What Is The Vertex Of The Graph Of The Function Below?${ Y = X^2 + 4x - 8 }$A. { (-2, 4)$}$B. { (2, 4)$}$C. { (2, 0)$}$D. { (-2, -12)$}$

Understanding the Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola. In this form, the vertex is the point on the graph where the parabola changes direction, and it is the minimum or maximum point of the function, depending on the value of $a$. If $a$ is positive, the vertex is the minimum point, and if $a$ is negative, the vertex is the maximum point.

Converting the Given Function to Vertex Form

The given function is $y = x^2 + 4x - 8$. To convert this function to vertex form, we need to complete the square. We start by factoring out the coefficient of $x^2$, which is 1 in this case.

y = x^2 + 4x - 8

Next, we add and subtract the square of half the coefficient of $x$ inside the parentheses. Half of 4 is 2, and the square of 2 is 4. We add 4 inside the parentheses and subtract 4 outside the parentheses to keep the equation balanced.

y = (x^2 + 4x + 4) - 8 - 4

Now, we can simplify the expression inside the parentheses by combining the like terms.

y = (x + 2)^2 - 12

Identifying the Vertex

Now that we have the function in vertex form, we can identify the vertex by looking at the expression inside the parentheses. The vertex is given by the point $(h, k)$, where $h$ is the value inside the parentheses and $k$ is the value outside the parentheses.

In this case, the vertex is $(h, k) = (-2, -12)$.

Conclusion

Therefore, the vertex of the graph of the function $y = x^2 + 4x - 8$ is $(h, k) = (-2, -12)$.

Answer

The correct answer is:

• D. $(-2, -12)$

Why is this the correct answer?

This is the correct answer because we converted the given function to vertex form and identified the vertex as $(-2, -12)$. This is the point on the graph where the parabola changes direction, and it is the minimum point of the function.

What is the significance of the vertex?

The vertex is the point on the graph where the parabola changes direction. It is the minimum or maximum point of the function, depending on the value of $a$. In this case, the vertex is the minimum point, and it is given by the coordinates $(-2, -12)$.

How to find the vertex of a quadratic function?

To find the vertex of a quadratic function, we need to convert the function to vertex form by completing the square. This involves factoring out the coefficient of $x^2$, adding and subtracting the square of half the coefficient of $x$ inside the parentheses, and simplifying the expression.

What is the vertex form of a quadratic function?

The vertex form of a quadratic function is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola.

Why is the vertex form useful?

The vertex form is useful because it allows us to identify the vertex of the parabola, which is the point where the parabola changes direction. It is also useful for graphing the parabola, as it gives us the coordinates of the vertex.

How to graph a quadratic function?

To graph a quadratic function, we need to identify the vertex and the direction of the parabola. We can use the vertex form to find the vertex and the direction of the parabola.

What is the significance of the vertex in real-world applications?

The vertex is significant in real-world applications because it represents the minimum or maximum point of a function. For example, in economics, the vertex represents the minimum or maximum price of a product. In physics, the vertex represents the minimum or maximum point of a trajectory.

Conclusion

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Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the parabola changes direction. It is the minimum or maximum point of the function, depending on the value of $a$.

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

A: To find the vertex of a quadratic function, you need to convert the function to vertex form by completing the square. This involves factoring out the coefficient of $x^2$, adding and subtracting the square of half the coefficient of $x$ inside the parentheses, and simplifying the expression.

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

A: The vertex form of a quadratic function is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola.

Q: Why is the vertex form useful?

A: The vertex form is useful because it allows us to identify the vertex of the parabola, which is the point where the parabola changes direction. It is also useful for graphing the parabola, as it gives us the coordinates of the vertex.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to identify the vertex and the direction of the parabola. You can use the vertex form to find the vertex and the direction of the parabola.

Q: What is the significance of the vertex in real-world applications?

A: The vertex is significant in real-world applications because it represents the minimum or maximum point of a function. For example, in economics, the vertex represents the minimum or maximum price of a product. In physics, the vertex represents the minimum or maximum point of a trajectory.

Q: Can I find the vertex of a quadratic function without converting it to vertex form?

A: Yes, you can find the vertex of a quadratic function without converting it to vertex form. You can use the formula $h = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then substitute this value into the function to find the y-coordinate.

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

A: The formula for finding the vertex of a quadratic function is $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

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

A: To use the formula to find the vertex of a quadratic function, you need to identify the coefficients $a$ and $b$ of the function. Then, you can plug these values into the formula to find the x-coordinate of the vertex. Finally, you can substitute this value into the function to find the y-coordinate of the vertex.

Q: Can I use the formula to find the vertex of a quadratic function with a negative leading coefficient?

A: Yes, you can use the formula to find the vertex of a quadratic function with a negative leading coefficient. The formula will still work, but the vertex will be the maximum point of the function instead of the minimum point.

Q: What is the difference between the vertex and the axis of symmetry?

A: The vertex and the axis of symmetry are related but distinct concepts. The vertex is the point on the graph where the parabola changes direction, while the axis of symmetry is the vertical line that passes through the vertex and is perpendicular to the x-axis.

Q: How do I find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, you need to identify the vertex of the function. Then, you can use the formula $x = h$ to find the x-coordinate of the axis of symmetry, where $h$ is the x-coordinate of the vertex.

Q: Can I use the formula to find the axis of symmetry of a quadratic function with a negative leading coefficient?

A: Yes, you can use the formula to find the axis of symmetry of a quadratic function with a negative leading coefficient. The formula will still work, but the axis of symmetry will be the vertical line that passes through the maximum point of the function instead of the minimum point.