Consider The Two Functions Given Below. F(x)= -3x+5 G(x)= F(x)+9 What Is The Y-intercept Of G? A. (0,9)
Understanding the concepts of functions, especially linear functions, is essential for mastering algebra and related mathematical disciplines. In this article, we will analyze two functions, F(x) = -3x + 5 and G(x) = F(x) + 9, to determine the y-intercept of G. This exploration will not only help you find the specific point where G crosses the y-axis but will also deepen your comprehension of how functions relate to each other and how to interpret their graphs.
Introduction to Functions and Their Graphs
In mathematics, a function describes a relationship between an input and an output, often represented as f(x). When graphing functions, we plot points (x, f(x)) on the coordinate plane to visualize this relationship. For linear functions, the graph is a straight line characterized by its slope and y-intercept.
What Is a Y-Intercept?
The y-intercept of a function is the point where its graph crosses the y-axis (vertical axis). At this point, the value of x is zero. Finding the y-intercept involves evaluating the function at x = 0.
Given Functions and Their Roles
Let's analyze the functions provided:
- F(x) = -3x + 5
- G(x) = F(x) + 9
Our goal is to determine the y-intercept of G, which requires understanding how G relates to F and how to evaluate G at x = 0.
Understanding F(x) = -3x + 5
This is a linear function with:
- Slope (m): -3
- Y-intercept (b): 5
Graphically, this line crosses the y-axis at (0, 5) and has a negative slope, indicating it decreases as x increases.
Understanding G(x) = F(x) + 9
G(x) is derived by adding 9 to F(x), which means G(x) is a vertically shifted version of F(x). Specifically, G(x) is shifted upward by 9 units.
Calculating the Y-Intercept of G
To find the y-intercept of G, evaluate G(x) at x = 0:
G(0) = F(0) + 9
Since F(x) = -3x + 5, then:
F(0) = -3(0) + 5 = 0 + 5 = 5
Therefore,
G(0) = 5 + 9 = 14
Thus, the y-intercept of G is at the point (0, 14).
Interpreting the Results
From our calculation:
- The point where G crosses the y-axis is at (0, 14).
- This is because G(x) is a vertical shift of F(x) by +9 units.
- The original y-intercept of F(x) was at (0, 5), and shifting upward by 9 moves it to (0, 14).
Common Mistakes and Clarifications
When solving problems like this, students often make some common errors:
- Confusing the y-intercept of F(x) with that of G(x): Remember that G(x) is shifted vertically, so its y-intercept is different unless the shift is zero.
- Forgetting to evaluate at x=0: The y-intercept always occurs at x=0, so plugging in x=0 is essential.
- Misinterpreting the added constant: Adding 9 to the entire function shifts the graph upward by 9 units, affecting the y-intercept accordingly.
Real-World Applications of Linear Functions and Their Intercepts
Understanding how to find the y-intercept of a function has practical implications in various fields:
- Economics: Determining the starting point of a cost or revenue function.
- Physics: Identifying initial conditions such as initial velocity or position.
- Biology: Modeling growth rates where the intercept indicates the starting population.
- Engineering: Analyzing systems where initial states are crucial for understanding behavior.
Step-by-Step Summary for Finding the Y-Intercept of G
To summarize the process:
- Identify the given functions: F(x) = -3x + 5 and G(x) = F(x) + 9.
- Recognize that G(x) is a vertical shift of F(x) by +9 units.
- Calculate F(0): 0, 5.
- Add 9 to F(0) to find G(0): 5 + 9 = 14.
- Conclude that the y-intercept of G is at (0, 14).
Final Answer and Explanation
Based on the calculations above, the y-intercept of G(x) = F(x) + 9 is at the point (0, 14). The multiple-choice options provided in typical quiz formats might include (0, 9), but that would be incorrect unless the function shifted downward, which it does not in this case. Instead, the correct y-intercept is at (0, 14).
Note: The answer choice (0, 9) might be a distractor or a common misconception, but the precise calculation confirms (0, 14) as the correct y-intercept.
Conclusion
Understanding how to manipulate and analyze linear functions is fundamental in algebra. Recognizing how transformations like vertical shifts affect key points such as the y-intercept enables students to interpret and graph functions accurately. By evaluating functions at specific points, especially x=0, you can determine intercepts efficiently and apply this knowledge across various scientific and practical contexts. Remember, for G(x) = F(x) + 9, the y-intercept is at (0, 14), confirming the shift upward of the original function F(x).
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