Analyzing the Graph of y -2x^2 - 5x - 3 and Understanding its Characteristics

In this article, we will explore the graph of the quadratic equation y -2x^2 - 5x - 3. Understanding the characteristics of quadratic functions is essential in algebra and has practical applications in various fields. We will delve into the vertex form, vertex, and the overall shape of the parabola.

Understanding Quadratic Functions

A quadratic equation is generally expressed in the form:

y ax^2 bx c

In this case, a -2, b -5, and c -3. The sign of the a coefficient determines the direction in which the parabola opens. Since a -2 is negative, the parabola opens downwards.

Vertex Form and Vertex of the Parabola

The vertex form of a quadratic equation is:

y a(x - h)^2 k

In this form, the vertex of the parabola is (h, k). To convert our equation into vertex form, we need to complete the square:

y -2x^2 - 5x - 3

Step 1: Factor out -2 from the first two terms:

y -2(x^2 frac{5}{2}x) - 3

Step 2: Add and subtract the square of half of the coefficient of x (which is frac{-5}{4}right)^2):

y -2(x^2 frac{5}{2}x frac{25}{16} - frac{25}{16}) - 3

Step 3: Group the perfect square trinomial and simplify:

y -2left(x^2 frac{5}{2}x frac{25}{16}right) - 2left(frac{25}{16}right) - 3

Step 4: The expression inside the parentheses is now a perfect square:

y -2left(x frac{5}{4}right)^2 - frac{25}{8} - 3

Step 5: Combine the constants:

y -2left(x frac{5}{4}right)^2 - frac{25}{8} - frac{24}{8}

y -2left(x frac{5}{4}right)^2 - frac{49}{8}

Thus, the equation in vertex form is:

y -2left(x frac{5}{4}right)^2 - frac{49}{8}

The vertex of the parabola is:

(h, k) left(-frac{5}{4}, -frac{49}{8}right)

Graphing the Parabola

The graph is an upside-down parabola (since the coefficient of (x^2) is negative) with the vertex at (left(-frac{5}{4}, -frac{49}{8}right)). To draw the graph, we can plot the vertex and find the y-intercept and another point to determine the shape.

The y-intercept can be found by setting (x 0):

y -2(0)^2 - 5(0) - 3 -3

The y-intercept is ((0, -3)).

Another point can be found by choosing a convenient value for (x), for example, (x -1):

y -2(-1)^2 - 5(-1) - 3 -2 5 - 3 0

The point ((-1, 0)) lies on the graph.

Conclusion

Understanding the characteristics of the quadratic equation y -2x^2 - 5x - 3 is crucial for graph analysis. By converting it to vertex form and completing the square, we can determine the vertex, axis of symmetry, and overall shape of the parabola. The vertex form reveals the maximum point of the parabola (since it opens downwards), and the y-intercept and another point help in sketches.

Quadratic equations are not only fundamental in algebra but also have practical applications in physics, engineering, and economics. Mastering the graph of quadratic equations is an important step in mathematical understanding and problem-solving.