Comparing Linear Functions: A And B

Let's dive into comparing two linear functions: Function A and Function B. We'll explore their properties and see what makes each one unique. Function A is presented as an equation, while Function B is given as a table of values. Our goal is to understand how these functions behave and relate to each other. So, buckle up, guys, and let's get started!

Function A: The Equation

Function A is defined by the equation:

$y = \frac{1}{4}x + 2$

Linear functions like Function A are characterized by a constant rate of change, which means that for every unit increase in x, the value of y changes by a fixed amount. In this case, the equation is in slope-intercept form, which is $y = mx + b$, where:

• m represents the slope of the line.
• b represents the y-intercept (the point where the line crosses the y-axis).

In Function A, the slope (m) is $\frac{1}{4}$, and the y-intercept (b) is 2. This tells us a lot about the function:

• Slope: The slope of $\frac{1}{4}$ means that for every 4 units we move to the right on the x-axis, the line goes up 1 unit on the y-axis. A positive slope indicates that the line is increasing as we move from left to right.
• Y-intercept: The y-intercept of 2 means that the line crosses the y-axis at the point (0, 2). This is the value of y when x is 0.

To further analyze Function A, we can find additional points on the line by plugging in different values for x and solving for y. For example:

• If x = 4, then $y = \frac{1}{4}(4) + 2 = 1 + 2 = 3$. So, the point (4, 3) is on the line.
• If x = -4, then $y = \frac{1}{4}(-4) + 2 = -1 + 2 = 1$. So, the point (-4, 1) is on the line.

By plotting these points and drawing a line through them, we can visualize Function A. The linear function's slope and y-intercept give us a clear understanding of its behavior.

Function B: The Table

Function B is defined by the following table of values:

Unlike Function A, which is given as an equation, Function B is presented as a set of points. To analyze Function B, we need to determine if these points lie on a straight line and, if so, find the equation of that line.

To check if the points are linear, we can calculate the slope between each pair of points. If the slope is constant, then the function is linear.

1. Slope between (-2, -5) and (1, 1):

   $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-5)}{1 - (-2)} = \frac{6}{3} = 2$

2. Slope between (1, 1) and (4, 7):

   $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 1}{4 - 1} = \frac{6}{3} = 2$

Since the slope is the same (2) between both pairs of points, Function B is indeed a linear function. Now that we know it's linear, we can find its equation in slope-intercept form ($y = mx + b$). We already know the slope (m) is 2. To find the y-intercept (b), we can use one of the points from the table and plug it into the equation.

Let's use the point (1, 1):

$1 = 2(1) + b$ $1 = 2 + b$ $b = -1$

So, the equation for Function B is:

$y = 2x - 1$

This equation now allows us to analyze Function B in the same way we analyzed Function A. We know that:

• Slope: The slope is 2, which means that for every 1 unit we move to the right on the x-axis, the line goes up 2 units on the y-axis.
• Y-intercept: The y-intercept is -1, meaning the line crosses the y-axis at the point (0, -1).

Comparing Function A and Function B

Now that we have the equations for both functions, we can compare them directly:

• Function A: $y = \frac{1}{4}x + 2$
• Function B: $y = 2x - 1$

Here are some key differences and similarities:

1. Slope:

   • Function A has a slope of $\frac{1}{4}$.
   • Function B has a slope of 2.

   This means that Function B is much steeper than Function A. For every unit increase in x, Function B increases 2 units, while Function A only increases $\frac{1}{4}$ of a unit. The slope of a linear function determines its steepness.

2. Y-intercept:

   • Function A has a y-intercept of 2.
   • Function B has a y-intercept of -1.

   This means that Function A crosses the y-axis at (0, 2), while Function B crosses the y-axis at (0, -1). The y-intercept of a linear function is where the line intersects the y-axis.

3. Increasing/Decreasing:

   • Both functions are increasing because their slopes are positive.
   • However, Function B increases much faster than Function A due to its larger slope.

4. Points of Intersection: To find the point(s) where the two lines intersect (if any), we can set their equations equal to each other and solve for x:

$\frac{1}{4}x + 2 = 2x - 1$

To solve for x, we can first subtract $\frac{1}{4}x$ from both sides:

$2 = \frac{7}{4}x - 1$

Next, add 1 to both sides:

$3 = \frac{7}{4}x$

Finally, multiply both sides by $\frac{4}{7}$:

$x = \frac{12}{7}$

Now that we have the x-coordinate of the intersection point, we can plug it back into either equation to find the y-coordinate. Let's use Function B:

$y = 2(\frac{12}{7}) - 1 = \frac{24}{7} - \frac{7}{7} = \frac{17}{7}$

So, the point of intersection is $(\frac{12}{7}, \frac{17}{7})$.

Visualizing the Functions

To get a better understanding of the two functions, it's helpful to visualize them on a graph. Function A has a shallow positive slope and crosses the y-axis at 2. Function B has a steeper positive slope and crosses the y-axis at -1. The point where the two lines intersect is approximately (1.71, 2.43).

By visualizing these linear functions, we can quickly grasp their behavior and how they relate to each other. It's like seeing the difference between a gentle hill (Function A) and a much steeper climb (Function B).

Conclusion

Comparing Function A and Function B has shown us how to analyze linear functions presented in different forms—one as an equation and the other as a table of values. We found that Function B is steeper and has a lower y-intercept compared to Function A. By understanding the slope and y-intercept of each function, we can easily compare their behavior and even find their point of intersection. Keep exploring these concepts, guys, and you'll become masters of linear functions in no time!