Question

Consider the formula \(y=x^{2}+3 x+2\) a. Complete the table of values for this formula. b. Which variable is the output variable? c. Plot the points that represent the input/output pairs of your table on the coordinate system.

Short answer

a: Table complete, b: Output variable is y, c: Points plotted on parabola.

Step-by-step solution

Understanding the Formula

The given formula is \(y = x^2 + 3x + 2\). Here, \(x\) is the input variable (independent variable), and \(y\) is the output variable (dependent variable). Our task is first to calculate \(y\) for different values of \(x\).

Choosing Values for x

Let's choose a set of integer values for \(x\) to easily complete the table. Common choices for a smooth quadratic function like this are \(-3, -2, -1, 0, 1, 2, 3\).

Calculating y for Each x

Plug each chosen value of \(x\) into the formula to find \(y\):- \(x = -3\): \(y = (-3)^2 + 3(-3) + 2 = 9 - 9 + 2 = 2\)- \(x = -2\): \(y = (-2)^2 + 3(-2) + 2 = 4 - 6 + 2 = 0\)- \(x = -1\): \(y = (-1)^2 + 3(-1) + 2 = 1 - 3 + 2 = 0\)- \(x = 0\): \(y = (0)^2 + 3(0) + 2 = 0 + 0 + 2 = 2\)- \(x = 1\): \(y = (1)^2 + 3(1) + 2 = 1 + 3 + 2 = 6\)- \(x = 2\): \(y = (2)^2 + 3(2) + 2 = 4 + 6 + 2 = 12\)- \(x = 3\): \(y = (3)^2 + 3(3) + 2 = 9 + 9 + 2 = 20\)

Completing the Table

Using the calculations from Step 3, we fill the table:\[\begin{array}{|c|c|}\hlinex & y \\hline-3 & 2 \-2 & 0 \-1 & 0 \0 & 2 \1 & 6 \2 & 12 \3 & 20 \\hline\end{array}\]

Identifying the Output Variable

In the formula \(y = x^2 + 3x + 2\), \(x\) is the input variable, and \(y\) is the output variable.

Plotting the Points

Plot the input-output pairs on a coordinate system using the table: - (-3, 2), (-2, 0), (-1, 0), (0, 2), (1, 6), (2, 12), (3, 20). This forms a parabolic curve on the graph, typical of quadratic functions.

Key concepts

Independent Variable

In the context of the given exercise, the independent variable is essentially the input for our function. It is symbolized by \( x \) in quadratic equations. The magic of the independent variable is its freedom.

• It can take on various values, such as -3, -2, -1, and so on.
• These values are usually picked to make calculations easy, and they are not affected by any other variable from the problem.

When working with equations, think of the independent variable as a starting point for calculations or, in mathematical terms, the 'cause' in cause-effect relationships. You choose it, plug it into the equation, and see what happens next!
All the magic of results begins with this important player.

Dependent Variable

Contrary to the independent variable, the dependent variable changes according to the input values. In our quadratic equation, the dependent variable is represented by \( y \). It's highly significant because its value showcases the outcome or result once the independent variable goes through the formula.

• The dependent variable is often referred to as the 'output.' It's what you're solving for.
• For each value of \( x \), there is a definite value of \( y \) based on the equation.

In our case, as \( x \) changes, plugging each value into the equation beautifully transforms it into different \( y \) values, showing the relationship and resulting effect of the function. The dependent variable embodies the core purpose of our calculations, often telling an insightful story about the relationship portrayed by the equation.

Parabolic Curve

A parabolic curve is the graphical representation resulting from a quadratic equation, such as \( y = x^2 + 3x + 2 \). When plotted on a coordinate plane, you will notice this unique shape:

• It's symmetrical, often looking like a 'U' shape either opening upwards or downwards.
• In our exercise, the values of \( y \) plotted against every \( x \) result in this charming curve that elegantly connects the points surveyed.

The parabolic curve is more than a shape; it visually conveys how the value of the dependent variable changes as the independent variable shifts its value.
The fascinating aspect of the parabolic curve is how it artfully demonstrates the mathematical bond between the inputs and outputs through its graceful bend.

Coordinate System

The coordinate system is the mathematical grid where we plot our equation's points. It uses axes to help us organize our data visually.

Key features include:

• A horizontal axis known as the x-axis, where independent variables are placed.
• A vertical axis called the y-axis, displaying the dependent variables.

Each point on the system corresponds to an \( (x, y) \) pair from our table. As you plot each point, you create a visual representation of the equation's data, turning numbers into a pattern.
Visualizing values helps to see the relationship and behavior of quadratic functions easily. The coordinate system ties abstract numbers into a cohesive picture, grounding our understanding of mathematical relationships.