Problem 1 - Intermediate Algebra

Question

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$y=x^{2}-7 x+5$$

Step-by-Step Solution

Answer

The parabola opens upwards.

Step 1: Identify the Standard Form

First, identify the standard form of the parabola. The given equation is \( y = x^2 - 7x + 5 \). This equation is in the form \( y = ax^2 + bx + c \), which is the standard form for a parabola opening either upwards or downwards.

Step 2: Determine the Direction of Opening

Check the coefficient \( a \) in the standard form \( y = ax^2 + bx + c \). Here, \( a = 1 \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), the parabola opens downwards. Since \( a = 1 \), which is positive, the parabola opens upwards.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). This equation forms the basis for understanding parabolas in algebra.

Quadratic equations can have several characteristics, such as:

• The variable \( x \) is squared, which defines the quadratic nature of the equation.
• The equation can be solved using different methods like factoring, completing the square, or the quadratic formula.
• Solutions to quadratic equations are the x-values where the graph of the equation (parabola) intersects the x-axis, often referred to as roots or solutions.

Solving these equations is crucial in many areas of mathematics and applied sciences as they model a wide variety of physical phenomena.

Parabola Opening Direction

The direction in which a parabola opens is determined by the coefficient of the squared term in its equation. For quadratic equations in the standard form \( y = ax^2 + bx + c \):

• If the coefficient \( a \) of \( x^2 \) is positive, the parabola opens upward.
• If the coefficient \( a \) is negative, the parabola opens downward.

This behavior is because the sign of \( a \) determines how the values of \( y \) behave as \( x \) becomes very large or very small.
Upward-opening parabolas have a minimum point called the vertex, whereas downward-opening parabolas have a vertex that represents their maximum point. Comprehending the opening direction is helpful for sketching the graph without plotting multiple points.

Standard Form of a Parabola

The standard form of a parabola concerning quadratic equations is expressed as \( y = ax^2 + bx + c \). This form is a simple representation that helps identify many characteristics of the parabola effortlessly.

Characteristics of this form of a parabola include:

• \( a \): Coefficient that defines the direction of opening and width of the parabola.
• \( b \): Influences the position of the vertex along the x-axis.
• \( c \): Represents the y-intercept, the point where the parabola crosses the y-axis.

Recognizing the standard form allows for easy transitions to other forms such as vertex form, which can further illuminate the properties of a parabola.