Problem 51 - Elementary Algebra

Question

Determine the maximum or minimum \(y\) -value. $$ y=-x 2-6 x+1 $$

Step-by-Step Solution

Answer

The maximum y-value is 10.

Step 1: Identify the Form of the Quadratic Equation

The given equation is in the form \( y = ax^2 + bx + c \) where \( a = -1 \), \( b = -6 \), and \( c = 1 \).

Step 2: Determine the Direction of the Parabola

Since \( a = -1 \) is negative, the parabola opens downward. This means the vertex represents a maximum point.

Step 3: Find the Vertex Using the Formula for the x-coordinate

The x-coordinate of the vertex \( x = -\frac{b}{2a} \). Substitute \( a = -1 \) and \( b = -6 \) into the formula: \( x = -\frac{-6}{2(-1)} = \frac{6}{-2} = -3 \).

Step 4: Find the y-coordinate of the Vertex

Substitute \( x = -3 \) back into the equation to find \( y \): \( y = -(-3)^2 - 6(-3) + 1 = -9 + 18 + 1 = 10 \).

Step 5: State the Maximum y-value

The maximum \( y \)-value occurs at the vertex, which is \( y = 10 \).

Key Concepts

Vertex Form

Quadratic functions can be expressed in a specific way called the vertex form, which is particularly useful for revealing the vertex of parabolas. The vertex form is given by: \[ y = a(x - h)^2 + k \]where

• \( (h, k) \) denotes the coordinates of the vertex.
• the parameter \( a \) illustrates the direction and the width of the parabola.

Employing the vertex form gives an immediate insight into the vertex's position and can help simplify many calculations. However, many quadratic equations are given in the standard form: \[ y = ax^2 + bx + c \] To convert from the standard form to the vertex form, one typically completes the square or uses the vertex formula. This transformation can be crucial in precisely identifying the key feature, which is the vertex, especially useful for graphing and analyzing the properties of parabolas.

Maximum and Minimum Values

Quadratic equations feature either a maximum or minimum value which is determined by the direction in which the parabola opens. This is contingent on the sign of the parameter \( a \).

• If \( a > 0 \), the parabola opens upwards, yielding a minimum value at the vertex.
• If \( a < 0 \), the parabola opens downward, meaning the vertex gives the maximum value.

The vertex formula, \( x = -\frac{b}{2a} \), calculates the x-coordinate of the vertex. Subsequently plugging this x-value into the quadratic equation provides the corresponding y-coordinate, thus granting the maximum or minimum value of the function.

Knowing whether a parabola opens up or down not only helps in understanding its shape but also offers insight into the overall behavior of the function. In practical terms, figuring out the maximal or minimal points is essential in numerous applications, including profit maximization, minimizing costs, and physics problems.

Graphing Parabolas

Graphing parabolas is about depicting the curve formed by quadratic functions on a Cartesian plane. To graph a parabola effectively, understanding its fundamental components is invaluable.

• The vertex is a starting point, indicating the highest or lowest tip of the parabola.
• The intercepts, points where the parabola crosses the axes, further illustrate its placement relative to its surroundings.
• The axis of symmetry, a vertical line through the vertex, divides the parabola into two reflective halves.

To produce the graph:1. Identify the vertex using the vertex form or by converting the standard form into one.2. Determine if the parabola opens upwards or downwards based on the sign of \( a \).3. Calculate the y-intercept by setting \( x = 0 \). Similarly, find the x-intercepts (if they exist) by solving the equation \( y = 0 \).4. Draw the axis of symmetry through the vertex.

Graphing accurately is an essential skill in algebra that helps visualize solutions and better understand the interplay between algebraic expressions and their geometric representations. It assists in solving real-world problems by providing a tangible picture of otherwise abstract mathematical concepts.