Introduction:

The graphical representation of mathematical functions is a fundamental tool in understanding the behavior and properties of equations. In this article, we will explore the graphical representation of the function y = 2x + 4. Through a detailed analysis, we will uncover the key characteristics and patterns exhibited by this particular function.

I. Understanding Linear Functions

A. Definition of Linear Function

A linear function is a mathematical relationship between two variables that can be represented by a straight line on a graph. It follows the general form y = mx + b, where m represents the slope or gradient, and b represents the y-intercept.

B. Slope-Intercept Form

The slope-intercept form of a linear function provides crucial insights into its graphical representation. In this form, y = mx + b, m determines the inclination or steepness of the line, while b indicates its vertical position at x = 0.

II. Analyzing y = 2x + 4

A. Determining Slope

For our given function, y = 2x + 4, we can observe that the coefficient of x is 2, which indicates that the slope is positive. This means that as x increases, so does y at twice the rate.

B. Calculating Y-Intercept

To find the value of b in our equation, we substitute x = 0 into y = 2x + 4:

y = (2 * 0) + 4

y = 0 + 4

y = 4

Hence, our y-intercept is located at (0,4), indicating that when x equals zero, y equals four.

III. Plotting Points and Drawing Lines

A. Plotting Key Points

To graphically represent our function on a Cartesian plane, we can start by plotting the y-intercept at (0,4). Additionally, we can determine other key points by selecting values for x and solving for y using the equation.

B. Selecting x-values

Let’s choose three arbitrary values for x: -2, 1, and 3.

1. For x = -2:

y = (2 * -2) + 4

y = -4 + 4

y = 0

Thus, the point (-2,0) belongs to our function’s graph.

2. For x = 1:

y = (2 * 1) + 4

y = 2 + 4

y = 6

Therefore, the point (1,6) is on the graph of our function.

3. For x = 3:

y = (2 * 3) + 4

y = 6 + 4

y =10

Consequently, the point (3,10) is part of the function’s graph as well.

C. Drawing the Line

By connecting these plotted points on our Cartesian plane using a straight line with a ruler or graphing software, we can visualize and better understand how our function behaves across all real values of x.

IV. Key Characteristics

A. Steepness and Direction

Since our function has a positive slope of m = 2, it means that as x increases by one unit, y increases by two units. This indicates a relatively steep incline on the graph as we move from left to right.

B. Root of Function

The root or zero of a function corresponds to an x-value that results in y equaling zero. For our equation y=2x+4,

we can find this value by setting y equal to zero and solving for x:

0=2x+4 => -4=2x => -2=x.

Hence, the root of our function is (-2,0).

C. Domain and Range

The domain represents the set of all possible x-values for which the function is defined. In this case, the domain is all real numbers since there are no restrictions on x.

The range, on the other hand, represents the set of all possible y-values that the function can produce. Since our function has a slope of 2, y will always be greater than or equal to 4. Thus, the range is y ≥ 4.

Conclusion:

In conclusion, understanding and representing functions graphically allows us to visualize their behavior and grasp essential properties such as slope, intercepts, and key characteristics. By analyzing and plotting points for y = 2x + 4, we have successfully illustrated its linear nature with a positive slope and a y-intercept at (0,4). Through careful examination of these graphical representations, we can extract valuable insights into various mathematical relationships.