Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is a line through the origin with intercepts at (0, 0).

Step 1: Create a Table of Values

To create a table of values, we will choose several values for \(x\) and use the equation \(y = 2x\) to find the corresponding \(y\) values. | \(x\) | \(y\) = 2\(x\) ||-----|------------|| -2 | 2\(-2\) = -4 || -1 | 2\(-1\) = -2 || 0 | 2\(0\) = 0 || 1 | 2\(1\) = 2 || 2 | 2\(2\) = 4 |

Step 2: Sketch the Graph

Using the table of values from Step 1, plot the points \((-2, -4), (-1, -2), (0, 0), (1, 2), (2, 4)\) on a coordinate plane. Draw a straight line through these points to create the graph of the equation \(y = 2x\). Since this is a linear equation, the graph will be a straight line through the origin with a slope of 2.

Step 3: Find the x-intercept

The x-intercept occurs where the graph crosses the x-axis, which is where \(y = 0\). Set \(y = 0\) in the equation \(y = 2x\) and solve for \(x\): \[ 0 = 2x \]\[ x = 0 \]So, the x-intercept is at the point \((0, 0)\).

Step 4: Find the y-intercept

The y-intercept occurs where the graph crosses the y-axis, which is where \(x = 0\). Substitute \(x = 0\) into the equation \(y = 2x\):\[ y = 2(0) = 0 \]So, the y-intercept is at the point \((0, 0)\).

Key Concepts

Graphing

Graphing is a powerful visual representation of relationships in mathematics. When you look at a graph, you see how values change and relate to one another. For the equation \( y = 2x \), graphing helps us understand its nature and behavior. Begin by plotting points on a coordinate plane. Each point is comprised of an \( x \) and a \( y \) value determined by our equation. For example, using the table of values, you might plot (-2,-4), (-1,-2), (0,0), (1,2), and (2,4). These give a set of points that align neatly. Draw a line through the plotted points. Since this equation represents a linear equation, the graph will form a straight line. The line provides clear insight into how the pair of values are connected and shows the linear growth pattern of the equation. In our case, you’ll see a direct positive correlation where \( y \) increases as \( x \) increases.

x-intercepts

The x-intercept of a graph is where the line crosses the x-axis. At this point, the \( y \) value is zero; essentially, the line is standing perfectly still on the x-axis. It's a key feature that helps us understand the behavior of the graph. To find the x-intercept of \( y = 2x \), set \( y = 0 \) and solve for \( x \). Solving \( 0 = 2x \) yields \( x = 0 \), meaning the x-intercept is at the origin, (0, 0). Why is this important? The x-intercept signifies where the output of the function is zero, which can help in contextual problems to determine points of equilibrium or transition.

y-intercepts

Y-intercepts are found where the graph crosses the y-axis. At this junction, the x-value is zero, showing where the line meets the y-axis directly. This can signal the starting value of the equation.In the equation \( y = 2x \), find the y-intercept by plugging \( x = 0 \) into the equation. Doing so calculates \( y = 2(0) = 0 \). Thus, the y-intercept is also at the point (0, 0).Usually, the y-intercept gives us a really clear starting point for graphing and assessing conditions when x is zero. In multiple real-world scenarios, it can represent an initial condition, such as a starting point with no time elapsed.

Table of Values

A table of values is a foundational tool in graphing. It acts like a blueprint, mapping different values for \( x \) and the resulting values of \( y \). This simplification is a first step to effectively plotting a graph.Here's how you create one: Choose several x-values, both positive and negative, and substitute them into the equation. For example, in \( y = 2x \), use \( x = -2 \), \( -1 \), 0, 1, and 2, and calculate the resulting \( y \) values. The results in our table include points like (-2, -4) and (2, 4). This table isn't just data points—it's the critical preparation for understanding the equation's behavior visually and conceptually. Once you have this table, it’s much easier to draw an accurate graph and visualize where the line falls on the coordinate plane.