The Function F F F Is Given By F ( X ) = ( X + 3 ) 2 − 1 F(x)=(x+3)^2-1 F ( X ) = ( X + 3 ) 2 − 1 .1. Write The Coordinates Of The Vertex Of The Graph Of F F F .

Introduction

In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. One of the most important properties of a quadratic equation is its graph, which is a parabola. In this article, we will discuss how to find the vertex of the graph of a quadratic equation, specifically the function $f(x) = (x+3)^2 - 1$.

Understanding the Function

The given function is $f(x) = (x+3)^2 - 1$. This is a quadratic function in the form of $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this case, the vertex is $(h,k) = (-3, -1)$.

Finding the Vertex

To find the vertex of the parabola, we need to rewrite the function in the form $f(x) = a(x-h)^2 + k$. We can do this by expanding the squared term:

$f(x) = (x+3)^2 - 1$ $f(x) = x^2 + 6x + 9 - 1$ $f(x) = x^2 + 6x + 8$

Now, we can see that the vertex is at the point where the squared term is equal to zero. In other words, we need to find the value of $x$ that makes the squared term equal to zero. We can do this by setting the squared term equal to zero and solving for $x$:

$x^2 + 6x + 8 = 0$

Unfortunately, this equation does not factor easily, so we will need to use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = 6$, and $c = 8$. Plugging these values into the quadratic formula, we get:

$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(8)}}{2(1)}$ $x = \frac{-6 \pm \sqrt{36 - 32}}{2}$ $x = \frac{-6 \pm \sqrt{4}}{2}$ $x = \frac{-6 \pm 2}{2}$

Simplifying, we get two possible values for $x$:

$x = \frac{-6 + 2}{2} = -2$ $x = \frac{-6 - 2}{2} = -4$

Now that we have found the values of $x$, we can plug them back into the original function to find the corresponding values of $y$:

$f(-2) = (-2+3)^2 - 1 = 1 - 1 = 0$ $f(-4) = (-4+3)^2 - 1 = 1 - 1 = 0$

However, we are looking for the vertex of the parabola, not the x-intercepts. To find the vertex, we need to find the value of $y$ that corresponds to the value of $x$ that makes the squared term equal to zero. In other words, we need to find the value of $y$ that corresponds to $x = -3$.

$f(-3) = (-3+3)^2 - 1 = 0 - 1 = -1$

Therefore, the coordinates of the vertex of the graph of $f$ are $(-3, -1)$.

Conclusion

In this article, we discussed how to find the vertex of the graph of a quadratic equation, specifically the function $f(x) = (x+3)^2 - 1$. We showed that the vertex is at the point where the squared term is equal to zero, and we used the quadratic formula to find the value of $x$ that makes the squared term equal to zero. We then plugged this value of $x$ back into the original function to find the corresponding value of $y$. Finally, we concluded that the coordinates of the vertex of the graph of $f$ are $(-3, -1)$.

Example Problems

1. Find the vertex of the graph of the function $f(x) = (x-2)^2 + 4$.
2. Find the vertex of the graph of the function $f(x) = (x+1)^2 - 3$.
3. Find the vertex of the graph of the function $f(x) = (x-4)^2 + 2$.

Solutions

1. The vertex of the graph of the function $f(x) = (x-2)^2 + 4$ is at the point where the squared term is equal to zero. In other words, we need to find the value of $x$ that makes the squared term equal to zero. We can do this by setting the squared term equal to zero and solving for $x$:

$(x-2)^2 = 0$

Simplifying, we get:

$x-2 = 0$ $x = 2$

Now that we have found the value of $x$, we can plug it back into the original function to find the corresponding value of $y$:

$f(2) = (2-2)^2 + 4 = 0 + 4 = 4$

Therefore, the coordinates of the vertex of the graph of $f$ are $(2, 4)$.

