Problem 3 - 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. $$y=2 x^{2}+3 x-4$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Identify the Equation Type

Recognize that the given equation is in the form of a quadratic equation: \( y = 2x^2 + 3x - 4 \). In general, quadratic equations of the form \( y = ax^2 + bx + c \) represent parabolas.

Step 2: Determine Conic Section from Quadratic

Quadratic equations like \( y = 2x^2 + 3x - 4 \) have the general form of a parabola. The presence of the \( x^2 \) term and the absence of \( y^2 \) or any \( xy \) cross-product terms confirm that this is a parabola.

Key Concepts

Parabolas

A parabola is a U-shaped curve that can open upwards or downwards along the y-axis, or sideways with respect to the x-axis. This depends on the equation form and the orientation of the coefficients. It's one of the simplest forms of conic sections, which makes it fundamental in the study of geometry and algebra.
Parabolas have certain key features:

• Vertex: The highest or lowest point of a parabola, depending on its orientation.
• Axis of Symmetry: A line through the vertex that divides the parabola into two mirror-image halves.
• Focus and Directrix: Points related to the parabola that help in understanding its geometric nature.

Understanding the equation helps in determining these features. In the equation \(y = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). Once you have x, find y using the equation to get the vertex point \((x, y)\). Here's an important note: if the coefficient \(a\) is positive, the parabola opens upwards, and if it is negative, it opens downwards.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These sections include parabolas, circles, ellipses, and hyperbolas, each with distinct properties and equations.
For a better understanding of these sections:

• Parabolas are characterized by having only one exponentially increasing term, like \(x^2\), on either side of the equation.
• Ellipses and circles have both \(x^2\) and \(y^2\) terms with coefficients of the same sign.
• Hyperbolas involve \(x^2\) and \(y^2\) terms, but with opposite signs.

Identifying which type of conic section you have is crucial, especially without graphing. By recognizing the term structure and coefficient styles, you can predict the graph's nature. In the given equation \(y = 2x^2 + 3x - 4\), the presence of the \(x^2\) term suggests it's a parabola, because there are no squared \(y\) terms or cross-products like \(xy\).

Equation Forms

The form of a quadratic equation helps to predict the shape and properties of its graph. The general standard form for a quadratic equation is \(y = ax^2 + bx + c\).
This form is widely used because:

• It allows easy identification of the parabola's orientation and whether it opens up or down based on the sign of \(a\).
• It simplifies finding the vertex, which is crucial for sketching and understanding the graph.
• It distinguishes from other conic sections by having no squared \(y\) terms or cross-products.

Equation forms guide you on how to approach problems involving conic sections efficiently. For example, converting equations to the standard form, when necessary, can reveal easier ways to deduce the graph's features. Recognizing the specific form of \(y = ax^2 + bx + c\) ensures that you are dealing with a parabola, as in our original exercise, without needing to plot each point.