Problem 12 - A Graphical Approach to Precalculus with Limits

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 conic section is a parabola.

Step 1: Identify the Form

Recognize that the given equation is \( y + 7 = 4(x + 3)^2 \). It should be compared to the standard forms of conic sections like ellipses, hyperbolas, parabolas, and circles to determine its type.

Step 2: Rearrange the Equation

Rearrange the given equation to make it more recognizable. By bringing the constant term to the other side, the equation becomes \( y = 4(x + 3)^2 - 7 \) which looks similar to the general form of a parabola.

Step 3: Compare with Standard Parabola Form

The standard form of a vertical parabola is \( y = a(x - h)^2 + k \). Compare this with \( y = 4(x + 3)^2 - 7 \). Here, \( a = 4 \), \( h = -3 \), and \( k = -7 \).

Step 4: Confirm the Conic Section Type

Since the equation \( y = 4(x + 3)^2 - 7 \) matches the standard form of a vertical parabola \( y = a(x - h)^2 + k \), it confirms that the conic section is a parabola.

Key Concepts

parabolas

Parabolas are a type of conic section, which means they are one of the curves formed by intersecting a plane with a cone. A parabola is best known for its "U" shape, whether opening upwards, downwards, or sideways.

• In mathematics, parabolas can be described through their standard form equations.
• One key feature of parabolas is their axis of symmetry, which is a vertical or horizontal line running through the middle.
• They also have a vertex, the point where the parabola changes direction.

Understanding the vertex can help determine other important properties such as the focus and directrix, which define how a parabola behaves and its width. Let's dive deeper into their equations to see exactly how they are represented.

equations of conic sections

Equations of conic sections include those for circles, ellipses, hyperbolas, and parabolas. Each conic section's equation follows a unique pattern;

• Circles have the form: \[ (x - h)^2 + (y - k)^2 = r^2 \]
• Ellipses have the form: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
• Hyperbolas use: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]
• Parabolas follow either \[ y = a(x - h)^2 + k \] or \[ x = a(y - k)^2 + h \]

The equation provided in the original problem \( y = 4(x + 3)^2 - 7 \) matches the standard form of a vertical parabola, \( y = a(x - h)^2 + k \). Recognizing these forms helps quickly identify the type of conic section without graphing. For parabolas specifically, the focus on the vertex form makes them much easier to work with when solving real-world problems.

precalculus

Precalculus is a foundational mathematical course that often serves as preparation for calculus. It revisits many algebraic principles, provides a deeper exploration into functions, and introduces conic sections.

• Focusing on conic sections in precalculus gives students the skills needed to handle graphing, transformations, and equation manipulation without a calculator.
• Understanding the distinguishing equations for different conic sections is crucial for solving more complex problems later on, such as those encountered in calculus.
• In precalculus, students learn to perform and interpret transformations using standard and vertex forms of parabolas extensively, setting a groundwork for later work.

Through these topics, precalculus equips students with the necessary tools to confidently progress in their study of mathematics, intertwining algebraic concepts with geometric shapes like parabolas.