Problem 76 - Elementary Algebra

Question

The height in feet of an object dropped from a 16 -foot ladder is given by the function \(h(t)=-16 t 2+16,\) where \(t\) is the time in seconds after it is tossed. Write the function in factored form.

Step-by-Step Solution

Answer

The function in factored form is \[ h(t) = -16(t - 1)(t + 1) \].

Step 1: Identify the Quadratic Function

The given function is a quadratic function: \[ h(t) = -16t^2 + 16 \] We need to express this in factored form.

Step 2: Factor Out the Common Factor

To factor the quadratic function, start by factoring out the greatest common factor (GCF) from the terms in the expression.The terms \(-16t^2\) and \(+16\) both have \(-16\) as a common factor, so factor \(-16\) out:\[ h(t) = -16(t^2 - 1) \]

Step 3: Recognize the Difference of Squares

The expression inside the parentheses, \(t^2 - 1\), is a difference of squares. We can apply the identity for the difference of squares:\[ a^2 - b^2 = (a - b)(a + b) \]In this case, \(t^2 - 1\) can be written as:\[(t)^2 - (1)^2 = (t - 1)(t + 1) \]

Step 4: Write the Function in Factored Form

Substitute the factored expression for \(t^2 - 1\) back into the function:\[ h(t) = -16(t - 1)(t + 1) \]This is the function in its factored form.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function. They have a specific form that features the square of the variable as the highest power. The standard form of a quadratic function is given by:

• \( f(x) = ax^2 + bx + c \)

Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The coefficient \(a\) is crucial because it determines the direction of the parabola (upward if \(a > 0\) or downward if \(a < 0\)).
Quadratic functions yield a graph that is a parabola. It can open up or down depending on the leading coefficient. In our exercise, the function \( h(t) = -16t^2 + 16 \) is a downward-opening parabola because of the negative leading coefficient (\(-16\)).
Understanding the basic structure of these functions helps us manipulate and convert them into different forms, such as factored form, which is often useful for solving and analyzing them.

Difference of Squares

The difference of squares is a specific algebraic pattern used to simplify expressions and factor quadratic functions. This method applies when you have two perfect squares subtracted from each other. It follows this identity:

• \( a^2 - b^2 = (a - b)(a + b) \)

In our exercise, the expression \( t^2 - 1 \) fits this pattern. It's a difference of squares because \( t^2 \) is a perfect square and \( 1 \) is also a perfect square \((1^2)\).
To factor it, you identify the square roots of each term, which are \(t\) and \(1\) respectively, then write it as the product of two binomials: \((t - 1)(t + 1)\).
Using the difference of squares makes factoring easier and often provides strategies to simplify further mathematical operations on the expressions. When you spot this pattern, you can quickly break down complex equations into simpler parts.

Factored Form

The factored form of a quadratic equation presents the function as a product of its factors. This form can be very useful for solving equations, finding roots, and understanding the properties of the function.
For a quadratic equation, the factored form can be expressed as:

• \( f(x) = a(x - r_1)(x - r_2) \)

Here, \(r_1\) and \(r_2\) are the roots or solutions where the function equals zero. In the case of the exercise problem, identifying the difference of squares helped to express \( h(t) = -16t^2 + 16 \) in its factored form: \(-16(t - 1)(t + 1)\).
Factored form reveals the symmetry of quadratic functions and aids in quickly determining zero values. By recognizing the patterns in quadratic equations, you can more easily navigate through problems, optimizing the way you tackle math challenges.