Problem 58 - Algebra A Combined Function

Question

Fill in each table so that each ordered pair is a solution of the given function. $$ \begin{aligned} &f(x)=-3 x^{2}\\\ &\begin{array}{|r|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & \\ \hline 1 & \\ \hline-1 & \\ \hline 2 & \\ \hline-2 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The completed table is: (0,0), (1,-3), (-1,-3), (2,-12), (-2,-12).

Step 1: Understand the Function

The function provided is \( f(x) = -3x^2 \). This means that for any given value of \( x \), we substitute it into the equation to find \( y \).

Step 2: Calculate y for x = 0

Substitute \( x = 0 \) into the function: \( y = -3(0)^2 = -3 imes 0 = 0 \). So, \( y = 0 \) when \( x = 0 \).

Step 3: Calculate y for x = 1

Substitute \( x = 1 \) into the function: \( y = -3(1)^2 = -3 imes 1 = -3 \). So, \( y = -3 \) when \( x = 1 \).

Step 4: Calculate y for x = -1

Substitute \( x = -1 \) into the function: \( y = -3(-1)^2 = -3 imes 1 = -3 \). So, \( y = -3 \) when \( x = -1 \).

Step 5: Calculate y for x = 2

Substitute \( x = 2 \) into the function: \( y = -3(2)^2 = -3 imes 4 = -12 \). So, \( y = -12 \) when \( x = 2 \).

Step 6: Calculate y for x = -2

Substitute \( x = -2 \) into the function: \( y = -3(-2)^2 = -3 imes 4 = -12 \). So, \( y = -12 \) when \( x = -2 \).

Step 7: Fill the Table

Using the calculations from the previous steps, fill in the table: \[\begin{array}{|r|r|} \hline \boldsymbol{x} & \boldsymbol{y} \ \hline 0 & 0 \ \hline 1 & -3 \ \hline -1 & -3 \ \hline 2 & -12 \ \hline -2 & -12 \ \hline \end{array}\]

Key Concepts

Function Evaluation for Quadratic Functions

When dealing with quadratic functions like the one given, our main task is to evaluate or "find" the value of the function for different inputs. This means substituting specific values of the variable, in this case, \( x \), into the function equation to find corresponding \( y \) values. To evaluate the function \( f(x) = -3x^2 \), follow these simple steps:

• Substitute the given \( x \) value into the equation.
• Calculate \( x^2 \) (square the \( x \) value).
• Multiply this result by \(-3\) as the function defines.

This gives you the \( y \) value or the output of the function when it "receives" an input \( x \). The computation is straightforward if you follow each step with care. For example, if \( x = 0 \), then \( y = -3(0)^2 = 0 \). This simple process allows us to evaluate the function for any given \( x \). It is a crucial part of understanding how quadratic functions behave.

Understanding Ordered Pairs

An ordered pair is simply a pair of numbers that go together, denoted as \((x, y)\), where \( x \) is typically the input and \( y \) is the output from evaluating a function. For the function \( f(x) = -3x^2 \), each ordered pair we find tells us something specific about the function's behavior.Remember, the order matters -- \((x, y)\) is not the same as \((y, x)\).

• The first number, \( x \), is the "input" or the independent variable.
• The second number, \( y \), is the "output" or the dependent variable since it depends on \( x \).

For example, the pair \((0, 0)\) shows that when \( x = 0 \), the function outputs \( y = 0 \). Similarly, \((1, -3)\) tells us that when \( x = 1 \), \( y \) becomes \(-3\) as per the function's rule. Understanding these pairs is essential to interpreting graphs and plotting results of functions.

Graphing Parabolas from Quadratic Functions

A parabola is the graph you get from a quadratic function, which, in this case, is \( f(x) = -3x^2 \). The shape of this graph is a "U" or upside-down "U", depending on the sign of the coefficient of \( x^2 \). Here, the graph opens downwards because the coefficient is negative (-3).Each ordered pair from the function represents a point on the graph. Once you have several points calculated from our previous steps, you plot them on a coordinate plane. For example:

• \((0, 0)\)
• \((1, -3)\)
• \((-1, -3)\)
• \((2, -12)\)
• \((-2, -12)\)

Connecting these points in a smooth curve will give you the parabola. Remember, the vertex of this parabola is at the origin, \((0,0)\), for this function. The symmetry is another characteristic of parabolas -- in this exercise, notice how points \((-1, -3)\) and \((1, -3)\), as well as \((-2, -12)\) and \((2, -12)\), reflect across the y-axis. Understanding how to draw and recognize parabolas is fundamental in visually interpreting quadratic functions.