Problem 20 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=3 x^{2}-2 x+7 $$

Step-by-Step Solution

Answer

The parabola given by the equation \( y=3x^2-2x+7 \) opens up because the coefficient of \( x^2 \) is positive.

Step 1: Identify the terms in the equation

Firstly, look at the provided equation \( y=3x^2-2x+7 \) and identify the coefficients. Here, 'a' is 3, 'b' is -2, and 'c' is 7.

Step 2: Determine the sign of 'a'

Next, look at the sign of the coefficient 'a', which is 3 in this case. Since 'a' is positive, the parabola opens upwards.

Step 3: Conclude the direction of the parabola

So, with 'a' being positive, the conclusion can be drawn that the parabola opens up.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial that can be described by the general form \( ax^2 + bx + c = 0 \). They are called "quadratic" because they involve terms raised to the power of two, which is the essence of everything quadratic.
In this form:

• \( a \) is the coefficient of the quadratic term \( x^2 \)
• \( b \) is the coefficient of the linear term \( x \)
• \( c \) is the constant term

The solutions to these equations, or the values of \( x \) that make the equation true, are often called the "roots" and can be found using techniques such as factoring, completing the square, or applying the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Quadratic equations often appear in many real-world scenarios like projectile motion, where the path of the projectile forms a parabola.

Coefficients in Quadratic Expressions

Coefficients in quadratic expressions play a crucial role in determining the properties and shape of the parabola represented by the equation. The expression is typically written in the form \( ax^2 + bx + c \). Each coefficient controls a different aspect of the parabola.

• \( a \) - The Leading Coefficient: This coefficient determines whether the parabola opens upwards or downwards, and it also affects the "width" or "narrowness" of the parabola. A larger absolute value of \( a \) makes the parabola narrower.
• \( b \) - The Linear Coefficient: This coefficient impacts the symmetry and vertex position of the parabola but does not affect whether it opens upward or downward.
• \( c \) - The Constant Term: This value represents the y-intercept of the parabola - the point where the parabola crosses the y-axis. It is useful for sketching the graph of the equation quickly.

Understanding these coefficients allows you to swiftly identify key characteristics of the quadratic function without necessarily having to plot every point.

Graph Direction of Parabolas

The direction in which a parabola opens is fundamentally controlled by the sign of the coefficient \( a \) in the quadratic expression \( ax^2 + bx + c \).

• If \( a > 0 \), the parabola opens upwards, forming a "U" shape. This means the vertex is the lowest point on the graph, known as the minimum.
• If \( a < 0 \), the parabola opens downwards, forming an upside-down "U". Here, the vertex is the highest point on the graph, known as the maximum.

This simple property of the coefficient \( a \) provides a quick way to visualize the basic structure of any parabola given by the particular quadratic equation. Always remember to check the sign of \( a \) first when analyzing the orientation of a parabola.