Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Match each equation with the appropriate description . Do not use a calculator. $$y=2 x^{2}$$ A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward

Step-by-Step Solution

Answer

C. Parabola; opens upward.

Step 1: Identify the Equation Type

The given equation is \(y = 2x^2\). We need to identify the type of curve this equation represents. Parabolic equations generally follow the form of \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\), which describes a parabola.

Step 2: Analyze the Parabola Orientation

The equation \(y = 2x^2\) is in the form \(y = ax^2\), where \(a > 0\). A positive \(a\) indicates that the parabola opens upwards.

Step 3: Match Description with Equation

From the list of options, the description that matches a parabola opening upwards is "C. Parabola; opens upward." Therefore, the equation \(y = 2x^2\) matches with option C.

Key Concepts

Equation identification

When you come across the equation \(y = 2x^2\), the task is to identify what type of graph this equation represents. This process often involves comparing the given equation to standard forms you may have learned in class. Let's break it down. In mathematics, equations can represent different types of curves, such as circles, ellipses, hyperbolas, or parabolas. Here, the equation \(y = 2x^2\) closely matches the general form for parabolas, which is \(y = ax^2 + bx + c\). Since in this exercise, \(b\) and \(c\) are zero, we have the simplest form \(y = ax^2\), confirming the equation defines a parabola.
Recognizing the form of an equation helps us classify the type of curve without any computational tools like a calculator. It's always useful to keep in mind various standard forms of equations to quickly identify the nature of the curve you're dealing with.

Graph description

Now that we know \(y = 2x^2\) represents a parabola, it is important to describe its graph. Parabolas are U-shaped curves and can either open upwards, downwards, or sideways. To describe this graph, focus on its symmetry and how the coefficient in front of \(x^2\) affects its orientation and width.
In our equation \(y = 2x^2\), there is no term attached to \(x\) (no \(b\) or \(c\)), indicating that the parabola is symmetric about the y-axis. The coefficient 2 in front of \(x^2\) influences how "narrow" or "wide" the parabola is. The value 2 makes the parabola narrower than the basic parabola \(y = x^2\), where the leading coefficient is 1.
Describing a graph isn't just about identifying the shape; it's about understanding its attributes from the equation, which also helps us draw it accurately by hand or imagine its form.

Parabola orientation

Determining the orientation of a parabola is crucial for graphing it correctly. The orientation indicates which way the parabola "opens." In the equation \(y = 2x^2\), the key factor in finding the orientation is the sign of the coefficient in front of \(x^2\). Here, the coefficient is 2, which is positive. This automatically tells us that the parabola opens upwards. If this coefficient were negative, the parabola would open downward.
Understanding the orientation also helps in distinguishing between different parabola descriptions given in exercises, like knowing that a positive leading coefficient means the graph opens "upward," compared to a negative one that opens "downward." This kind of insight makes analyzing and answering problems related to parabolas more intuitive.
Recognizing the right orientation assists in easily matching equations with given graph descriptions, which is a frequent requirement in mathematical problems.