Problem 167 - Elementary Algebra

Question

In the following exercises, determine if the parabola opens up or down. $$ y=4 x^{2}+x-4 $$

Step-by-Step Solution

Answer

The parabola opens upwards.

Step 1 - Identify the Standard Form

The given quadratic equation is written in the standard form \[ y = ax^2 + bx + c \]. Identify the coefficients: \[ a = 4, \, b = 1, \, c = -4 \].

Step 2 - Analyze the Coefficient of the Quadratic Term

Determine the coefficient of the quadratic term, which is the value of \(a\). Here, \( a = 4 \).

Step 3 - Determine the Direction of the Parabola

If the coefficient \(a\) is positive (\(a > 0\)), the parabola opens upwards. If \(a\) is negative (\(a < 0\)), the parabola opens downwards. Since \( a = 4 \) is positive, the parabola opens upwards.

Key Concepts

parabola

A parabola is a U-shaped curve that can open either upwards or downwards. The general shape of a parabola depends on the quadratic equation from which it originates. Parabolas are fundamental in algebra and appear in physics, engineering, and many other fields. In the context of a quadratic equation, the parabola's orientation is determined by the coefficient of the quadratic term.

standard form

The standard form of a quadratic equation is represented as \[ y = ax^2 + bx + c \] This form is essential for identifying the key components of the equation, such as its coefficients. Whether you are solving for the roots, determining the vertex, or figuring out the direction of the parabola, starting with the equation in standard form is vital. The coefficients, designated as \(a\), \(b\), and \(c\), play specific roles in shaping the graph of the quadratic equation. Always ensure your quadratic equation is in the standard form before performing any further calculations.

coefficients

In the quadratic equation \[ y = ax^2 + bx + c \], the coefficients \(a\), \(b\), and \(c\) are critical elements. The coefficient \(a\) is particularly important as it determines the direction of the parabola.

When \(a > 0\), the parabola opens upwards, forming a U-shape. When \(a < 0\), the parabola opens downwards, resembling an inverted U.

Aside from \(a\), the coefficient \(b\) affects the position of the parabola along the x-axis, and \(c\) shifts it along the y-axis. Understanding these coefficients gives you control over predicting and modifying the graph's shape.