Problem 39 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given by first writing the equation in the form \(y=a x^{2}+b x+c\) $$y=(x+5)^{2}-4$$

Step-by-Step Solution

Answer

The vertex of the parabola \(y=(x+5)^{2}-4\) is \((-5, -4)\).

Step 1: Rewrite the equation in standard form

First, expand the given equation \(y=(x+5)^{2}-4\) to bring it to the standard form \(y=a x^{2}+b x+c\). This will give us \(y=x^2+10x+25-4\), which simplifies to \(y=x^2+10x+21\).

Step 2: Identify the coefficients to determine the vertex

To find the vertex when a quadratic equation is in the standard form, we use the formulas for the vertex \((h, k)\), where \(h=-b/2a\) and \(k=c-(b^2/4a)\). In this case, our coefficients \(a\), \(b\) and \(c\) from the equation \(y=x^2+10x+21\) are 1, 10 and 21 respectively.

Step 3: Plug the coefficients into the vertex formulas

Doing this, we have \(h=-b/2a = -10/{2*1} = -5\), and \(k=c-(b^2/4a) = 21 - (10^2/4*1) = 21-25 = -4\). Therefore, the vertex of the parabola is \((-5, -4)\).

Key Concepts

Quadratic Equation

Understanding quadratic equations is a fundamental skill in algebra, which opens the door to various applications in mathematics and analytics. A quadratic equation is an expression featuring an unknown variable that is squared, written in the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \) is non-zero.

It represents a parabola in coordinate geometry, which could open upwards or downwards depending on the sign of the coefficient \( a \). The solutions to a quadratic equation, known as the roots, can be real or complex and are pivotal in determining the x-intercepts of the graph of the equation. These roots can be found using methods such as factoring, completing the square, graphing, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
To further aid understanding, here's a practical example: given the equation \( y = (x+5)^2 - 4 \), turning it into standard form can illuminate the characteristics of the parabola, including its vertex, axis of symmetry, and direction of opening.

Standard Form of a Quadratic

The standard form of a quadratic equation plays a vital role in graphing and analyzing the properties of a parabola. It is expressed as \( y = ax^2 + bx + c \), where each term contributes to the shape and position of the parabola: \( a \) influences the direction and width, \( b \) affects the symmetry and orientation, and \( c \) determines the vertical position.

When reconfiguring a quadratic expression, like \( y = (x+5)^2 - 4 \), into the standard form, we expand it to obtain \( y = x^2 + 10x + 21 \). This expanded form is more straightforward for identifying the coefficients necessary for further analysis, such as locating its vertex, focusing on the symmetry axis, and plotting its graph on a coordinate plane.
Visualizing the graph and recognizing its basic traits are much easier when you have the equation structured in this way. It's an essential step in the process, as shown in the solution to our example problem where the equation was expanded to find the parabola's vertex.

Vertex Formula

The vertex is the point where a parabola changes direction; it is either the highest or lowest point on the graph, making it a central concept in understanding quadratic functions. To find the vertex of a parabola given by the equation \( y = ax^2 + bx + c \), we use the vertex formula, which is based on the coefficients from the standard form of a quadratic equation.

The formula for the vertex \( (h, k) \) is: \( h = -\frac{b}{2a} \) and \( k = c - \frac{b^2}{4a} \). Here, \( h \) gives the x-coordinate of the vertex, and \( k \) represents the y-coordinate.
In our example, where we expanded \( y = (x+5)^2 - 4 \) to \( y = x^2 + 10x + 21 \), applying the vertex formulas rendered the coordinates \( h = -5 \) and \( k = -4 \). Therefore, the vertex of the parabola in question is (-5, -4). This valuable point gives us a visual anchor for graphing the parabola and serves as a reference for various applications, such as maximizing or minimizing values in real-world problems.