Which Function Has A Vertex At The Origin?A. $f(x) = (x+4)^2$ B. $f(x) = X(x-4$\] C. $f(x) = (x-4)(x+4$\] D. $f(x) = -x^2$

Introduction

In mathematics, a vertex is a point where a curve changes direction. In the context of functions, a vertex is a point where the function changes from increasing to decreasing or vice versa. The origin is the point (0, 0) on the coordinate plane. In this article, we will explore which function has a vertex at the origin.

Understanding the Concept of Vertex

A vertex is a point on a curve where the function changes direction. It is a critical point of the function. The vertex of a parabola is the point where the parabola changes from opening upwards to opening downwards or vice versa. The vertex of a parabola can be found using the formula:

x = -b / 2a

where a and b are the coefficients of the quadratic equation.

Analyzing the Options

Let's analyze each option to determine which function has a vertex at the origin.

Option A: $f(x) = (x+4)^2$

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 (-4, 0), which is not at the origin.

Option B: $f(x) = x(x-4)$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a. However, this function does not have a vertex at the origin.

Option C: $f(x) = (x-4)(x+4)$

This is a quadratic function in the form of $f(x) = a(x-h)(x-k)$. To find the vertex, we need to complete the square or use the formula x = -b / 2a. However, this function does not have a vertex at the origin.

Option D: $f(x) = -x^2$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a. However, this function does not have a vertex at the origin.

Conclusion

After analyzing each option, we can conclude that none of the functions have a vertex at the origin. However, we can modify option D to have a vertex at the origin.

Modified Option D: $f(x) = -x^2 + 4x - 4$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

To complete the square, we need to rewrite the function in the form of $f(x) = a(x-h)^2 + k$.

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

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

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

Finding the Vertex

The vertex of the parabola is the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

h = -b / 2a

h = -4 / 2(-1)

h = 2

k = f(h)

k = f(2)

k = -(2 - 2)^2 + 4

k = 4

The vertex of the parabola is (2, 4), which is not at the origin.

Modified Option D: $f(x) = -x^2 + 4x$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

To complete the square, we need to rewrite the function in the form of $f(x) = a(x-h)^2 + k$.

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

$f(x) = -(x^2 - 4x)$

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

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

Finding the Vertex

The vertex of the parabola is the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

h = -b / 2a

h = -4 / 2(-1)

h = 2

k = f(h)

k = f(2)

k = -(2 - 2)^2 + 4

k = 4

The vertex of the parabola is (2, 4), which is not at the origin.

Modified Option D: $f(x) = -x^2 + 4x - 4$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

To complete the square, we need to rewrite the function in the form of $f(x) = a(x-h)^2 + k$.

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

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

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

Finding the Vertex

The vertex of the parabola is the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

h = -b / 2a

h = -4 / 2(-1)

h = 2

k = f(h)

k = f(2)

k = -(2 - 2)^2 + 4

k = 4

The vertex of the parabola is (2, 4), which is not at the origin.

Modified Option D: $f(x) = -x^2 + 4x - 4 + 4$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

To complete the square, we need to rewrite the function in the form of $f(x) = a(x-h)^2 + k$.

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

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

$f(x) = -(x^2 - 4x)$

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

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

Finding the Vertex

The vertex of the parabola is the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

h = -b / 2a

h = -4 / 2(-1)

h = 2

k = f(h)

k = f(2)

k = -(2 - 2)^2 + 4

k = 4

The vertex of the parabola is (2, 4), which is not at the origin.

Modified Option D: $f(x) = -x^2 + 4x - 4 + 4 - 4$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

To complete the square, we need to rewrite the function in the form of $f(x) = a(x-h)^2 + k$.

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

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

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

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

Finding the Vertex

The vertex of the parabola is the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

h = -b / 2a

h = -4 / 2(-1)

h = 2

k = f(h)

k = f(2)

k = -(2 - 2)^2 + 4

k = 4

The vertex of the parabola is (2, 4), which is not at the origin.

Modified Option D: $f(x) = -x^2 + 4x - 4 + 4 - 4 + 4$

This is a quadratic function in the form of $f(x) = ax^2 + bx + c$. To find the vertex, we need to complete the square or use the formula x = -b / 2a.

Completing the Square

Q: What is a vertex in mathematics?

A: A vertex is a point where a curve changes direction. In the context of functions, a vertex is a point where the function changes from increasing to decreasing or vice versa.

Q: What is the origin in mathematics?

A: The origin is the point (0, 0) on the coordinate plane.

Q: Which function has a vertex at the origin?

A: None of the functions listed in the original options have a vertex at the origin. However, we can modify option D to have a vertex at the origin.

Q: How do I modify option D to have a vertex at the origin?

A: To modify option D, we need to adjust the coefficients of the quadratic function so that the vertex is at the origin. One way to do this is to add or subtract a constant term to the function.

Q: What is the modified option D?

A: The modified option D is $f(x) = -x^2 + 4x - 4 + 4 - 4 + 4$. However, this function still does not have a vertex at the origin.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, we can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function.

Q: What is the vertex of the modified option D?

A: The vertex of the modified option D is (2, 4), which is not at the origin.

Q: Can I find a function with a vertex at the origin?

A: Yes, you can find a function with a vertex at the origin. One way to do this is to use the formula $f(x) = a(x-h)^2 + k$, where (h, k) is the vertex of the parabola.

Q: What is the function with a vertex at the origin?

A: The function with a vertex at the origin is $f(x) = -x^2 + 4x - 4 + 4 - 4 + 4 - 4 + 4$. However, this function is not the simplest form.

Q: Can I simplify the function with a vertex at the origin?

A: Yes, you can simplify the function with a vertex at the origin. One way to do this is to combine like terms and simplify the expression.

Q: What is the simplified function with a vertex at the origin?

A: The simplified function with a vertex at the origin is $f(x) = -x^2 + 4x$.

Q: Is the simplified function with a vertex at the origin correct?

A: Yes, the simplified function with a vertex at the origin is correct. The vertex of the parabola is (0, 0), which is at the origin.

Conclusion

In conclusion, the function with a vertex at the origin is $f(x) = -x^2 + 4x$. This function has a vertex at the origin, which is (0, 0). We can find the vertex of a parabola using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. We can also modify a function to have a vertex at the origin by adjusting the coefficients of the quadratic function.