Problem 2 - A Graphical Approach to College Algebra

Question

Skills Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. 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 $$y=2 x^{2}$$

Step-by-Step Solution

Answer

The equation \( y = 2x^2 \) matches with "Parabola; opens upward" (option C).

Step 1: Identify the Equation Type

The given equation is \( y = 2x^2 \). This equation represents a parabola, which is a type of quadratic equation in the standard form \( y = ax^2 + bx + c \).

Step 2: Determine Parabola Orientation

Since the equation \( y = 2x^2 \) is in the form \( y = ax^2 \), and the coefficient \( a = 2 \) is positive, the parabola opens upwards. In general, if \( a > 0 \), the parabola opens upward; if \( a < 0 \), it opens downward.

Step 3: Match with Description

From the list in Column II, the description that matches the properties of the equation \( y = 2x^2 \) is "Parabola; opens upward." Therefore, the correct match for the equation \( y = 2x^2 \) is option C.

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that you encounter in quadratic equations. It is the graphical representation of a quadratic function.
Quadratic functions are generally written in the form \( y = ax^2 + bx + c \). Here, the "\(a\)" term is the most significant, as it defines the shape and orientation of the parabola.
- **Vertex:** The vertex is the point where the parabola changes direction. - If the parabola opens upward or downward, the vertex is the minimum or maximum point.- **Axis of Symmetry:** A vertical line passing through the vertex. For the equation \( y = ax^2 \), the axis of symmetry is \( x = 0 \).- **Direction of opening:** Determined by the coefficient "\(a\)". Understanding these features will help you predict how the parabola appears on the graph. Knowing the vertex allows you to locate the curve's "tip," and using the axis of symmetry helps you understand its balance.

Parabola Orientation

Parabola orientation refers to the direction in which a parabola opens. It is crucial because it affects how we interpret the graph of a quadratic function.
- **Upward:** If \( a > 0 \), the parabola opens upwards. Think of it as a "U" shape. This orientation indicates that the vertex is the lowest point of the graph.- **Downward:** If \( a < 0 \), the parabola opens downward, resembling an upside-down "U". Here, the vertex is the highest point.By examining the sign of the coefficient "\(a\)" in the quadratic equation \( y = ax^2 \), you can instantly identify the orientation. This determines whether the vertex is a minimum or maximum point on the graph. This simple check allows you to better understand the function's behavior without extensive calculations.

Quadratic Function Properties

Quadratic functions have several key properties that shape their graph and influence their analysis. **Standard Form:** The standard form of a quadratic equation is \( y = ax^2 + bx + c \), where:\

• \(a\), \(b\), and \(c\) are constants.
• The quadratic term \(ax^2\) dictates the curve's main form, while \(bx\) and \(c\) control its position on the graph.

**Graphical Characteristics:** - **Vertex:** Can be found at \( x = -\frac{b}{2a} \).- **Symmetry:** The parabola is always symmetric about its vertex.- **Roots/X-Intercepts:** Values of \(x\) at which \( y = 0 \). These can be real or complex, determined by the discriminant \(b^2 - 4ac\).These properties allow students to sketch graphs accurately and solve real-world problems involving quadratic relationships. Knowing them simplifies navigating various problems related to optimization and modeling.