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

Step-by-Step Solution

Answer

The equation \( y = 2x^2 + 3x - 4 \) represents a parabola.

Step 1: Identify the Form

The given equation is \( y = 2x^2 + 3x - 4 \). This is in the form \( y = ax^2 + bx + c \), which represents a quadratic equation.

Step 2: Recognize the Equation Type

Quadratic equations in the form \( y = ax^2 + bx + c \) are equations of a parabola. The key identifying feature is the \( x^2 \) term.

Key Concepts

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic equation, like the one given in the exercise: \(y = 2x^2 + 3x - 4\). It can open upwards or downwards depending on the coefficient of the \(x^2\) term. In this equation, the coefficient of \(x^2\) is positive \(2\), which means the parabola opens upwards.

Key characteristics of a parabola include:

• Vertex: The highest or lowest point on the parabola, where it changes direction.
• Axis of Symmetry: A vertical line passing through the vertex, dividing the parabola into two mirror-image halves.
• Direction: Decided by the sign of the \(a\) coefficient, positive for upward and negative for downward directions.

Understanding a parabola is not just about recognizing the equation form, it's about visualizing its shape and how it behaves, which is crucial for solving problems that involve graphing or finding intersections with other graphs.

Conic Sections

Conic sections are shapes created as a plane intersects a cone. They include parabolas, ellipses, circles, and hyperbolas. Each shape is defined by different types of equations.A parabola is one of these classic shapes. It arises when the plane cuts parallel to the edge of the cone. In standard form, quadratic equations, like \(y = ax^2 + bx + c\), represent parabolas which are fundamental to conic sections.

Here’s a quick breakdown of the curve types:

• Circular and Elliptical: When the cone is cut parallel or at an angle less steep than the edge of the cone.
• Parabolic: When the cut is parallel to the edge of the cone, representing quadratic relations.
• Hyperbolic: When the plane cuts both halves of the cone.

Knowing conic sections helps in understanding the geometric nature of various algebraic equations.

Graphing Equations

Graphing equations is one of the most effective tools for visualizing and solving quadratic equations like \(y = 2x^2 + 3x - 4\). When we graph, we turn algebra into a picture. This provides a clear understanding of the structure and solution of equations.

For a quadratic equation, key steps in graphing include:

• Finding the Vertex: Using the vertex formula \(x = -\frac{b}{2a}\), which helps locate the turning point of the parabola.
• Plotting the Axis of Symmetry: This vertical line \(x = -\frac{b}{2a}\) divides the parabola into two equal halves.
• Identifying Points: Calculating several \(x\) values and finding their corresponding \(y\) values to plot the curve.

Graphing gives insight into intercepts, direction, and can reveal the number of real solutions through the points where the graph crosses the x-axis. This visual method complements algebraic techniques and enhances comprehension.