Problem 81 - Precalculus : Building Concepts and Connections

Question

Use the function \(f(x)=a x^{2}+2 x+1\) where a is a real number. Find the discriminant \(b^{2}-4 a c\)

Step-by-Step Solution

Answer

The discriminant of the function \(f(x) = ax^2 + 2x + 1\) is \(4 - 4a\).

Step 1: Identifying a, b, and c

In the given quadratic function \(f(x)=ax^2 + 2x + 1\), the coefficients of the terms are the values of \(a\), \(b\), and \(c\). Here, \(a\) is an unspecified real number, \(b\) is the coefficient of \(x\) which is 2, and \(c\) is the constant term which is 1.

Step 2: Substituting values into the formula for the discriminant

Using the formula for the discriminant \(b^2 - 4ac\), substitute the values of \(a\), \(b\), and \(c\) from the equation into the formula to calculate the discriminant.

Step 3: Calculating the Discriminant

After substitution, the formula becomes \((2)^2 - 4*a*1\). The value of the discriminant is thus \(4 - 4a\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable (usually denoted as \(x\)) is 2. This means the largest exponent on \(x\) is \(x^2\). Quadratic functions are commonly written in the form:

• \( f(x) = ax^2 + bx + c \)

Here, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\) to ensure the equation is indeed quadratic.

The graph of a quadratic function is a parabola, which can open upwards or downwards. This opening direction is determined by the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

Quadratic functions are commonly used to model physical scenarios such as projectile motion, where they can represent paths that change direction, forming a distinct curve.

Coefficients in Quadratic Equations

In a quadratic function of the form \(f(x) = ax^2 + bx + c\), the coefficients play a critical role in determining the shape and position of the parabola.

• \(a\) is the coefficient of \(x^2\) and is known as the leading coefficient. It affects the width of the parabola and its direction (up or down).
• \(b\) is the coefficient of \(x\) and influences the location of the vertex horizontally and the symmetry of the parabola.
• \(c\) is the constant term and represents the \(y\)-intercept, where the graph crosses the \(y\)-axis.

Understanding these coefficients helps in graphing the parabola and predicting its behavior.

For the discriminant, these coefficients are essential as they feed into the formula \(b^2 - 4ac\) to determine the nature of the roots of the quadratic equation. This nature can be characterized as having:

• Two distinct real roots if the discriminant is positive.
• One real root (a repeated root) if the discriminant is zero.
• Two complex roots if the discriminant is negative.

Real Number Parameter

In the exercise, the parameter \(a\) is a real number. The term "real number" refers to any number that can be found on the number line, including all positive and negative integers, fractions, and irrational numbers like \( \sqrt{2} \), and \( \pi \).

When dealing with quadratic equations, having the parameter \(a\) as a real number ensures that the calculations and derivations of the quadratic function remain logical and applicable in real-world scenarios. It helps maintain the continuity and feasibility in computations.

The value of \(a\) significantly affects the shape and direction of the parabola in the quadratic function \(f(x) = ax^2 + 2x + 1\). In the discriminant \(b^2 - 4ac\), the term \(-4ac\) directly depends on \(a\). By exploring values of \(a\), one can determine how this affects the nature and number of roots of the quadratic equation.