Select The Correct Answer.The Quadratic Function $f$ Has A Vertex At \[$(3, 4)\$\] And Opens Upward. The Quadratic Function $g$ Is Shown Below.$g(x) = 2(x-4)^2 + 3$Which Statement Is True?A. The Minimum Value Of

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will explore the properties of quadratic functions, specifically the vertex form and the behavior of quadratic functions that open upward or downward.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

$$f(x) = a(x-h)^2 + k$$

where $(h,k)$ is the vertex of the parabola. The vertex form is useful for identifying the vertex and the direction of the parabola.

The Quadratic Function $f$

The quadratic function $f$ has a vertex at $(3,4)$ and opens upward. This means that the parabola has a minimum value at the vertex, and the function increases as we move away from the vertex.

The Quadratic Function $g$

The quadratic function $g$ is given by:

$$g(x) = 2(x-4)^2 + 3$$

This function has a vertex at $(4,3)$ and opens upward. Since the coefficient of the squared term is positive, the parabola opens upward, and the function increases as we move away from the vertex.

Comparing the Two Quadratic Functions

Now, let's compare the two quadratic functions $f$ and $g$. Both functions have a vertex, but the vertices are located at different points. The function $f$ has a vertex at $(3,4)$, while the function $g$ has a vertex at $(4,3)$.

The Minimum Value of $f$

The minimum value of a quadratic function that opens upward is the value of the function at the vertex. Since the vertex of the function $f$ is at $(3,4)$, the minimum value of $f$ is $4$.

The Minimum Value of $g$

The minimum value of the quadratic function $g$ is the value of the function at the vertex. Since the vertex of the function $g$ is at $(4,3)$, the minimum value of $g$ is $3$.

Which Statement is True?

Now, let's consider the statements given in the problem:

A. The minimum value of $f$ is greater than the minimum value of $g$.

B. The minimum value of $f$ is less than the minimum value of $g$.

C. The minimum value of $f$ is equal to the minimum value of $g$.

D. The minimum value of $f$ is not equal to the minimum value of $g$.

Based on our analysis, we can conclude that:

• The minimum value of $f$ is $4$, and the minimum value of $g$ is $3$.
• Therefore, the minimum value of $f$ is greater than the minimum value of $g$.

Conclusion

In conclusion, the correct statement is:

A. The minimum value of $f$ is greater than the minimum value of $g$.

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In this article, we will answer some frequently asked questions about quadratic functions, including their properties, vertex form, and behavior.

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

A: The vertex form of a quadratic function is given by:

$$f(x) = a(x-h)^2 + k$$

where $(h,k)$ is the vertex of the parabola.

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

A: The vertex form is useful for identifying the vertex and the direction of the parabola. It also helps in determining the minimum or maximum value of the function.

Q: What is the difference between a quadratic function that opens upward and one that opens downward?

A: A quadratic function that opens upward has a positive coefficient of the squared term, while a quadratic function that opens downward has a negative coefficient of the squared term.

Q: How do you determine the minimum or maximum value of a quadratic function?

A: To determine the minimum or maximum value of a quadratic function, you need to find the value of the function at the vertex. If the function opens upward, the minimum value is at the vertex, and if the function opens downward, the maximum value is at the vertex.

Q: What is the relationship between the vertex and the minimum or maximum value of a quadratic function?

A: The vertex of a quadratic function is the point where the function has its minimum or maximum value. The value of the function at the vertex is the minimum or maximum value of the function.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can have only one vertex. The vertex form of a quadratic function is unique, and it represents the point where the function has its minimum or maximum value.

Q: How do you graph a quadratic function?

A: To graph a quadratic function, you need to plot the vertex and then use the symmetry of the parabola to plot the other points. You can also use the x-intercepts and the y-intercept to graph the function.

Q: What is the significance of the x-intercepts and the y-intercept of a quadratic function?

A: The x-intercepts of a quadratic function are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis. These points are useful in graphing the function and determining its behavior.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the function has a negative leading coefficient and the vertex is above the x-axis.

Q: Can a quadratic function have no y-intercept?

A: No, a quadratic function always has a y-intercept. The y-intercept is the point where the function crosses the y-axis, and it is always present in a quadratic function.

Q: What is the relationship between the quadratic function and its inverse?

A: The inverse of a quadratic function is a function that undoes the action of the original function. The inverse of a quadratic function is also a quadratic function, and it has the same vertex as the original function.

Q: Can a quadratic function have an inverse that is not a quadratic function?

A: No, a quadratic function can only have an inverse that is also a quadratic function. The inverse of a quadratic function is always a quadratic function, and it has the same vertex as the original function.