Find The \[$ X \$\]-intercept(s) And The Coordinates Of The Vertex For The Parabola \[$ Y = -x^2 + 2x + 8 \$\].If There Is More Than One \[$ X \$\]-intercept, Separate Them With Commas.\[$ X \$\]-intercept(s):
Introduction
In mathematics, a parabola is a type of quadratic function that can be represented in the form { y = ax^2 + bx + c $}$, where { a $}$, { b $}$, and { c $}$ are constants. The parabola { y = -x^2 + 2x + 8 $}$ is a specific example of a quadratic function, where { a = -1 $}$, { b = 2 $}$, and { c = 8 $}$. In this article, we will focus on finding the { x $}$-intercept(s) and the coordinates of the vertex for the given parabola.
Finding the { x $}$-intercept(s)
The { x $}$-intercept(s) of a parabola are the points where the parabola intersects the { x $}$-axis. In other words, the { x $}$-intercept(s) are the values of { x $}$ for which { y = 0 $}$. To find the { x $}$-intercept(s) of the parabola { y = -x^2 + 2x + 8 $}$, we need to set { y = 0 $}$ and solve for { x $}$.
Step 1: Set { y = 0 $}$
{ 0 = -x^2 + 2x + 8 $}$
Step 2: Rearrange the equation
{ x^2 - 2x - 8 = 0 $}$
Step 3: Solve the quadratic equation
We can solve the quadratic equation using the quadratic formula:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
In this case, { a = 1 $}$, { b = -2 $}$, and { c = -8 $}$. Plugging these values into the quadratic formula, we get:
{ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} $}$
{ x = \frac{2 \pm \sqrt{4 + 32}}{2} $}$
{ x = \frac{2 \pm \sqrt{36}}{2} $}$
{ x = \frac{2 \pm 6}{2} $}$
Therefore, we have two possible values for { x $}$:
{ x = \frac{2 + 6}{2} = 4 $}$
{ x = \frac{2 - 6}{2} = -2 $}$
Step 4: Write the { x $}$-intercept(s)
The { x $}$-intercept(s) of the parabola { y = -x^2 + 2x + 8 $}$ are { x = 4 $}$ and { x = -2 $}$.
Finding the coordinates of the vertex
The vertex of a parabola is the point where the parabola changes direction. The coordinates of the vertex can be found using the formula:
{ x = -\frac{b}{2a} $}$
In this case, { a = -1 $}$ and { b = 2 $}$. Plugging these values into the formula, we get:
{ x = -\frac{2}{2(-1)} $}$
{ x = -\frac{2}{-2} $}$
{ x = 1 $}$
To find the { y $}$-coordinate of the vertex, we need to plug the value of { x $}$ into the equation of the parabola:
{ y = -(1)^2 + 2(1) + 8 $}$
{ y = -1 + 2 + 8 $}$
{ y = 9 $}$
Therefore, the coordinates of the vertex are { (1, 9) $}$.
Conclusion
In this article, we found the { x $}$-intercept(s) and the coordinates of the vertex for the parabola { y = -x^2 + 2x + 8 $}$. The { x $}$-intercept(s) are { x = 4 $}$ and { x = -2 $}$, and the coordinates of the vertex are { (1, 9) $}$. These values can be used to graph the parabola and understand its behavior.
Final Answer
The { x $}$-intercept(s) of the parabola { y = -x^2 + 2x + 8 $}$ are { x = 4 $}$ and { x = -2 $}$. The coordinates of the vertex are { (1, 9) $}$.
Q: What is the { x $}$-intercept(s) of a parabola?
A: The { x $}$-intercept(s) of a parabola are the points where the parabola intersects the { x $}$-axis. In other words, the { x $}$-intercept(s) are the values of { x $}$ for which { y = 0 $}$.
Q: How do I find the { x $}$-intercept(s) of a parabola?
A: To find the { x $}$-intercept(s) of a parabola, you need to set { y = 0 $}$ and solve for { x $}$. This can be done by rearranging the equation and solving the resulting quadratic equation.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:
{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$
Q: How do I use the quadratic formula to find the { x $}$-intercept(s) of a parabola?
