Function A Vs. Function B: Which Is Greater?

When comparing different mathematical functions, it's crucial to understand how they behave at various points. Today, we're diving into a scenario comparing two functions, 'Function A' and 'Function B,' to determine which one yields a larger $y$-value at a specific $x$-value. This might sound like a simple task, but it involves understanding how to interpret given data and how to extrapolate or calculate values for functions.

Understanding Function A

Function A is presented to us in a tabular format, which is a common and straightforward way to represent discrete points of a function. We are given the following pairs of $x$ and $y$ values: $(-9, -2)$, $(-3, 2)$, and $(3, 6)$. To understand Function A better, we need to determine the relationship between $x$ and $y$. Let's examine the changes in $y$ relative to the changes in $x$. Between the first two points, as $x$ increases from $-9$ to $-3$ (an increase of 6), $y$ increases from $-2$ to $2$ (an increase of 4). The rate of change here is $\frac{4}{6} = \frac{2}{3}$. Between the second and third points, as $x$ increases from $-3$ to $3$ (an increase of 6), $y$ increases from $2$ to $6$ (an increase of 4). The rate of change is again $\frac{4}{6} = \frac{2}{3}$. Since the rate of change is constant, Function A is a linear function with a slope of $\frac{2}{3}$. We can represent this as $y = \frac{2}{3}x + b$. To find the y-intercept ($b$), we can plug in any of the given points. Using $(-3, 2)$: $2 = \frac{2}{3}(-3) + b 2 = -2 + b b = 4$. So, the equation for Function A is $y = \frac{2}{3}x + 4$. Now, let's find the $y$-value of Function A when $x = -6$. Plugging this into our equation: $y = \frac{2}{3}(-6) + 4 y = -4 + 4 y = 0$. So, for Function A, when $x = -6$, the $y$-value is $0$. This step is critical because it provides a concrete value for comparison.

Understanding Function B

Now, let's turn our attention to Function B. Unlike Function A, Function B is not explicitly defined by a table or an equation. Instead, it's presented as a question: "Which statement is true? The $y$-value of Function A when $x=-6$ is greater than the $y$-value of Function B when..." This implies that Function B must also have a defined $y$-value at $x=-6$ for the comparison to be meaningful. The structure of the problem suggests that there might be a missing piece of information for Function B, or that Function B is implicitly defined by the options that would follow this statement. However, since we are asked to determine which statement is true and we've already calculated the $y$-value for Function A at $x=-6$ to be $0$, we can infer that we need to evaluate Function B at $x=-6$. Without further information about Function B's definition (e.g., another table, an equation, or a description), we cannot definitively calculate its $y$-value at $x=-6$. The question as presented is incomplete for a full comparison. However, if we assume there are implied options or that this is a setup for a multiple-choice question where the properties of Function B are revealed later, we can proceed with the known value of Function A. The critical step here is recognizing what information is provided and what is needed. Since the problem is posed as a comparison, and we have one side of the comparison ($y=0$ for Function A at $x=-6$), the missing element is the specific definition or value for Function B at $x=-6$. This could be another linear function, a quadratic function, or any other type of function.

Making the Comparison

We have established that for Function A, when $x = -6$, the $y$-value is $0$. The question asks us to compare this value to the $y$-value of Function B when $x = -6$. Without the explicit definition of Function B, we cannot perform a direct calculation. However, the phrasing suggests a comparative statement is to be made. Let's assume, for the sake of illustrating the process, that Function B is defined by a different set of points or an equation. For instance, if Function B were given by the equation $y = x + 5$, then at $x = -6$, the $y$-value would be $y = -6 + 5 = -1$. In this hypothetical case, $0 > -1$, meaning the $y$-value of Function A ($0$) is greater than the $y$-value of Function B ($-1$) at $x=-6$. Alternatively, if Function B were given by $y = -x - 10$, then at $x = -6$, the $y$-value would be $y = -(-6) - 10 = 6 - 10 = -4$. Again, $0 > -4$, so Function A's $y$-value is greater. If Function B were $y = 2x + 10$, then at $x = -6$, $y = 2(-6) + 10 = -12 + 10 = -2$. In this scenario, $0 > -2$, and Function A is still greater. The core of this section is understanding that once both functions are evaluated at the same $x$-value, the comparison becomes a simple inequality check. The challenge here lies entirely in the incomplete definition of Function B. If Function B were defined, for example, by the points $(0, 10), (1, 12), (2, 14)$, it would also be a linear function with a slope of 2. Its equation would be $y = 2x + 10$. Evaluating at $x = -6$ would give $y = 2(-6) + 10 = -12 + 10 = -2$. Comparing Function A ($y=0$ at $x=-6$) with this Function B ($y=-2$ at $x=-6$) leads to the conclusion that Function A's $y$-value is greater.

Conclusion and Next Steps

In summary, we've successfully determined that for Function A, the $y$-value at $x = -6$ is $0$. The comparison hinges on the unknown $y$-value of Function B at the same $x$-value. Without a complete definition for Function B, we cannot definitively state which function's $y$-value is greater. The problem, as presented, likely expects a comparison against a set of predefined conditions or options for Function B. If you were presented with this problem in a test or assignment, ensure you have all the information for both functions before attempting to answer. To further explore concepts of functions and their comparisons, you can visit **

Khan Academy or **

Math is Fun for excellent resources and explanations.**