Problem 43 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given by first writing the equation in the form \(y=a x^{2}+b x+c\) $$y=-3(x+2)^{2}+5$$

Step-by-Step Solution

Answer

The vertex of the parabola given by the equation \( y=-3(x+2)^2+5 \) is at the point (-2, 5).

Step 1: Understanding the Problem

In standard form, the equation of a parabola is given as \( y = a(x-h)^2 + k \), where (h, k) are the coordinates of the vertex. Here, we already have this form represented by the equation \( y=-3(x+2)^2+5 \).

Step 2: Identify Vertex Coordinates

We can obtain the vertex coordinates by identifying the values of 'h' and 'k' in the given equation. 'h' corresponds to '+2' (note the switch in sign) and 'k' corresponds to '+5'. Therefore, the vertex of the parabola is (-2, 5).

Key Concepts

Quadratic Equations

At the core of algebra, quadratic equations form the bedrock of many mathematical concepts. A quadratic equation is any equation that can be written in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not equal to zero. This specific condition ensures that the equation is, indeed, quadratic and not linear.

These equations are essential because they describe parabolas—a type of curve on the graph. Solutions to quadratic equations, also known as the roots, can reveal where the parabola crosses the x-axis. When trying to find the vertex of the parabola, as in our exercise, the quadratic can also be manipulated into a vertex form that provides a clear path to the answer.

Standard Form of a Parabola

The standard form of a parabola is an intuitive way to visualize and comprehend its properties. It is represented as \( y = ax^2 + bx + c \), where 'a' determines the direction and width of the parabola, 'b' affects the location of its axis of symmetry, and 'c' indicates the y-intercept. When 'a' is positive, the parabola opens upwards, and when 'a' is negative, as in our example \( y = -3(x + 2)^2 + 5 \), it opens downwards.

Understanding the standard form is crucial because it gives insight into the parabola's behavior. The key to transforming a quadratic equation into vertex form lies in completing the square, a technique which rearranges the equation to isolate the perfect square term, allowing us to clearly identify the vertex coordinates.

Vertex Coordinates

The vertex of a parabola is the point where it turns; it is either the maximum or minimum point on the graph, depending on whether the parabola opens up or down. Identifying the vertex is straightforward when the quadratic equation is in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) are the vertex coordinates.

For the equation \( y = -3(x + 2)^2 + 5 \), the vertex form reveals that 'h' is the opposite sign of the constant inside the parentheses, and 'k' is the constant term outside. Therefore, our vertex coordinates are indeed \( (-2, 5) \), signifying the point where the parabola changes direction. Knowing how to find the vertex is incredibly useful for graphing quadratics and for solving various applied problems in physics, engineering, and economics, where this concept frequently occurs.