Problem 79 - Algebra 1

Question

VERTEX FORM The vertex form of a quadratic function is \(y=a(x-h)^{2}+k\). Its graph is a parabola with vertex at \((\boldsymbol{h}, \boldsymbol{k})\). Use completing the square to write the quadratic function in vertex form. Then give the coordinates of the vertex of the graph of the function. $$y=-x^{2}-5 x+6$$

Step-by-Step Solution

Answer

The vertex form of the given quadratic function is \(y=-(x+ \frac{5}{2})^{2}+\frac{89}{4}\). The coordinates of the vertex of the graph are \((- \frac{5}{2}, \frac{89}{4})\).

Step 1: Rewrite the Quadratic Equation in Vertex Form

To rewrite the equation in vertex form, we have to complete the square. We group the x terms together, extract the coefficient of \(x^{2}\) as a common factor and complete the square inside the parenthesis: \[\begin{aligned} y &= -(x^{2}+5x)+6 \ &= - \left( x^{2} + 5x + \left(\frac{5}{2}\right)^{2} - \left(\frac{5}{2}\right)^{2}\right) + 6 \ &= - \left( x^2 + 5x + \left(\frac{5}{2}\right)^2 - \left(\frac{5}{2}\right)^2 + \left(\frac{5}{2}\right)^2 \right) \ &= - \left( x + \frac{5}{2} \right)^2 + \left(\frac{5}{2}\right)^2 + 6 \ &= - \left( x + \frac{5}{2}\right)^2 + \frac{89}{4} . \end{aligned}\] This is the quadratic equation in vertex form.

Step 2: Identify the Coordinates of the Vertex

From the standard form of the vertex form equation, we can tell that the vertex is (-h, k). In our case, the coefficient of x inside the parenthesis is +5/2, which means that h is -5/2. The value of k equals 89/4. Thus, the coordinates of the vertex are \((- \frac{5}{2}, \frac{89}{4})\).

Key Concepts

Completing the Square

Completing the square is a valuable technique used to transform a quadratic equation from its standard form to vertex form. This transformation is crucial for understanding the characteristics of the equation's graph, particularly its vertex. To complete the square, we follow a specific process:

• First, identify the quadratic and linear coefficients from the equation.
• Next, factor out the coefficient of the squared term if it's not 1. This sets the stage for completing the square within the parentheses.
• Then, take half of the linear coefficient, square it, and add it inside the parentheses to form a perfect square trinomial.
• Don't forget to maintain balance in the equation. Subtract the same squared value outside the parentheses to account for the added term.

By following these steps, the quadratic function is nicely arranged into a square format that reveals the vertex of the graph.

Quadratic Function

A quadratic function is a polynomial function of degree 2, typically given in the form \[y = ax^2 + bx + c\]In this equation,

• \( a \), \( b \), and \( c \)are constants.
• \( a \) is referred to as the coefficient of the quadratic term.
• The function always graphs into a parabola.
• Its highest power of the variable is 2.
The vertex form of a quadratic function, represented as \[y = a(x-h)^2 + k\]is particularly useful because it explicitly indicates the vertex \((h, k)\). This form is achieved through completing the square, allowing for an easy read of the vertex's location and understanding of the graph's behavior.

Parabola

A parabola is the U-shaped graph of a quadratic function. It's essential in various fields, from physics to engineering, due to its reflective properties and symmetry. Parabolas have several defining characteristics:

• They open upwards if the coefficient \( a \) in the quadratic function is positive and downwards if it is negative.
• The vertex serves as the parabola's peak if it opens downward or its lowest point if facing upward, making it a critical feature of the graph.
Every parabola is symmetric with respect to a vertical line called the axis of symmetry, which passes through the vertex. The precise location of the vertex in the completed square form provides crucial information:
• It indicates the parabola's position.
• Shows its direction of opening.
Understanding these properties allows students to anticipate the behavior of quadratic functions and use them effectively in problem-solving.