Problem 51 - Elementary Algebra

Question

The height of a projectile dropped from a 64 -foot tower is given by the function \(h(t)=-16 t 2+64,\) where \(t\) represents the time in seconds after it is dropped. Rewrite this function in factored form. (Hint: Factor out -16 first.)

Step-by-Step Solution

Answer

The factored form is \( h(t) = -16(t-2)(t+2) \).

Step 1: Understand the Given Function

The function given is \( h(t) = -16t^2 + 64 \). This represents the height of a projectile over time and involves a quadratic term and a constant.

Step 2: Factor Out the Common Factor

We are given a hint to factor out \(-16\). So, take out \(-16\) from the expression. This means setting: \[ h(t) = -16(t^2 - 4) \]

Step 3: Apply Differences of Squares Factoring

The expression \( t^2 - 4 \) is a difference of squares which can be factored further. Recognize that \( t^2 - 4 \) is \( t^2 - 2^2 \), a classic difference of squares, and factor it as: \[ t^2 - 4 = (t - 2)(t + 2)\]

Step 4: Write the Factored Form of the Function

Substitute the factored form of \( t^2 - 4 \) back into the expression. Hence the factored form of \( h(t) \) becomes: \[ h(t) = -16(t - 2)(t + 2)\]

Key Concepts

Projectile Motion

Projectile motion describes the movement of an object that is thrown or propelled near the Earth's surface, moving along a curved path under the action of gravity alone. The height of such a projectile can be modeled using a quadratic function, which is a type of polynomial equation. For example, the function

• \( h(t) = -16t^2 + 64 \)

represents the height of a projectile dropped from a 64-foot tower over time \( t \). The term \(-16t^2\) accounts for the effect of gravity on the projectile, while the 64 represents its initial height.
This equation allows us to determine the height of the projectile at any given time, assuming no other forces act on it except gravity. By analyzing such equations, one can understand how quickly an object falls or how long it stays in the air.

Difference of Squares

The difference of squares is a common mathematical pattern used in algebra that can be factored easily. It takes the form \( a^2 - b^2 \), where both \( a \) and \( b \) are squared numbers. This expression can be uniquely factored into two binomials, \( (a-b)(a+b) \).
Let's consider the example from the exercise: the expression \( t^2 - 4 \) fits this pattern, since it equates to \( t^2 - 2^2 \). Therefore, it can be factored into:

• \( (t - 2)(t + 2) \)

Recognizing and applying this rule can significantly simplify quadratic expressions and make them easier to solve or graph. Factoring using the difference of squares often reveals the roots of the equation, helping us to understand more about the function's behavior and intercepts.

Quadratic Functions

Quadratic functions are polynomial functions, specifically of degree two, and they are pivotal in algebra. They are generally written in the standard form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. These functions are characterized by their distinctive "U-shaped" graph, known as a parabola.
The given quadratic function in the exercise, \( h(t) = -16t^2 + 64 \), is a simplified quadratic form where \( a = -16 \), \( b = 0 \), and \( c = 64 \). Various properties of quadratic functions include:

• The vertex, which represents the maximum or minimum point of the parabola.
• The axis of symmetry, a vertical line that divides the parabola into two symmetrical halves.
• The roots or zeros, which are the points where the parabola crosses the x-axis.

Converting quadratic functions into factored form, like \(-16(t-2)(t+2)\) obtained from the exercise, makes it easy to find the roots and understand the function’s graph.