Problem 8 - Algebra for College Students

Question

Complete each ordered pair so that it satisfies the given equation. $$f(x)=-x^{2} \quad(-9, \quad),(\quad,-4)$$

Step-by-Step Solution

Answer

(-9, -81) and (2, -4) or (-2, -4).

Step 1: Understand the function

The given function is $$f(x) = -x^2$$It takes an input value (x), squares it, and then takes the negative of that value.

Step 2: Complete the first ordered pair (-9, )

For the first ordered pair (-9, ), substitute $$x = -9$$ into the function. Calculate $$f(-9) = -(-9)^2 = -81.$$Therefore, the first ordered pair is $$(-9, -81).$$

Step 3: Complete the second ordered pair (, -4)

For the second ordered pair (, -4), set the function equal to -4 and solve for x. Solve the equation $$-x^2 = -4$$...by dividing both sides by -1, we get$$x^2 = 4$$Take the square root of both sides which gives two solutions:$$x = 2$$ and $$x = -2$$Thus, the second ordered pair is: $$ (2, -4)$$ and $$(-2, -4).$$

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function where the highest exponent of the variable is 2. It has the general form: \( f(x) = ax^2 + bx + c \).
In this case, the function is \( f(x) = -x^2 \), which is a specific type of quadratic function known as a downward-opening parabola because the coefficient of \( x^2 \) (which is -1 here) is negative.
This means that as \( x \) increases or decreases, the value of \( f(x) \) will become more negative. Most quadratic functions will have a graph that looks like a U-shape (parabola), but because the coefficient is negative here, the U is upside down.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. For the given function \( f(x) = -x^2 \), you simply substitute the input value \( x \) into the function and perform the necessary calculations.
For example, for the input \( x = -9 \), substituting it in, we get:

• \( f(-9) = -(-9)^2 \)
• \( = -81 \)

This means when \( x = -9 \), the output is \( -81 \), completing the first ordered pair to be \( (-9, -81) \).

Solving Equations

Solving equations involves finding the value(s) of the variable(s) that make the equation true. This often requires isolating the variable on one side of the equation.
In the second part of the exercise, we were given an output \( f(x) = -4 \) and needed to find the input value \( x \).
To solve:

• Set the function equal to -4: \( -x^2 = -4 \)
• Divide both sides by -1 to get: \( x^2 = 4 \)
• Take the square root of both sides, yielding two solutions: \( x = 2 \) and \( x = -2 \)

Therefore, the second ordered pair could be either \( (2, -4) \) or \( (-2, -4) \).