Problem 18 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=5 x+6 x^{2}-1 $$

Step-by-Step Solution

Answer

The parabola opens up.

Step 1: Identify the Coefficient of the \(x^{2}\) Term

Look at the equation \(y = 5x + 6x^{2} - 1\) and identify the coefficient of the \(x^{2}\) term, which here is 6.

Step 2: Determine the Direction of Parabola

Since the coefficient of the \(x^{2}\) term, which is 6, is a positive number, this means that the parabola opens up.

Key Concepts

Parabola

A parabola is a special curve that you'll often come across while studying quadratic functions. It's shaped like a "U" or an upside-down "U," depending on certain parameters in the quadratic equation. Generally, a quadratic function is expressed in the general form: \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

Understanding parabolas is fundamental to tackling quadratic equations as they represent the graph of the function. The vertex of the parabola is either the highest or lowest point on the graph. If it opens upwards, the vertex is the lowest point, and if it opens downwards, the vertex is the highest point. Parabolas are symmetric, which means they mirror themselves on either side of a line that runs through their vertex. This line is called the axis of symmetry.

• A parabola's width and direction are determined by the values of the coefficients in the quadratic equation.
• They can model real-world scenarios, such as projectile motion or the design of satellite dishes.

Coefficient

Coefficients are numerical values that multiply variables in an equation. In the realm of quadratic functions, the most critical coefficient to pay attention to is the one attached to the \(x^2\) term, also known as the "leading coefficient."

For the equation \( y = ax^2 + bx + c \), \(a\) is the coefficient of \(x^2\). It has a profound impact on the shape and direction of the parabola. A positive \(a\) results in a parabola that opens upwards, akin to a "smiling" face. Conversely, a negative \(a\) causes the parabola to open downwards, similar to a "frowning" face.

• The value of \(a\) dictates whether the parabola is steep or shallow. Larger values of \(a\) imply a steeper parabola.
• While \(b\) and \(c\) also play roles in the position of the parabola, \(a\) is solely responsible for its vertical orientation.

Direction of Parabola

Determining the direction of a parabola is an integral part of understanding quadratic functions. As you've learned, the direction is primarily dependent on the sign of the coefficient of the \(x^2\) term.

When tasked with deciding whether a parabola opens up or down, follow these simple steps:

• Identify the quadratic term—this is the term containing \(x^2\).


• Check the coefficient of \(x^2\):
  - If the coefficient is positive (\(a > 0\)), the parabola opens upwards.
  - If the coefficient is negative (\(a < 0\)), the parabola opens downwards.

For example, in the equation \( y = 5x + 6x^2 - 1 \): The coefficient of \(x^2\) is 6, which is positive, indicating that the parabola opens upwards. This determination helps predict and understand the behavior of the function when graphed.