A Function $f$ Is Defined By $f(x) = -3x + 6$.Part A:(a) Graph $f$ In The Coordinate Plane.Part B:(b) What Is The X-intercept Of The Graph Of $f$?$\square$Part C:(c) What Is The Y-intercept Of The Graph Of

A Function $f$ Defined by $f(x) = -3x + 6$

Part A: Graphing the Function $f$

To graph the function $f(x) = -3x + 6$, we need to understand the behavior of the function. The function is a linear function, which means it has a constant rate of change. The slope of the function is -3, which means that for every unit increase in $x$, the value of $f(x)$ decreases by 3 units.

To graph the function, we can start by finding the $y$-intercept, which is the point where the function intersects the $y$-axis. To find the $y$-intercept, we can substitute $x = 0$ into the function:

$f(0) = -3(0) + 6 = 6$

So, the $y$-intercept is $(0, 6)$.

Next, we can find the $x$-intercept, which is the point where the function intersects the $x$-axis. To find the $x$-intercept, we can substitute $f(x) = 0$ into the function:

$0 = -3x + 6$

Solving for $x$, we get:

$x = 2$

So, the $x$-intercept is $(2, 0)$.

Now that we have the $y$-intercept and the $x$-intercept, we can graph the function. We can start by plotting the $y$-intercept at $(0, 6)$. Then, we can plot the $x$-intercept at $(2, 0)$. Finally, we can draw a line through the two points to graph the function.

Part B: Finding the $x$-Intercept of the Graph of $f$

The $x$-intercept of the graph of $f$ is the point where the function intersects the $x$-axis. To find the $x$-intercept, we can substitute $f(x) = 0$ into the function:

$0 = -3x + 6$

Solving for $x$, we get:

$x = 2$

So, the $x$-intercept of the graph of $f$ is $(2, 0)$.

Part C: Finding the $y$-Intercept of the Graph of $f$

The $y$-intercept of the graph of $f$ is the point where the function intersects the $y$-axis. To find the $y$-intercept, we can substitute $x = 0$ into the function:

$f(0) = -3(0) + 6 = 6$

So, the $y$-intercept of the graph of $f$ is $(0, 6)$.

Graphing the Function $f$

To graph the function $f(x) = -3x + 6$, we can use the following steps:

1. Find the $y$-intercept by substituting $x = 0$ into the function.
2. Find the $x$-intercept by substituting $f(x) = 0$ into the function.
3. Plot the $y$-intercept and the $x$-intercept on the coordinate plane.
4. Draw a line through the two points to graph the function.

Properties of the Function $f$

The function $f(x) = -3x + 6$ has the following properties:

• The function is a linear function, which means it has a constant rate of change.
• The slope of the function is -3, which means that for every unit increase in $x$, the value of $f(x)$ decreases by 3 units.
• The $y$-intercept is $(0, 6)$, which means that the function intersects the $y$-axis at the point $(0, 6)$.
• The $x$-intercept is $(2, 0)$, which means that the function intersects the $x$-axis at the point $(2, 0)$.

Real-World Applications of the Function $f$

The function $f(x) = -3x + 6$ has several real-world applications, including:

• Modeling the cost of a product as a function of the number of units produced.
• Modeling the revenue of a company as a function of the number of units sold.
• Modeling the temperature of a substance as a function of time.

Conclusion

In conclusion, the function $f(x) = -3x + 6$ is a linear function that has a constant rate of change. The function has a $y$-intercept of $(0, 6)$ and an $x$-intercept of $(2, 0)$. The function has several real-world applications, including modeling the cost of a product, the revenue of a company, and the temperature of a substance.
A Function $f$ Defined by $f(x) = -3x + 6$: Q&A

Q: What is the definition of the function $f$?

A: The function $f$ is defined by the equation $f(x) = -3x + 6$. This means that for any input value of $x$, the function $f$ will output a value that is equal to $-3x + 6$.

Q: What type of function is $f$?

A: The function $f$ is a linear function. This means that it has a constant rate of change, and its graph is a straight line.

Q: What is the slope of the function $f$?

A: The slope of the function $f$ is -3. This means that for every unit increase in $x$, the value of $f(x)$ decreases by 3 units.

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

A: The $y$-intercept of the function $f$ is $(0, 6)$. This means that the function intersects the $y$-axis at the point $(0, 6)$.

Q: What is the $x$-intercept of the function $f$?

A: The $x$-intercept of the function $f$ is $(2, 0)$. This means that the function intersects the $x$-axis at the point $(2, 0)$.

Q: How do I graph the function $f$?

A: To graph the function $f$, you can use the following steps:

1. Find the $y$-intercept by substituting $x = 0$ into the function.
2. Find the $x$-intercept by substituting $f(x) = 0$ into the function.
3. Plot the $y$-intercept and the $x$-intercept on the coordinate plane.
4. Draw a line through the two points to graph the function.

Q: What are some real-world applications of the function $f$?

A: The function $f(x) = -3x + 6$ has several real-world applications, including:

• Modeling the cost of a product as a function of the number of units produced.
• Modeling the revenue of a company as a function of the number of units sold.
• Modeling the temperature of a substance as a function of time.

Q: How do I find the value of $f(x)$ for a given value of $x$?

A: To find the value of $f(x)$ for a given value of $x$, you can substitute the value of $x$ into the equation $f(x) = -3x + 6$ and solve for $f(x)$.

Q: What is the domain of the function $f$?

A: The domain of the function $f$ is all real numbers. This means that the function $f$ is defined for any value of $x$.

Q: What is the range of the function $f$?

A: The range of the function $f$ is all real numbers less than or equal to 6. This means that the function $f$ can output any value less than or equal to 6.

Q: How do I determine if the function $f$ is increasing or decreasing?

A: To determine if the function $f$ is increasing or decreasing, you can look at the slope of the function. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing. In this case, the slope of the function $f$ is -3, which means that the function is decreasing.

Q: How do I find the equation of the function $f$ if I know the $y$-intercept and the $x$-intercept?

A: To find the equation of the function $f$ if you know the $y$-intercept and the $x$-intercept, you can use the following steps:

1. Write the equation of the function in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
2. Substitute the value of the $x$-intercept into the equation and solve for $m$.
3. Substitute the value of the $y$-intercept into the equation and solve for $b$.
4. Write the final equation of the function in the form $f(x) = mx + b$.