Problem 22 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{2}+2\)

Step-by-Step Solution

Answer

The graph of the function \(y = x^{2} + 2\) is a parabola opening upwards, centered at the point (0,2). This can be seen by plotting the points derived from integer values of \(x\) from \(-3\) to 3, inclusive.

Step 1: Identify the equation

The quadratic function to be graphed is \(y = x^{2} + 2\).

Step 2: Choose \(x\) values

Choose integer values for \(x\) from \(-3\) to 3, inclusive.

Step 3: Calculate \(y\) values

Substitute each value of \(x\) into the equation to find the corresponding \(y\) value. The ordered pairs formed by the \(x\) and \(y\) values are points on the graph.

Step 4: Plot points

Plot the points derived from the chosen \(x\) values and their corresponding \(y\) values on the coordinate plane.

Step 5: Draw the graph

Draw the graph by connecting the plotted points with a curve.

Key Concepts

Quadratic Functions

Quadratic functions are a special type of polynomial function where the highest power of the variable is 2. This gives them a characteristic "U" shaped graph called a parabola. The simplest form of a quadratic function is given by the equation \[ y = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. In the exercise at hand, the quadratic function is \( y = x^2 + 2 \). This indicates a parabola that opens upwards, because the coefficient of \(x^2\) is positive. Understanding the basic structure of quadratic functions helps in predicting the shape and position of their graphs.

Coordinate Plane

A coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates are given in the form \( (x, y) \), where \(x\) is the horizontal axis, and \(y\) is the vertical axis. The plane is divided into four quadrants. When graphing quadratic functions, understanding the coordinate plane is crucial. It allows for precise plotting of points, where specific \(x\) values translate into corresponding \(y\) values.

Plotting Points

Plotting points requires choosing specific \(x\) values and calculating their corresponding \(y\) values using the equation of the function. From the exercise, we selected integers from \(-3\) to \(3\) for \(x\). Substituting these values into the quadratic equation \( y = x^2 + 2 \), we find the \(y\) values for each \(x\). The resulting ordered pairs, such as \((-3, 11)\), \((-2, 6)\), and so on, are then plotted on the coordinate plane. Plotting accurately is key to visualizing the function correctly.

Mathematical Graphing

Mathematical graphing involves connecting plotted points with a smooth curve to represent the function. For a quadratic function like \( y = x^2 + 2 \), the graph will be a parabola. The curve is continuous and smooth, reflecting the ongoing relation that \(x\) and \(y\) share in the function. After plotting the individual points, such as \((-3, 11)\), \( (0, 2)\), and \( (3, 11)\), connect these points gently. A helpful tip is to ensure the graph is symmetric around the axis of symmetry, which for this function is the \(y\)-axis.

Integer Selection

Integer selection is a crucial step in plotting functions. Choosing the right values for \(x\) ensures you capture the behavior and shape of the function effectively. In the exercise, we select integer values from \(-3\) to \(3\), which is a common range for such exercises. This range provides sufficient points to identify the key features of the parabola, like its vertex and symmetry. Because integers are whole numbers, they simplify the process of calculation and plotting, making your graphing job easier and more accurate.