Using the Given Information to Find the Equation

Oct 13, 2025 by miodeya.com 14 views

Understanding the Problem

When graphing a function, the x-intercepts represent the points where the function crosses the x-axis, and the y-intercept represents the point where the function crosses the y-axis. In this problem, we are given the x-intercepts at (2,0) and (4,0) and the y-intercept at (0,-16). We need to find the equation of the function that satisfies these conditions.

The Importance of x-Intercepts and y-Intercepts

The x-intercepts of a function are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by setting x equal to zero and solving for y.

Using the Given Information to Find the Equation

We are given the x-intercepts at (2,0) and (4,0), which means that the function must be equal to zero at these points. We can write this as:

f(2) = 0

f(4) = 0

We are also given the y-intercept at (0,-16), which means that the function must be equal to -16 when x is equal to zero. We can write this as:

f(0) = -16

Finding the Equation of the Function

To find the equation of the function, we can start by writing the function in factored form. Since the x-intercepts are at (2,0) and (4,0), we can write the function as:

f(x) = a(x-2)(x-4)

where a is a constant. We can find the value of a by using the y-intercept. When x is equal to zero, the function must be equal to -16, so we can write:

f(0) = a(0-2)(0-4) = -16

Simplifying this equation, we get:

-8a = -16

Dividing both sides by -8, we get:

a = 2

Substituting the Value of a into the Equation

Now that we have found the value of a, we can substitute it into the equation:

f(x) = 2(x-2)(x-4)

Simplifying the Equation

To simplify the equation, we can expand the factored form:

f(x) = 2(x^2 - 6x + 8)

f(x) = 2x^2 - 12x + 16

Comparing the Equation to the Options

Now that we have found the equation of the function, we can compare it to the options:

A. $$f(x)=-(x-2)(x-4)$$ B. $$f(x)=-(x+2)(x+4)$$ C. $$f(x)=2(x-2)(x-4)$$

Conclusion

Based on the given information, the equation of the function that has x-intercepts at (2,0) and (4,0) and a y-intercept of (0,-16) is:

f(x) = 2(x-2)(x-4)

This equation satisfies the conditions given in the problem, and it is the only option that does so.

Final Answer

The final answer is:

f(x) = 2(x-2)(x-4)

Graphing the Function

To graph the function, we can use the x-intercepts and the y-intercept to find the points on the graph. The x-intercepts are at (2,0) and (4,0), and the y-intercept is at (0,-16). We can plot these points on the graph and use them to find the equation of the function.

Using the Graph to Verify the Equation

Once we have graphed the function, we can use it to verify the equation. We can check that the function crosses the x-axis at the points (2,0) and (4,0), and that it crosses the y-axis at the point (0,-16). If the graph matches the equation, then we can be confident that the equation is correct.

Conclusion

In conclusion, the equation of the function that has x-intercepts at (2,0) and (4,0) and a y-intercept of (0,-16) is:

f(x) = 2(x-2)(x-4)

This equation satisfies the conditions given in the problem, and it is the only option that does so. We can use the graph to verify the equation and ensure that it is correct.

Final Answer

The final answer is:

f(x) = 2(x-2)(x-4)$$<br/> # **Q&A: Understanding the Equation of a Function with Given x-Intercepts and y-Intercept** ## **Q: What are x-intercepts and y-intercepts?** A: The x-intercepts of a function are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by setting x equal to zero and solving for y. ## **Q: How do I find the equation of a function with given x-intercepts and y-intercept?** A: To find the equation of a function with given x-intercepts and y-intercept, you can start by writing the function in factored form. Since the x-intercepts are at (2,0) and (4,0), you can write the function as: $$f(x) = a(x-2)(x-4)

where a is a constant. You can find the value of a by using the y-intercept. When x is equal to zero, the function must be equal to -16, so you can write:

f(0) = a(0-2)(0-4) = -16

Simplifying this equation, you get:

-8a = -16

Dividing both sides by -8, you get:

a = 2

Q: What if the x-intercepts are not at (2,0) and (4,0)?

A: If the x-intercepts are not at (2,0) and (4,0), you can still find the equation of the function by using the given x-intercepts. For example, if the x-intercepts are at (3,0) and (5,0), you can write the function as:

f(x) = a(x-3)(x-5)

where a is a constant. You can find the value of a by using the y-intercept.

Q: How do I know which option is correct?

A: To determine which option is correct, you can compare the equation you found to the options given. In this case, the correct option is:

f(x) = 2(x-2)(x-4)

This equation satisfies the conditions given in the problem, and it is the only option that does so.

Q: Can I use the graph to verify the equation?

