Problem 12 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$y+7=4(x+3)^{2}$$

Step-by-Step Solution

Answer

The graph is a vertically oriented parabola.

Step 1: Recognize the Form of the Equation

The given equation is \( y + 7 = 4(x + 3)^2 \). This resembles the standard form of a vertical parabola \( y = a(x - h)^2 + k \). The general form indicates a parabola opens vertically.

Step 2: Identify the Vertex Form Components

Compare the form \( y + 7 = 4(x + 3)^2 \) to \( y = a(x - h)^2 + k \). We find \( a = 4 \), \( h = -3 \), and \( k = -7 \). As it matches the form of \( y = a(x - h)^2 + k \), this confirms it is a parabola.

Step 3: Determine Orientation of the Parabola

Since the equation doesn’t have a negative coefficient for \( a \), the parabola opens upwards. If \( a \) were negative, the parabola would open downwards.

Key Concepts

Parabola

Parabolas are a fundamental type of conic section. They are defined as the set of all points that are equidistant from a fixed point known as the focus and a line called the directrix. These curved shapes are pivotal in understanding quadratic equations and their properties. Parabolas come up in various real-world contexts, such as the arcs in bridges or the paths of projectiles.

Key characteristics of a parabola include:

• Vertex: The highest or lowest point on the parabola, depending on its orientation.
• Axis of Symmetry: A line that runs down the 'middle' of the parabola, effectively splitting it into two mirror-image halves.
• Focus and Directrix: These define the parabola's shape and position, but are not required for finding the vertex form.

Understanding the nature of a parabola's direction—whether upwards or downwards—is essential. This direction is governed by the coefficient of the squared term. For instance, in the equation provided earlier, the positive coefficient indicates the parabola opens upwards.

Equation of a Conic Section

Conic sections are the curves obtained by intersecting a cone with a plane. These include ellipses, circles, hyperbolas, and parabolas. Each type of conic section has its own unique equation form which determines its appearance and properties.

For example, the general form of a conic section is \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Depending on the values of the coefficients, different conic shapes emerge. However, these can often be simplified into a more recognizable form, such as a parabola's vertex form. This simplification process helps identify the type of conic section quickly.

By observing the presence or absence of specific terms and coefficients, like whether the equation involves a squared term, the equation of a conic section reveals its type. In our case, recognizing that the equation is a parabolic form was based on its simplification directly aligning with the vertical axis parabola form.

Vertex Form

The vertex form of a quadratic or a parabola equation is one of the most user-friendly formats for understanding their structure and features. It is expressed as \[ y = a(x - h)^2 + k \]where \( (h, k) \) represents the vertex of the parabola.

Why is the vertex form so appealing? It clearly outlines the transformation components of the parabola,

• \( a \): This acts as a stretch or compression factor, influencing how "wide" or "narrow" the parabola appears. If \( a \) is positive, the parabola opens upwards. If negative, it opens downwards.
• \( h \): This determines the horizontal shift, indicating where along the x-axis the vertex of the parabola sits.
• \( k \): This points out the vertical movement, signifying how far up or down the vertex is located.

In the given problem, the vertex form allowed us to easily identify the parabola's features, determining it opens upwards and locates its vertex at \( (-3, -7) \). This clarity makes the vertex form an essential tool in graphing and understanding parabolas easily.