Problem 14 - College Algebra Essentials

Question

Graph each equation in Exercises \(13-28\). Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$ y=x^{2}+2 $$

Step-by-Step Solution

Answer

The graph is a parabola that opens upwards, the vertex is located at (0,2) and passes through the points (-3,11), (-2,6), (-1,3), (0,2), (1,3), (2,6), and (3,11).

Step 1: Substitution

Substitute each \(x\) value into the equation \(y=x^{2}+2\) and calculate the resulting \(y\) values.

Step 2: List the Points

List down all the calculated point pairs (x, y). For example, substituting -3 results in 11, thus (-3, 11) is one pair.

Step 3: Drawing the Graph

Plot all point pairs (x, y) on the \(x-y\) plane and draw the graph.

Key Concepts

Graphing Equations

Graphing quadratic equations like the one given in the exercise is a fundamental concept in algebra. The goal is to visualize how the equation behaves across different values of \(x\). In this exercise, each \(x\) value from -3 to 3 is plugged into the equation \(y = x^2 + 2\). This provides specific points, or coordinates, that can be plotted.

• Start by substituting each value of \(x\) into \(y = x^2 + 2\).
• Calculate the corresponding \(y\) for each \(x\).
• These form coordinate pairs (\(x, y\)).

Graphing equations helps to see the shape of the equation. Quadratic equations like this one create a parabolic curve, which is symmetric around the y-axis in this particular case. Drawing this graph gives a visual understanding of how \(y\) changes as \(x\) varies.

Substitution Method

The substitution method is commonly used in mathematics to find particular solutions at given points. In this exercise, substitution is used to find \(y\) values from specific \(x\) values. Here's a breakdown of how this applies:

• Select a specific \(x\) value from the given set \(-3, -2, -1, 0, 1, 2, 3\).
• Replace \(x\) in the equation \(y = x^2 + 2\) with the given \(x\) value.
• Calculate the \(y\) value, which completes your coordinate pair \((x, y)\).

This technique is foundational as it provides exact results for \(y\) given any \(x\). It's a helpful way to predict the outputs of an equation, ensuring that each pair is plotted accurately in creating the graph.

Coordinate Plane

Understanding the coordinate plane is crucial for graphing. The plane is divided into four quadrants by the x-axis and y-axis that intersect at the origin \((0,0)\). Here’s how it applies:

• The horizontal axis is known as the x-axis.
• The vertical axis is called the y-axis.
• Any point on this plane is identified by an \(x\) and \(y\) coordinate pair.

In the exercise, after calculating the coordinate pairs, these points need to be marked on the coordinate plane. Each coordinate \((x, y)\) corresponds to an exact location on the plane, allowing you to draw the graph accurately by joining these points.

Mathematical Graph Interpretation

Interpreting graphs helps to understand the behavior of equations. Once you have plotted all points from the substitution method on the coordinate plane, join them smoothly. This specific equation, \(y = x^2 + 2\), forms a parabola. Here's why interpreting this graph is beneficial:

• Recognize the shape and direction of the graph (opens upwards in this case).
• Understand where the vertex (the lowest point of the parabola) is located. In this equation, it occurs at \(x = 0\) with \(y = 2\).
• Notice the symmetry around the y-axis, which mirrors points on either side of the vertex.

Mathematical graph interpretation allows you to predict and analyze how the quadratic function behaves beyond the points you originally calculated, offering insights into real-world applications and deeper mathematical understanding.