A: Yes, you can use the graph to verify the equation. Once you have graphed the function, you can check that the function crosses the x-axis at the points (2,0) and (4,0), and that it crosses the y-axis at the point (0,-16). If the graph matches the equation, then you can be confident that the equation is correct.

Q: What if I get a different equation?

A: If you get a different equation, it may be because you made a mistake in finding the value of a or in writing the equation. You can try reworking the problem to see if you can find the correct equation.

Q: Can I use this method to find the equation of any function?

A: No, this method is only useful for finding the equation of a function with given x-intercepts and y-intercept. If you are given a different type of information, such as a table of values or a graph, you will need to use a different method to find the equation of the function.

Q: Is there a shortcut to finding the equation of a function?

A: Yes, there are shortcuts to finding the equation of a function. For example, you can use the fact that the x-intercepts are at (2,0) and (4,0) to write the function as:

f(x) = a(x-2)(x-4)

where a is a constant. You can then find the value of a by using the y-intercept.

Q: Can I use this method to find the equation of a quadratic function?

A: Yes, you can use this method to find the equation of a quadratic function. A quadratic function is a function of the form:

f(x) = ax^2 + bx + c

where a, b, and c are constants. You can use the given x-intercepts and y-intercept to find the equation of the quadratic function.

Q: Is there a way to check my work?

A: Yes, there are several ways to check your work. You can use a graphing calculator to graph the function and check that it matches the equation. You can also use a table of values to check that the function satisfies the given conditions.

Q: Can I use this method to find the equation of a function with complex x-intercepts?

A: No, this method is only useful for finding the equation of a function with real x-intercepts. If the x-intercepts are complex, you will need to use a different method to find the equation of the function.

Q: Is there a way to find the equation of a function with given x-intercepts and y-intercept without using the factored form?

A: Yes, there are several ways to find the equation of a function with given x-intercepts and y-intercept without using the factored form. For example, you can use the fact that the x-intercepts are at (2,0) and (4,0) to write the function as:

f(x) = a(x-2)(x-4)

where a is a constant. You can then find the value of a by using the y-intercept.

Q: Can I use this method to find the equation of a function with given x-intercepts and y-intercept in a different coordinate system?

A: No, this method is only useful for finding the equation of a function with given x-intercepts and y-intercept in the Cartesian coordinate system. If you are working in a different coordinate system, you will need to use a different method to find the equation of the function.

Q: Is there a way to find the equation of a function with given x-intercepts and y-intercept without using algebra?

A: Yes, there are several ways to find the equation of a function with given x-intercepts and y-intercept without using algebra. For example, you can use a graphing calculator to graph the function and check that it matches the equation. You can also use a table of values to check that the function satisfies the given conditions.

Q: Can I use this method to find the equation of a function with given x-intercepts and y-intercept in a different mathematical context?

A: No, this method is only useful for finding the equation of a function with given x-intercepts and y-intercept in the context of algebra and geometry. If you are working in a different mathematical context, you will need to use a different method to find the equation of the function.

Q: Is there a way to find the equation of a function with given x-intercepts and y-intercept without using a graphing calculator?

A: Yes, there are several ways to find the equation of a function with given x-intercepts and y-intercept without using a graphing calculator. For example, you can use a table of values to check that the function satisfies the given conditions. You can also use a graphing software to graph the function and check that it matches the equation.

Q: Can I use this method to find the equation of a function with given x-intercepts and y-intercept in a different mathematical discipline?

A: No, this method is only useful for finding the equation of a function with given x-intercepts and y-intercept in the context of algebra and geometry. If you are working in a different mathematical discipline, you will need to use a different method to find the equation of the function.

Q: Is there a way to find the equation of a function with given x-intercepts and y-intercept without using a table of values?

A: Yes, there are several ways to find the equation of a function with given x-intercepts and y-intercept without using a table of values. For example, you can use a graphing software to graph the function and check that it matches the equation. You can also use a graphing calculator to graph the function and check that it matches the equation.

Q: Can I use this method to find the equation of a function with given x-intercepts and y-intercept in a different mathematical context?

Q: Is there a way to find the equation of a function with given x-intercepts and y-intercept without using a graphing calculator or graphing software?

A: Yes, there are several ways to find the equation of a function with given x-intercepts and y-intercept without using a graphing calculator or graphing software. For example, you can use a table of values to check that the function satisfies the given conditions. You can also use algebraic methods to find the equation of the function.

Q: Can I use this method to find the equation of a function with given x-intercepts and y-intercept in a different mathematical discipline?

A: No, this method is only useful for finding the equation of a function with given x-intercepts and y-intercept in the context of algebra and geometry. If you are working in a different