If $f(x) = 4x + 12$ Is Graphed On A Coordinate Plane, What Is The $y$-intercept Of The Graph?A. 4 B. 8 C. 12 D. 16

The $y$-intercept of a linear function is the point at which the graph of the function crosses the $y$-axis. In other words, it is the value of $y$ when $x$ is equal to zero. In this article, we will explore how to find the $y$-intercept of a linear function, using the example of the function $f(x) = 4x + 12$.

What is a Linear Function?

A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope of the function and $b$ is the $y$-intercept. The slope of a linear function represents the rate of change of the function with respect to $x$, while the $y$-intercept represents the value of $y$ when $x$ is equal to zero.

Finding the $y$-Intercept of a Linear Function

To find the $y$-intercept of a linear function, we need to substitute $x = 0$ into the equation of the function. This will give us the value of $y$ when $x$ is equal to zero, which is the $y$-intercept.

Example: Finding the $y$-Intercept of $f(x) = 4x + 12$

Let's use the example of the function $f(x) = 4x + 12$ to find its $y$-intercept. To do this, we need to substitute $x = 0$ into the equation of the function.

f(x) = 4x + 12
f(0) = 4(0) + 12
f(0) = 0 + 12
f(0) = 12

Therefore, the $y$-intercept of the graph of $f(x) = 4x + 12$ is 12.

Conclusion

In conclusion, the $y$-intercept of a linear function is the point at which the graph of the function crosses the $y$-axis. To find the $y$-intercept of a linear function, we need to substitute $x = 0$ into the equation of the function. Using the example of the function $f(x) = 4x + 12$, we found that the $y$-intercept of the graph of this function is 12.

Why is the $y$-Intercept Important?

The $y$-intercept of a linear function is an important concept in mathematics because it represents the value of $y$ when $x$ is equal to zero. This is useful in a variety of applications, such as graphing linear functions and solving systems of linear equations.

Real-World Applications of the $y$-Intercept

The $y$-intercept of a linear function has a number of real-world applications. For example, in economics, the $y$-intercept of a linear function can represent the initial cost of a product or service. In physics, the $y$-intercept of a linear function can represent the initial velocity of an object.

Common Mistakes to Avoid When Finding the $y$-Intercept

When finding the $y$-intercept of a linear function, there are a number of common mistakes to avoid. One mistake is to substitute $x = 1$ into the equation of the function instead of $x = 0$. Another mistake is to forget to simplify the equation after substituting $x = 0$.

Tips for Finding the $y$-Intercept

When finding the $y$-intercept of a linear function, there are a number of tips to keep in mind. One tip is to make sure to substitute $x = 0$ into the equation of the function. Another tip is to simplify the equation after substituting $x = 0$.

Conclusion

In our previous article, we explored the concept of the $y$-intercept of a linear function and how to find it using the example of the function $f(x) = 4x + 12$. In this article, we will answer some common questions about the $y$-intercept of a linear function.

Q: What is the $y$-intercept of a linear function?

A: The $y$-intercept of a linear function is the point at which the graph of the function crosses the $y$-axis. It is the value of $y$ when $x$ is equal to zero.

Q: How do I find the $y$-intercept of a linear function?

A: To find the $y$-intercept of a linear function, you need to substitute $x = 0$ into the equation of the function. This will give you the value of $y$ when $x$ is equal to zero, which is the $y$-intercept.

Q: What is the difference between the $y$-intercept and the slope of a linear function?

A: The $y$-intercept of a linear function represents the value of $y$ when $x$ is equal to zero, while the slope of a linear function represents the rate of change of the function with respect to $x$.

Q: Can the $y$-intercept of a linear function be negative?

A: Yes, the $y$-intercept of a linear function can be negative. For example, if the equation of the function is $f(x) = -2x + 5$, the $y$-intercept would be 5, which is positive. However, if the equation of the function is $f(x) = -2x - 5$, the $y$-intercept would be -5, which is negative.

Q: How do I graph a linear function on a coordinate plane?

A: To graph a linear function on a coordinate plane, you need to plot two points on the graph. One point is the $y$-intercept, which is the point at which the graph crosses the $y$-axis. The other point is a point on the graph that is not on the $y$-axis. You can then draw a line through these two points to graph the function.

Q: Can the $y$-intercept of a linear function be a fraction?

A: Yes, the $y$-intercept of a linear function can be a fraction. For example, if the equation of the function is $f(x) = 1/2x + 3/4$, the $y$-intercept would be 3/4, which is a fraction.

Q: How do I find the $y$-intercept of a linear function with a negative slope?

A: To find the $y$-intercept of a linear function with a negative slope, you need to substitute $x = 0$ into the equation of the function, just like you would for a linear function with a positive slope. However, keep in mind that the $y$-intercept of a linear function with a negative slope will be negative.

Q: Can the $y$-intercept of a linear function be a decimal?

A: Yes, the $y$-intercept of a linear function can be a decimal. For example, if the equation of the function is $f(x) = 0.5x + 2.5$, the $y$-intercept would be 2.5, which is a decimal.

Conclusion

In conclusion, the $y$-intercept of a linear function is an important concept in mathematics that represents the value of $y$ when $x$ is equal to zero. We hope that this Q&A article has helped to clarify any questions you may have had about the $y$-intercept of a linear function.