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Magazine Chapter 0: Problem 16
Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$x=y^{2}+2$$
Short Answer
Expert verified
Plot points such as (6, -2), (3, -1), (2, 0), (3, 1), and (6, 2) and draw a right-opening parabola.
Step by step solution
01
- Understand the Equation
The given equation is in the form of a parabola. Notice that it is not in the standard form of a parabola opening upwards or downwards (i.e., y = ax^2 + bx + c), but rather it opens sideways. Our equation is x = y^2 + 2.
02
- Identify the Vertex and Orientation
Rewrite the equation as x = (y-0)^2 + 2. Comparing this to the standard form x = a(y - k)^2 + h, we see that the vertex is at (h, k) = (2, 0) and the parabola opens to the right since the coefficient of y^2 is positive.
03
- Create a Table of Values
Select a few values of y to determine corresponding x values. For example, if y = -2, -1, 0, 1, 2:- For y = -2: x = (-2)^2 + 2 = 4 + 2 = 6- For y = -1: x = (-1)^2 + 2 = 1 + 2 = 3- For y = 0: x = 0^2 + 2 = 0 + 2 = 2- For y = 1: x = 1^2 + 2 = 1 + 2 = 3- For y = 2: x = 2^2 + 2 = 4 + 2 = 6
04
- Plot the Points
Using the values from the table, plot the points (6, -2), (3, -1), (2, 0), (3, 1), and (6, 2) on the coordinate plane.
05
- Draw the Parabola
Connect the plotted points with a smooth curve to show the parabola opening to the right. Ensure the curve goes through the vertex at (2, 0) and includes points on both sides of the y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Plotting Points
Plotting points is the foundation of graphing any function, including parabolas. To plot points, we need pairs of coordinates (x, y). In our specific example, since the equation is given as x in terms of y ( x = y^2 + 2 ), we select values of y, compute the corresponding x-values, and form coordinate pairs.
Let's take y-values like -2, -1, 0, 1, and 2.
Let's take y-values like -2, -1, 0, 1, and 2.
- For y = -2, x = 6. So, point is (6, -2).
- For y = -1, x = 3. So, point is (3, -1).
- For y = 0, x = 2. So, point is (2, 0).
- For y = 1, x = 3. So, point is (3, 1).
- For y = 2, x = 6. So, point is (6, 2).
Parabola Vertex
The vertex of a parabola is a crucial point since it is the turning point of the curve. The vertex gives us a clear starting reference from which the parabola grows. In the equation x = y^2 + 2, we can rewrite it as x = (y - 0)^2 + 2, which shows that the vertex, denoted as (h, k), is at (2, 0).
Because the coefficient of y is positive, the parabola opens to the right. This information helps us understand that our plotted points will grow outward from this vertex point in the positive x-direction.
Because the coefficient of y is positive, the parabola opens to the right. This information helps us understand that our plotted points will grow outward from this vertex point in the positive x-direction.
Graph Orientation
Graph orientation indicates the direction in which the parabolic curve opens. For standard quadratic equations in the form y = ax^2 + bx + c, the parabola typically opens upwards or downwards. However, in our equation x = y^2 + 2, the parabolic curve opens sideways.
Since the coefficient of y^2 is positive, we know the parabola opens to the right. Understanding this helps us visualize and draw the curve correctly as we connect the plotted points. Always pay attention to the sign in front of the squared term to determine the direction of the graph.
Since the coefficient of y^2 is positive, we know the parabola opens to the right. Understanding this helps us visualize and draw the curve correctly as we connect the plotted points. Always pay attention to the sign in front of the squared term to determine the direction of the graph.
Table of Values
Creating a table of values helps in systematically determining the coordinates required for plotting the points. It involves selecting specific values for y and calculating the corresponding x-values. Let's see how to do this for our equation x = y^2 + 2.
Here鈥檚 how you do it:
Here鈥檚 how you do it:
- Choose several y-values (e.g., -2, -1, 0, 1, 2).
- For each y-value, calculate x by applying the equation.
- Form coordinate pairs. For instance, y = -2 results in the pair (6, -2).
Coordinate Plane
A coordinate plane is where graphing occurs, consisting of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is denoted by an ordered pair (x, y).
For our example, after computing the required points - (6, -2), (3, -1), (2, 0), (3, 1), and (6, 2) - we plot them where the x and y values intersect on the plane.
Always start by locating the vertex. Then, plot the other points around it, keeping in mind the orientation of the graph.
This way, you'll see the parabolic shape form as you draw a smooth curve through the plotted points.
For our example, after computing the required points - (6, -2), (3, -1), (2, 0), (3, 1), and (6, 2) - we plot them where the x and y values intersect on the plane.
Always start by locating the vertex. Then, plot the other points around it, keeping in mind the orientation of the graph.
This way, you'll see the parabolic shape form as you draw a smooth curve through the plotted points.
