Find The $y$-intercept Of The Parabola $y=\frac{31}{10} X^2+\frac{21}{10}$.Simplify Your Answer And Write It As A Proper Fraction, Improper Fraction, Or Integer. □ \square □

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Introduction

In mathematics, the $y$-intercept of a parabola is the point at which the parabola intersects the $y$-axis. It is a crucial concept in algebra and calculus, and is used to determine the behavior of a parabola. In this article, we will explore how to find the $y$-intercept of a parabola, using the equation $y=\frac{31}{10} x^2+\frac{21}{10}$ as an example.

What is a Parabola?

A parabola is a type of quadratic function that can be represented by the equation $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. The parabola opens upwards if $a>0$ and downwards if $a<0$. The $y$-intercept of a parabola is the point at which the parabola intersects the $y$-axis, which occurs when $x=0$.

Finding the $y$-intercept

To find the $y$-intercept of a parabola, we need to substitute $x=0$ into the equation of the parabola. This will give us the value of $y$ at the point where the parabola intersects the $y$-axis.

Let's use the equation $y=\frac{31}{10} x^2+\frac{21}{10}$ as an example. To find the $y$-intercept, we need to substitute $x=0$ into the equation:

y=3110(0)2+2110y=\frac{31}{10} (0)^2+\frac{21}{10}

y=3110(0)+2110y=\frac{31}{10} (0)+\frac{21}{10}

y=0+2110y=0+\frac{21}{10}

y=2110y=\frac{21}{10}

Therefore, the $y$-intercept of the parabola $y=\frac{31}{10} x^2+\frac{21}{10}$ is $\frac{21}{10}$.

Simplifying the Answer

In this case, the answer is already simplified, as it is a proper fraction. However, if the answer were an improper fraction or an integer, we would need to simplify it further.

For example, if the answer were $\frac{42}{10}$, we could simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

4210=215\frac{42}{10}=\frac{21}{5}

Therefore, the simplified answer is $\frac{21}{5}$.

Conclusion

In conclusion, finding the $y$-intercept of a parabola is a straightforward process that involves substituting $x=0$ into the equation of the parabola. By following these steps, we can easily find the $y$-intercept of any parabola, and simplify the answer to its simplest form.

Example Problems

Here are a few example problems to help you practice finding the $y$-intercept of a parabola:

  1. Find the $y$-intercept of the parabola $y=2x^2+3x+1$.
  2. Find the $y$-intercept of the parabola $y=-x^2+4x-3$.
  3. Find the $y$-intercept of the parabola $y=\frac{1}{2} x^2+\frac{3}{2} x-\frac{1}{2}$.

Answer Key

  1. y=1y=1

  2. y=−3y=-3

  3. y=−12y=-\frac{1}{2}

Tips and Tricks

Here are a few tips and tricks to help you find the $y$-intercept of a parabola:

  • Make sure to substitute $x=0$ into the equation of the parabola.
  • Simplify the answer to its simplest form.
  • Use the equation $y=ax^2+bx+c$ to find the $y$-intercept of a parabola.
  • Practice finding the $y$-intercept of different parabolas to become more comfortable with the process.

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about finding the $y$-intercept of a parabola, along with their answers:

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

A: The $y$-intercept of a parabola is the point at which the parabola intersects the $y$-axis. It is the value of $y$ when $x=0$.

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

A: To find the $y$-intercept of a parabola, you need to substitute $x=0$ into the equation of the parabola. This will give you the value of $y$ at the point where the parabola intersects the $y$-axis.

Q: What if the equation of the parabola is not in the form $y=ax^2+bx+c$?

A: If the equation of the parabola is not in the form $y=ax^2+bx+c$, you can still find the $y$-intercept by substituting $x=0$ into the equation. However, you may need to simplify the equation first.

Q: Can I use a calculator to find the $y$-intercept of a parabola?

A: Yes, you can use a calculator to find the $y$-intercept of a parabola. Simply enter the equation of the parabola into the calculator and substitute $x=0$ to find the value of $y$.

Q: What if the $y$-intercept of a parabola is not a whole number?

A: If the $y$-intercept of a parabola is not a whole number, you can still simplify it to its simplest form. For example, if the $y$-intercept is $\frac{3}{4}$, you can simplify it to $\frac{3}{4}$.

Q: Can I find the $y$-intercept of a parabola with a negative coefficient?

A: Yes, you can find the $y$-intercept of a parabola with a negative coefficient. Simply substitute $x=0$ into the equation and simplify the result.

Q: What if I get a negative value for the $y$-intercept of a parabola?

A: If you get a negative value for the $y$-intercept of a parabola, it means that the parabola opens downwards. This is because the $y$-intercept is the point at which the parabola intersects the $y$-axis, and a negative value indicates that the parabola is below the $x$-axis.

Q: Can I use the $y$-intercept of a parabola to determine its behavior?

A: Yes, you can use the $y$-intercept of a parabola to determine its behavior. If the $y$-intercept is positive, the parabola opens upwards. If the $y$-intercept is negative, the parabola opens downwards.

Common Mistakes

Here are some common mistakes to avoid when finding the $y$-intercept of a parabola:

  • Not substituting $x=0$ into the equation of the parabola.
  • Not simplifying the answer to its simplest form.
  • Using the wrong equation of the parabola.
  • Not checking the sign of the $y$-intercept.

Conclusion

In conclusion, finding the $y$-intercept of a parabola is a crucial concept in algebra and calculus. By following the steps outlined in this article, you can easily find the $y$-intercept of any parabola, and simplify the answer to its simplest form. With practice and patience, you will become more comfortable with finding the $y$-intercept of a parabola, and be able to apply this concept to a wide range of mathematical problems.

Additional Resources

Here are some additional resources to help you learn more about finding the $y$-intercept of a parabola:

  • Khan Academy: Finding the $y$-intercept of a parabola
  • Mathway: Finding the $y$-intercept of a parabola
  • Wolfram Alpha: Finding the $y$-intercept of a parabola

Practice Problems

Here are some practice problems to help you practice finding the $y$-intercept of a parabola:

  1. Find the $y$-intercept of the parabola $y=2x^2+3x+1$.
  2. Find the $y$-intercept of the parabola $y=-x^2+4x-3$.
  3. Find the $y$-intercept of the parabola $y=\frac{1}{2} x^2+\frac{3}{2} x-\frac{1}{2}$.

Answer Key

  1. y=1y=1

  2. y=−3y=-3

  3. y=−12y=-\frac{1}{2}