Problem 21 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given. $$y=2 x^{2}+4 x-6$$

Step-by-Step Solution

Answer

The vertex of the parabola \(y = 2x^2 + 4x - 6\) is (-1, -4).

Step 1: Determine the x-coordinate of the vertex

Use the formula \(x = -b/(2a)\) to determine the x-coordinate of the vertex. In the given equation, \(b = 4\) and \(a = 2\). So, \(x = -(4)/(2*2) = -1\)

Step 2: Determine the y-coordinate of the vertex

Substitute the x-coordinate of the vertex into the equation to find the y-coordinate. So \(y = 2*(-1)^2 + 4*(-1) -6 = -4\)

Step 3: Present the vertex

The vertex is the point with coordinates obtained from the previous two steps: Vertex (-1, -4)

Key Concepts

Quadratic Equations

Quadratic equations, a fundamental concept in algebra, are polynomial equations of the second degree that generally have the standard form of \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants with \( a \) not equal to zero.

The solutions to these equations, known as the roots, can be found using various methods such as factoring, completing the square, and the quadratic formula. Calculating the roots is crucial for graphing the parabola, as they represent where the parabola intersects the x-axis. These points are called x-intercepts or zeros.

A notable point related to the quadratic equation is the vertex of the parabola it represents. The vertex is a maximum or minimum point on the graph, depending on the direction the parabola opens. To find the vertex, you can use the vertex formula \( x = -\frac{b}{2a} \), which yields the x-coordinate of the vertex. Subsequently, you substitute this value back into the original equation to determine the corresponding y-coordinate.

Coordinates in Algebra

Coordinates in algebra are the set of values that determine the location of a point on a two-dimensional plane, usually defined by a pair \( (x, y) \). The concept of coordinates is essential when it comes to numerically representing algebraic expressions graphically.

In algebra, finding the coordinates often involves solving for 'x' and 'y'. However, the context dictates the method used - sometimes you'll solve for 'y' by setting 'x' to a particular value, and other times you'll evaluate both 'x' and 'y' simultaneously, like when determining the vertex of a parabola.

The vertex of a parabola is a prime example of an instance where coordinates serve a specific purpose. By determining the x-coordinate using \( x = -\frac{b}{2a} \), and then computing the corresponding y-coordinate by plugging that value back into the quadratic equation, you can pinpoint this crucial point on the graph which represents either the highest or lowest point depending on the parabola's orientation.

Graphing Parabolas

Graphing parabolas is an illustrative way to understand quadratic functions visually. When graphing a parabola, we typically start by identifying the vertex, which acts as a central reference point before plotting the rest of the curve.

The vertex is crucial because it represents the peak or trough of the parabola, depending on whether the coefficient of the \( x^2 \) term is positive or negative. A positive coefficient means the parabola opens upwards and the vertex is the minimum point. Conversely, a negative coefficient makes the parabola open downwards with the vertex being the maximum point.

Once the vertex is found, as in our exercise where the vertex is at \( (-1, -4) \), we can plot that point on a coordinate grid. Afterward, we find additional points on the parabola by substituting different 'x' values into the equation. This helps in drawing a symmetrical curve that represents the quadratic function. Remember that the axis of symmetry of the parabola passes through the vertex, which implies that points on opposite sides of the vertex are mirror images of each other.

Understanding how to graph parabolas with reference to their vertices helps students to visualize the function's maximum or minimum value and prepares them for more advanced concepts in calculus like finding the derivative.