Select The Correct Answer.Which Of These Is The Standard Form Of The Following Function? F ( X ) = 7 ( X − 1 ) 2 + 3 F(x)=7(x-1)^2+3 F ( X ) = 7 ( X − 1 ) 2 + 3 A. F ( X ) = 7 X 2 − 14 X + 10 F(x)=7x^2-14x+10 F ( X ) = 7 X 2 − 14 X + 10 B. F ( X ) = − 7 X 2 − 14 X − 10 F(x)=-7x^2-14x-10 F ( X ) = − 7 X 2 − 14 X − 10 C. F ( X ) = 7 X 2 − 14 X − 10 F(x)=7x^2-14x-10 F ( X ) = 7 X 2 − 14 X − 10 D. F ( X ) = − 7 X 2 − 14 X + 10 F(x)=-7x^2-14x+10 F ( X ) = − 7 X 2 − 14 X + 10

When it comes to quadratic functions, understanding the standard form is crucial for simplifying and solving equations. In this article, we will explore the standard form of a quadratic function and apply it to the given function $f(x)=7(x-1)^2+3$ to determine the correct answer.

What is the Standard Form of a Quadratic Function?

The standard form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. This form is also known as the general form of a quadratic function. The standard form is useful for identifying the vertex, axis of symmetry, and other key features of the quadratic function.

Expanding the Given Function

To determine the standard form of the given function $f(x)=7(x-1)^2+3$, we need to expand the squared term using the formula $(x-h)^2=x^2-2hx+h^2$. In this case, $h=1$, so we have:

$f(x)=7(x-1)^2+3$ $f(x)=7(x^2-2x+1)+3$ $f(x)=7x^2-14x+7+3$ $f(x)=7x^2-14x+10$

Comparing with the Options

Now that we have expanded the given function, we can compare it with the options provided:

A. $f(x)=7x^2-14x+10$ B. $f(x)=-7x^2-14x-10$ C. $f(x)=7x^2-14x-10$ D. $f(x)=-7x^2-14x+10$

Based on our expansion, we can see that option A matches the standard form of the given function.

Conclusion

In conclusion, the standard form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. By expanding the given function $f(x)=7(x-1)^2+3$, we determined that the standard form is $f(x)=7x^2-14x+10$. Therefore, the correct answer is option A.

Key Takeaways

• The standard form of a quadratic function is $f(x)=ax^2+bx+c$.
• To determine the standard form of a quadratic function, expand the squared term using the formula $(x-h)^2=x^2-2hx+h^2$.
• Compare the expanded function with the options provided to determine the correct answer.

Practice Problems

1. Find the standard form of the quadratic function $f(x)=3(x+2)^2-5$.
2. Determine the standard form of the quadratic function $f(x)=2(x-3)^2+1$.
3. Find the standard form of the quadratic function $f(x)=5(x-2)^2-2$.

Solutions

1. $f(x)=3x^2+12x+13$
2. $f(x)=2x^2-12x+7$
3. $f(x)=5x^2-20x+6$

In our previous article, we explored the standard form of a quadratic function and applied it to the given function $f(x)=7(x-1)^2+3$ to determine the correct answer. In this article, we will provide a Q&A section to help you better understand the standard form of a quadratic function.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants.

Q: How do I determine the standard form of a quadratic function?

A: To determine the standard form of a quadratic function, expand the squared term using the formula $(x-h)^2=x^2-2hx+h^2$. Then, simplify the expression to obtain the standard form.

Q: What is the formula for expanding a squared term?

A: The formula for expanding a squared term is $(x-h)^2=x^2-2hx+h^2$.

Q: How do I compare the expanded function with the options provided?

A: To compare the expanded function with the options provided, simply substitute the values of $a$, $b$, and $c$ into the standard form equation and compare it with the options.

Q: What are some common mistakes to avoid when determining the standard form of a quadratic function?

A: Some common mistakes to avoid when determining the standard form of a quadratic function include:

• Not expanding the squared term correctly
• Not simplifying the expression correctly
• Not comparing the expanded function with the options provided correctly

Q: How can I practice determining the standard form of a quadratic function?

A: You can practice determining the standard form of a quadratic function by working through practice problems, such as the ones provided in our previous article. You can also try creating your own practice problems to challenge yourself.

Q: What are some real-world applications of the standard form of a quadratic function?

A: The standard form of a quadratic function has many real-world applications, including:

• Modeling the trajectory of a projectile
• Determining the maximum or minimum value of a function
• Finding the vertex of a parabola

Q: Can you provide some examples of quadratic functions in real-world applications?

A: Here are some examples of quadratic functions in real-world applications:

• The trajectory of a baseball thrown from a height of 3 meters with an initial velocity of 20 meters per second: $h(t)=-4.9t^2+20t+3$
• The cost of producing x units of a product: $C(x)=2x^2+10x+50$
• The height of a ball thrown from the ground with an initial velocity of 15 meters per second: $h(t)=-4.9t^2+15t$

Conclusion

In conclusion, the standard form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. By expanding the squared term using the formula $(x-h)^2=x^2-2hx+h^2$ and simplifying the expression, you can determine the standard form of a quadratic function. We hope this Q&A section has helped you better understand the standard form of a quadratic function and its applications.

Key Takeaways

• The standard form of a quadratic function is $f(x)=ax^2+bx+c$.
• To determine the standard form of a quadratic function, expand the squared term using the formula $(x-h)^2=x^2-2hx+h^2$.
• Compare the expanded function with the options provided to determine the correct answer.
• Practice determining the standard form of a quadratic function by working through practice problems.
• The standard form of a quadratic function has many real-world applications, including modeling the trajectory of a projectile and determining the maximum or minimum value of a function.