Problem 18 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=x^{2}+2 $$

Step-by-Step Solution

Answer

The y-intercept is (0, 2), and there are no real x-intercepts.

Step 1: Create a Table of Values

Choose a range of x-values around zero to see how the parabola behaves, such as -3 to 3. Compute the corresponding y-values for each x-value using the equation \( y = x^2 + 2 \). | \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 ||------|----|----|----|---|---|---|---|| \(y\) | 11 | 6 | 3 | 2 | 3 | 6 | 11 |Fill in the table with these computed values.

Step 2: Sketch the Graph

Using the table of values, plot each (x, y) point on a coordinate plane. Connect these points with a smooth curve, forming a U-shaped parabola. The graph opens upwards because the coefficient of \( x^2 \) is positive.

Step 3: Find the y-intercept

Set \( x = 0 \) to find the \( y \)-intercept. Substitute 0 for \( x \) in the equation: \[ y = (0)^2 + 2 = 2 \] So, the \( y \)-intercept is the point (0, 2).

Step 4: Find the x-intercept(s)

To find the \( x \)-intercepts, set \( y = 0 \) and solve for \( x \):\[ 0 = x^2 + 2 \]\[ x^2 = -2 \] Since you cannot take the square root of a negative number and get a real result, there are no real \( x \)-intercepts for this equation.

Key Concepts

Graphing Parabolas

When dealing with quadratic functions, one of the most visually insightful methods of understanding them is by graphing parabolas. A parabola is the U-shaped curve you obtain from plotting quadratic equations. For our specific case, the equation is \( y = x^2 + 2 \). Here, the parabola opens upwards because the coefficient before \( x^2 \) is positive.

To graph a parabola, we typically start by choosing a set of \( x \)-values. We calculate corresponding \( y \)-values using the given equation. Once we have a table of these values, they become our guide to plotting. Placing these pairs on a coordinate plane and connecting them with a smooth curve shows us the shape of the parabola. In this process, we visualize how the equation behaves as \( x \) changes.

For example, suppose we choose \( x \)-values from -3 to 3. By plugging each value into the equation \( y = x^2 + 2 \), we calculate the \( y \)-values and plot the points \((-3, 11), (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)\). Through practice with graphing, students improve their intuitive grasp of quadratic functions, essential for both conceptual understanding and problem-solving.

X-Intercepts

X-intercepts are points where the graph of an equation crosses the x-axis. At these points, the value of \( y \) is zero. To find the x-intercepts of a quadratic equation like \( y = x^2 + 2 \), you set the entire equation equal to zero and solve for \( x \).

For our equation, that means solving \( 0 = x^2 + 2 \). Rearranging gives us \( x^2 = -2 \). However, when we attempt to take the square root of both sides to solve for \( x \), we encounter a problem; we cannot take the square root of a negative number in the real number system. Therefore, this equation has no real x-intercepts.

This finding indicates that the graph of the parabola does not touch or cross the x-axis. The vertex of the parabola and the rest of the curve stay above the x-axis. Understanding the concept of x-intercepts helps us learn about the solutions and roots of quadratic equations, which are critical in various mathematical applications.

Y-Intercepts

Y-intercepts are where the graph crosses the y-axis. At this intersection, the value of \( x \) is zero. To find the y-intercept for the quadratic equation \( y = x^2 + 2 \), substitute zero for \( x \).

Doing the computation, we find:

• \( y = (0)^2 + 2 \)
• \( y = 2 \)

Thus, the y-intercept is at the point \((0, 2)\). When graphing, this point is where the curve crosses the y-axis.

Knowing how to quickly find the y-intercept is helpful because it provides a starting point when sketching the graph. For many linear and quadratic equations, the y-intercept provides a direct insight into the function's behavior and serves as a foundational reference you can always return to while graphing or analyzing the information.

Coordinate Plane

The coordinate plane is the flat, two-dimensional surface on which we plot points to graph equations. It is defined by the perpendicular intersection of two axes: the x-axis (horizontal) and the y-axis (vertical). This setup allows us to accurately map and interpret mathematical functions.

To construct a graph, each point is specified by an ordered pair \((x, y)\) that corresponds to a location in the coordinate plane. In our example with the equation \( y = x^2 + 2 \), points such as \((0, 2)\) are identified and plotted, helping us to visualize the function as they form a coherent graphical line or curve.

Understanding how to utilize the coordinate plane is crucial. It allows us to gain insights into the nature of mathematical relationships by providing a visual representation. This tool is particularly helpful when working with quadratic equations and graphing parabolas, as it illustrates how the equation behaves across different values.