graphs of equations. Parabolas can open in one of four directions: up, down, left, or right.

• When concerning vertical parabolas, an upwards-opening parabola has a positive \(a\) value in the vertex form, while a downwards-opening one has a negative \(a\) value.
• For horizontal parabolas, if the parabola opens to the right, \(p\) is positive; if it opens to the left, \(p\) is negative.

In the provided exercise solution, the parabola opens to the left, indicated by the focus being to the left of the vertex at \((-3,0)\). This requires a negative \(p\) value in its equation to correctly reflect the direction. Such clarity between horizontal and vertical parabolas aids in determining accurate graphical representations and crafting the correct algebraic formulas. By checking the direction of opening, one can anticipate the behavior and set the associated equations appropriately.