Problem 22 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1 : Calculate x-coordinate of Vertex

Calculate the x-coordinate of the vertex using the formula \(-b/2a\). Here \(a=-2\) and \(b=-4\). Substituting those values into the formula, the result is 4 divided by -4 which equals -1.

Step 2 : Calculate y-coordinate of Vertex

To get the y-coordinate of the vertex, substitute the -1 (which we got from step 1 as the x-coordinate) into the given equation. This gives you \(y = -2*(-1)^2 - 4*(-1) -2\), which simplifies to -2.

Step 3 : Identify the Vertex

Now that we have both x and y coordinates for the vertex, we can see the vertex of the parabola is at (-1, -2).

Key Concepts

Vertex Formula

Understanding the vertex formula is essential when dealing with quadratic equations and their graphs. The vertex of a parabola is a point where the curve changes direction and is considered the maximum or minimum point of the function. It's also the point that provides a lot of information about the parabola, like its axis of symmetry and the optimal value of the quadratic function.

To find the vertex of a parabola represented by the equation of the form \(y = ax^2 + bx + c\), we use the vertex formula \(x = -\frac{b}{2a}\). This gives the x-coordinate of the vertex. To find the corresponding y-coordinate, simply plug the value of \(x\) back into the original equation. As shown in the exercise solution, knowing how to manipulate this formula is crucial for quickly identifying the vertex and hence understanding the properties of the parabola.

Parabola Equation

The parabola equation standard form, \(y = ax^2 + bx + c\), may seem simple, yet it holds all the secrets of the parabola's shape and position. The coefficients \(a\), \(b\), and \(c\) dictate not only the width and direction of the parabola but also its position in the coordinate plane. The coefficient \(a\) determines the parabola's concavity — if \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, like in our exercise, it opens downwards.

Moreover, understanding the influence of \(b\) and \(c\) helps with graphing the parabola and finding its x-intercepts and y-intercept, respectively, which are essential for sketching the graph. Exploring the parabola equation and its implications enables students to confidently deal with parabolic graphs and apply them to real-world problems, such as projectile motion.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to describe the position of points, lines, and curves within the Cartesian plane. This field of mathematics is fundamental when it comes to analyzing and understanding the behavior of quadratic functions like parabolas.

The precise location of any point can be represented by an ordered pair of numbers, also known as coordinates, which are measured along perpendicular axes. In the case of the vertex of a parabola, like in the example provided, coordinate geometry allows us to define the vertex's position as \((-1, -2)\), where -1 is the x-coordinate, and -2 is the y-coordinate.

By mastering coordinate geometry concepts, students gain a powerful tool for visualizing and interpreting mathematical relationships, ultimately leading to a stronger grasp of functions and their graphs.