Problem 2 - Introductory Algebra for College Students

Question

Determine if the parabola whose equation is given opens upward or downward. $$y=x^{2}-6 x+5$$

Step-by-Step Solution

Answer

The given parabola opens upwards.

Step 1: Identify the Equation of the Parabola

The provided equation is \(y=x^{2}-6x+5\). This is a quadratic equation of the form \(ax^{2}+bx+c\) where \(a=1\), \(b=-6\), and \(c=5\).

Step 2: Identify the coefficient of \(x^{2}\)

The coefficient of \(x^{2}\) in the equation of the parabola is \(a\). In this case, \(a=1\).

Step 3: Determine the Direction of the Parabola

The coefficient \(a>0\), which means that the parabola opens in the upward direction.

Key Concepts

Quadratic equation

In mathematics, a quadratic equation is any equation that can be transformed into a standard form: \[ ax^2 + bx + c = 0 \]where:- \( a \), \( b \), and \( c \) are constants.- \( a \) is not equal to zero.
In the standard form, \( ax^2 \) is known as the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term. The term quadratic comes from "quad" meaning square, because the variable \( x \) is squared.
Quadratic equations can be solved using various methods, including:

• Factoring
• Using the quadratic formula
• Completing the square
• Graphing

Each method offers a unique approach to finding the solutions, or roots, of the equation, which can be real or complex numbers. Understanding how to manipulate and solve a quadratic equation is foundational in algebra and is critical for higher-level mathematics learning.

Coefficient

In a quadratic equation, the coefficient is a numerical or constant factor that multiplies the variable. Consider the standard form of a quadratic equation, \( ax^2 + bx + c \), where:- \( a \) is the coefficient of the quadratic term \( x^2 \).- \( b \) is the coefficient of the linear term \( x \).- \( c \) is the constant term and does not have a variable.
Coefficients define the properties of the quadratic equation, including its shape and the direction in which the parabola opens. The coefficient \( a \) is the most critical when determining these traits. If \( a \) is greater than zero, the parabola opens upwards. If \( a \) is less than zero, the parabola opens downwards.
In the equation \( y=x^2-6x+5 \), the coefficients are:- \( a = 1 \)- \( b = -6 \)- \( c = 5 \)This indicates a positive coefficient \( a \), which influences the orientation of the parabola.

Direction of parabola

The direction in which a parabola opens is directly influenced by the coefficient \( a \) in the quadratic equation \( ax^2 + bx + c \). Understanding this helps in graphing parabolas correctly and predicting their behavior.
Key principles include:

• If \( a > 0 \), the parabola opens upwards. This means that as \( x \) moves away from the vertex, both arms of the parabola extend upwards.
• If \( a < 0 \), the parabola opens downwards, with both arms stretching downwards as \( x \) moves away from the vertex.

The direction is crucial for identifying the range of the function and determining the vertex, which is the highest or lowest point on the graph, depending on the parabola's direction.
For example, in the quadratic equation \( y=x^2-6x+5 \), the coefficient \( a = 1 \) (\( a > 0 \)), indicating an upward-opening parabola. This means the vertex represents the minimum point of the curve. Knowing this helps in further analyzing and solving quadratic functions in various applications.