Problem 86 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The minimum y-value is \(-\frac{9}{8}\).

Step 1: Rewrite the equation in standard form

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

Step 2: Identify the type of function

Since \( a = 2 \) is positive, the parabola opens upwards, indicating a minimum value for \( y \).

Step 3: Calculate the vertex x-value

Use the vertex formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex: \( x = -\frac{-1}{2 \times 2} = \frac{1}{4} \).

Step 4: Calculate the minimum y-value

Substitute \( x = \frac{1}{4} \) into the original equation to find the minimum y-value: \[ y = 2\left(\frac{1}{4}\right)^2 - \frac{1}{4} - 1 = 2\left(\frac{1}{16}\right) - \frac{1}{4} - 1 \].\ This simplifies to \[ \frac{1}{8} - \frac{1}{4} - 1 = -\frac{9}{8} \]."

Key Concepts

Vertex Formula

The vertex formula is an essential tool in understanding quadratic functions. It helps us find the vertex of a parabola, which is a crucial point indicating either the maximum or minimum value on the curve. For a quadratic function in standard form, given by \( y = ax^2 + bx + c \), the vertex \( (x, y) \) can be calculated using:

• \( x = -\frac{b}{2a} \)

This formula gives you the x-coordinate of the vertex. Once you have the x-value, you can plug it back into the original equation to find the corresponding y-value. This y-value tells you either the peak or the bottom of the parabola, depending on the sign of \( a \). If \( a \) is positive, the parabola opens upward, and this y-value is a minimum. Conversely, if \( a \) is negative, the parabola opens downward, making this y-value a maximum. Always remember to compute both coordinates to understand the full benefits of the vertex and its implications on the quadratic function's graph.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is fundamental to solving and graphing quadratic functions. It is expressed as \( y = ax^2 + bx + c \). The variables \( a \), \( b \), and \( c \) are constants where:

• \( a \) determines the direction and width of the parabola.
• \( b \) influences the position and tilt of the curve.
• \( c \) represents the y-intercept, the point where the parabola crosses the y-axis.

The standard form makes it easier to identify these characteristics directly from the equation. By examining \( a \), you can determine if the parabola opens up or down. Positive \( a \) means it opens upwards, while a negative \( a \) means it opens downwards. This form is useful not only for graphing but also for applying the vertex formula and finding the axis of symmetry, enhancing your understanding of the parabola’s shape and location.

Parabola Properties

Parabolas are the unique shapes graphed from quadratic functions, boasting several interesting properties. Understanding these properties is key to mastering quadratic equations. Some essential features include:

• **Vertex:** As the highest or lowest point on the curve, it illustrates either the maximum or minimum value of the function.
• **Axis of Symmetry:** A vertical line running through the vertex, dividing the parabola into two mirror images, given by the equation \( x = \frac{-b}{2a} \).
• **Direction of Opening:** Determined by the coefficient \( a \) in the standard form. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• **Y-intercept:** The point where the parabola meets the y-axis, which occurs at \( y = c \).
• **End Behavior:** As \( x \) approaches infinity, the arms of the parabola move up if \( a \) is positive, and down if \( a \) is negative.

These properties help us when graphing quadratic functions or solving practical problems. By identifying how the parabola opens, its line of symmetry, and vertex, you have a comprehensive understanding of the quadratic's birthing equation, leading to more intuitive problem-solving techniques often needed in algebra and calculus applications.