Problem 30 - Precalculus : Building Concepts and Connections

Question

Find a possible expression for a quadratic function \(f(x)\) having the given zeros. There can be more than one correct answer. \(x=-5\) is the only zero

Step-by-Step Solution

Answer

Because the given zero is \(x = -5\), a possible quadratic function could be \(f(x) = a(x+5)^2\), where \(a\) is an undetermined real number.

Step 1: Identifying the given zero

The problem has given that the only zero of the function is \(x = -5\).

Step 2: Formulating the quadratic function

We can plug \(h = -5\) into the system \(f(x) = a(x-h)^2 + k\) to form the quadratic function. But as k and a are not given, they can be any real numbers. So, one possible function is \(f(x) = a(x+5)^2\).

Key Concepts

Zeros of a Function

A zero of a function, often called a root, is a value of \( x \) where the function equals zero. In simpler terms, it's where the graph of the function touches or crosses the x-axis. For a quadratic function like \( f(x) \), the zero can be found by solving \( f(x) = 0 \).

For the exercise, the given zero is \( x = -5 \). This means for this particular quadratic function, substituting \( x = -5 \) makes the function equal to zero. Knowing a zero helps in creating the function, as it tells you one of the pivotal points of the parabola that represents the quadratic equation.

Zeros are important because they help define the shape and position of the parabola in a graph. In the case of having one zero like \( x = -5 \), the function's graph will be symmetric around that zero when it's in the form \( (x - h)^2 \).

Formulating Quadratic Expressions

Formulating a quadratic expression involves building an equation that represents a parabola. It usually takes the general form \( f(x) = ax^2 + bx + c \). But if you know the zeros of the equation, it's often simpler to use the vertex form \( f(x) = a(x-h)^2 + k \), where \( h \) is the x-coordinate of the vertex.

In our exercise, because the zero is \( x = -5 \), the quadratic can be arranged as \( f(x) = a(x + 5)^2 \). Here's why:

• Replacing \( h \) with \(-5\), results in \((x + 5)^2\) because subtracting a negative is the same as adding a positive.
• \( a \) is a coefficient that affects the "width" and direction of the parabola. Without further information, \( a \) is a real number, meaning it can be any non-irrational, non-imaginary number.

This formulation allows a lot of flexibility in defining a specific function, depending on other conditions like vertex position or direction of the parabola's opening.

Real Numbers in Quadratic Functions

Real numbers are a vast category including both rational and irrational numbers. In quadratic functions, these real numbers substantially affect the function's graph and properties.

The coefficients and constants \( a \), \( b \), and \( k \) in a quadratic expression like \( f(x) = ax^2 + bx + c \) or \( a(x-h)^2 + k \) can all be real numbers.

• \( a \): Determines the direction and "spread" of the parabola. If \( a > 0 \), the parabola opens upwards, while \( a < 0 \) makes it open downwards.
• \( h \) & \( k \): Define the vertex. In this exercise with one zero \( x = -5 \), \( h \) is directly related to the zero.
• \( k \): Represents the vertical shift of the parabola, or the y-coordinate of the vertex. It can shift the entire graph up or down, depending on its value.

In our exercise, while constructing the quadratic formula, both \( a \) and \( k \) can be any real numbers. This flexibility shows the rich possibilities in quadratic expressions when manipulated under different conditions.