2. The vertex of the graph of the function $f(x) = (x+1)^2 - 3$ is at the point where the squared term is equal to zero. In other words, we need to find the value of $x$ that makes the squared term equal to zero. We can do this by setting the squared term equal to zero and solving for $x$:

$(x+1)^2 = 0$

Simplifying, we get:

$x+1 = 0$ $x = -1$

Now that we have found the value of $x$, we can plug it back into the original function to find the corresponding value of $y$:

$f(-1) = (-1+1)^2 - 3 = 0 - 3 = -3$

Therefore, the coordinates of the vertex of the graph of $f$ are $(-1, -3)$.

3. The vertex of the graph of the function $f(x) = (x-4)^2 + 2$ is at the point where the squared term is equal to zero. In other words, we need to find the value of $x$ that makes the squared term equal to zero. We can do this by setting the squared term equal to zero and solving for $x$:

$(x-4)^2 = 0$

Simplifying, we get:

$x-4 = 0$ $x = 4$

Now that we have found the value of $x$, we can plug it back into the original function to find the corresponding value of $y$:

$f(4) = (4-4)^2 + 2 = 0 + 2 = 2$

Therefore, the coordinates of the vertex of the graph of $f$ are $(4, 2)$.

Final Thoughts

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Q&A: Finding the Vertex of a Quadratic Equation

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph of the equation where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on whether the parabola opens upward or downward.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to rewrite the equation in the form $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. You can do this by expanding the squared term and then completing the square.

Q: What is the formula for finding the vertex of a quadratic equation?

A: The formula for finding the vertex of a quadratic equation is:

$h = -\frac{b}{2a}$

This formula gives you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you need to plug the value of $h$ back into the original equation.

Q: How do I use the quadratic formula to find the vertex of a quadratic equation?

A: To use the quadratic formula to find the vertex of a quadratic equation, you need to first rewrite the equation in the form $ax^2 + bx + c = 0$. Then, you can plug the values of $a$, $b$, and $c$ into the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula gives you the x-coordinates of the vertex. To find the y-coordinates of the vertex, you need to plug the values of $x$ back into the original equation.

Q: What is the difference between the x-intercepts and the vertex of a quadratic equation?

A: The x-intercepts of a quadratic equation are the points on the graph of the equation where the parabola crosses the x-axis. The vertex of a quadratic equation is the point on the graph of the equation where the parabola changes direction.

Q: Can I find the vertex of a quadratic equation without using the quadratic formula?

A: Yes, you can find the vertex of a quadratic equation without using the quadratic formula. You can rewrite the equation in the form $f(x) = a(x-h)^2 + k$ and then find the values of $h$ and $k$ by expanding the squared term and completing the square.

Q: How do I know if a quadratic equation has a minimum or maximum vertex?

A: To determine if a quadratic equation has a minimum or maximum vertex, you need to look at the coefficient of the squared term. If the coefficient is positive, the vertex is a minimum point. If the coefficient is negative, the vertex is a maximum point.

Q: Can I use the vertex form of a quadratic equation to find the x-intercepts?

A: Yes, you can use the vertex form of a quadratic equation to find the x-intercepts. To do this, you need to set the squared term equal to zero and solve for $x$.

Q: How do I graph a quadratic equation using the vertex form?

A: To graph a quadratic equation using the vertex form, you need to first find the vertex of the equation. Then, you can use the vertex as the center of the parabola and draw the parabola using the equation.

Q: Can I use a calculator to find the vertex of a quadratic equation?

A: Yes, you can use a calculator to find the vertex of a quadratic equation. Most graphing calculators have a built-in function for finding the vertex of a quadratic equation.

Q: How do I use a calculator to find the vertex of a quadratic equation?

A: To use a calculator to find the vertex of a quadratic equation, you need to first enter the equation into the calculator. Then, you can use the calculator's built-in function to find the vertex of the equation.

Conclusion

In this article, we discussed how to find the vertex of a quadratic equation. We covered the basics of quadratic equations, including the vertex form and the quadratic formula. We also answered some common questions about finding the vertex of a quadratic equation. Whether you're a student or a teacher, this article should give you a good understanding of how to find the vertex of a quadratic equation.