Problem 44 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated half of the parabola. Left half of \((x+3)^{2}=y+2\)

Step-by-Step Solution

Answer

The equation for the left half is \(x = -3 - \sqrt{y+2}\).

Step 1: Recognize the Parabola Form

First, recognize that the given equation \((x+3)^{2}=y+2\) is in the form of \((x-h)^{2}=a(y-k)\). This is a standard form of a parabola that opens vertically (either up or down). Here, the vertex form is \((x+3)^2 = y+2\).

Step 2: Identify the Vertex and Axis of Symmetry

The vertex of the parabola can be found from the equation \((x+3)^2 = y+2\) as \((h, k) = (-3, -2)\). Since the term \((x+3)\) is squared, the axis of symmetry is vertical and given by the line \(x = -3\).

Step 3: Determine the Direction of Opening

Since \((x+3)^2\) equals \(y+2\), the parabola opens upwards. This means that for any given \(x\), there is one corresponding \(y\) value above \(y = -2\).

Step 4: Select the Left Half of the Parabola

To find the left half of the parabola, we consider the values of \(x\) that are less than or equal to the axis of symmetry. Therefore, we restrict the parabola to \(x \leq -3\).

Step 5: Express Parabola Equation for the Left Side

Since we need the equation as a function of \(y\), solve \(x+3 = \pm\sqrt{y+2}\). For the left half, select the negative square root: \(x = -3 - \sqrt{y+2}\). This represents the left side of the parabola.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is an alternative way to express a quadratic equation. It helps us easily identify key features of the parabola. This form is useful because it focuses on the parabola's vertex, which is its highest or lowest point, depending on how it opens. The standard vertex form is given by \[ (x-h)^2 = a(y-k) \] or \[ (y-k)^2 = a(x-h) \] depending on the orientation of the parabola.

• Here, \( (x-h)^2 = a(y-k) \) defines a parabola opening upwards or downwards, with \( h \) and \( k \) being the coordinates of the vertex.
• The value of \( a \) determine how stretched or compressed the parabola is. It also influences the direction of opening.

To see how this works, consider the equation \( (x+3)^2 = y+2 \). This equation matches the vertex form with a vertex at \( (-3, -2) \). Understanding the vertex form makes it easier to analyze and graph parabolas.

Axis of Symmetry

The axis of symmetry is a crucial feature of parabolas. It is a line that divides the parabola into two mirror-image halves. Imagine a line that cuts the parabola right in the middle. This line goes through the vertex and every point is equidistant from both sides of the parabola.

• For parabolas in the form \( (x-h)^2 = a(y-k) \), the axis of symmetry is vertical and given by \( x = h \).
• For our particular example, with a vertex form \( (x+3)^2 = y+2 \), the axis of symmetry is \( x = -3 \).

Recognizing the axis of symmetry not only helps divide the parabola into left and right halves but also aids in graphing, as it simplifies locating other points by reflection across this line.

Direction of Opening

Pretty much every parabola opens in a specific direction, either up, down, left, or right. The vertex form of a parabola gives insight into this direction based on its equation structure.

• An equation in the form \( (x-h)^2 = a(y-k) \) will open vertically. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
• In our example, \( (x+3)^2 = y+2 \), there's an implicit positive \( a \)= 1 (before the square term), indicating the parabola opens upwards.

This means for every point on the x-axis, there's a corresponding y point above the vertex \((-3, -2)\). Understanding the direction of opening is essential for sketching the parabola and solving practical problems involving its shape.

Half of a Parabola

Parabolas can be split into left and right halves by their axis of symmetry. Each half can represent a specific mathematical situation. For instance, in some problems, you might only want to graph or analyze one half.

• In our example, we focused on the left half of the parabola. With an axis of symmetry at \( x = -3 \), this means considering values \( x \leq -3 \).
• To represent this left half, when solving \( x+3 = \pm\sqrt{y+2} \), choose the negative square root: \( x = -3 - \sqrt{y+2} \).

This equation form describes only the left side of the parabola, simplifying problems where complete symmetry isn't necessary. Understanding how to isolate one half is beneficial for focused graphing and interpreting the parabolic behavior in different scenarios.