A: To use the quadratic formula to find the { x $}$-intercept(s) of a parabola, you need to plug in the values of { a $}$, { b $}$, and { c $}$ into the formula and simplify.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the parabola changes direction. The coordinates of the vertex can be found using the formula:
{ x = -\frac{b}{2a} $}$
Q: How do I find the coordinates of the vertex of a parabola?
A: To find the coordinates of the vertex of a parabola, you need to plug the value of { x $}$ into the equation of the parabola and simplify.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola is significant because it represents the point where the parabola changes direction. This point is also known as the turning point of the parabola.
Q: Can you provide an example of how to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, let's consider the parabola { y = -x^2 + 2x + 8 $}$. To find the { x $}$-intercept(s), we need to set { y = 0 $}$ and solve for { x $}$. This can be done by rearranging the equation and solving the resulting quadratic equation. The quadratic formula can be used to solve the quadratic equation.
Q: What is the final answer for the { x $}$-intercept(s) and the coordinates of the vertex of the parabola { y = -x^2 + 2x + 8 $}$?
A: The { x $}$-intercept(s) of the parabola { y = -x^2 + 2x + 8 $}$ are { x = 4 $}$ and { x = -2 $}$. The coordinates of the vertex are { (1, 9) $}$.
Q: Can you provide a summary of the steps to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, here is a summary of the steps to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola:
- Set { y = 0 $}$ and solve for { x $}$.
- Rearrange the equation and solve the resulting quadratic equation.
- Use the quadratic formula to solve the quadratic equation.
- Find the coordinates of the vertex using the formula { x = -\frac{b}{2a} $}$.
- Plug the value of { x $}$ into the equation of the parabola and simplify.
Q: What are some common mistakes to avoid when finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Some common mistakes to avoid when finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola include:
- Not setting { y = 0 $}$ and solving for { x $}$.
- Not rearranging the equation and solving the resulting quadratic equation.
- Not using the quadratic formula to solve the quadratic equation.
- Not finding the coordinates of the vertex using the formula { x = -\frac{b}{2a} $}$.
- Not plugging the value of { x $}$ into the equation of the parabola and simplifying.
Q: Can you provide some tips for finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, here are some tips for finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola:
- Make sure to set { y = 0 $}$ and solve for { x $}$.
- Rearrange the equation and solve the resulting quadratic equation.
- Use the quadratic formula to solve the quadratic equation.
- Find the coordinates of the vertex using the formula { x = -\frac{b}{2a} $}$.
- Plug the value of { x $}$ into the equation of the parabola and simplify.
- Double-check your work to avoid mistakes.
Q: Can you provide some examples of how to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, here are some examples of how to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola:
- Example 1: Find the { x $}$-intercept(s) and the coordinates of the vertex of the parabola { y = x^2 + 2x + 1 $}$.
- Example 2: Find the { x $}$-intercept(s) and the coordinates of the vertex of the parabola { y = -x^2 + 3x - 2 $}$.
- Example 3: Find the { x $}$-intercept(s) and the coordinates of the vertex of the parabola { y = x^2 - 4x + 4 $}$.
Q: Can you provide some real-world applications of finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, here are some real-world applications of finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola:
- Physics: The { x $}$-intercept(s) and the coordinates of the vertex of a parabola can be used to model the trajectory of a projectile.
- Engineering: The { x $}$-intercept(s) and the coordinates of the vertex of a parabola can be used to design the shape of a parabolic mirror.
- Economics: The { x $}$-intercept(s) and the coordinates of the vertex of a parabola can be used to model the demand and supply curves of a product.
Q: Can you provide some additional resources for learning about finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola?
A: Yes, here are some additional resources for learning about finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola:
- Online tutorials: There are many online tutorials available that provide step-by-step instructions for finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola.
- Textbooks: There are many textbooks available that provide detailed explanations and examples of how to find the { x $}$-intercept(s) and the coordinates of the vertex of a parabola.
- Online communities: There are many online communities available where you can ask questions and get help from others who are learning about finding the { x $}$-intercept(s) and the coordinates of the vertex of a parabola.