Gerald Graphs The Function F ( X ) = ( X − 3 ) 2 − 1 F(x)=(x-3)^2-1 F ( X ) = ( X − 3 ) 2 − 1 . Which Statements Are True About The Graph? Select Three Options.A. The Domain Is { X ∣ X ≥ 3 } \{x \mid X \geq 3\} { X ∣ X ≥ 3 } .B. The Range Is { Y ∣ Y ≥ − 1 } \{y \mid Y \geq -1\} { Y ∣ Y ≥ − 1 } .C. The Function Decreases Over The

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will focus on the graph of the function f(x) = (x - 3)^2 - 1, and we will determine which statements are true about the graph.

The Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function produces a real number as output. For the function f(x) = (x - 3)^2 - 1, we can see that there are no restrictions on the values of x, as long as x is a real number. Therefore, the domain of the function is the set of all real numbers, which can be represented as (-∞, ∞).

The Range of a Function

The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible y-values for which the function produces a real number as output. For the function f(x) = (x - 3)^2 - 1, we can see that the minimum value of the function is -1, which occurs when x = 3. As x increases or decreases from 3, the value of the function increases without bound. Therefore, the range of the function is the set of all real numbers greater than or equal to -1, which can be represented as [-1, ∞).

The Behavior of the Function

To determine the behavior of the function, we need to examine its derivative. The derivative of a function represents the rate of change of the function with respect to the input variable. For the function f(x) = (x - 3)^2 - 1, the derivative is f'(x) = 2(x - 3). We can see that the derivative is zero when x = 3, which means that the function has a local minimum at x = 3. As x increases or decreases from 3, the derivative is positive, which means that the function is increasing. Therefore, the function does not decrease over any interval.

Conclusion

In conclusion, the statements about the graph of the function f(x) = (x - 3)^2 - 1 are:

• The domain of the function is the set of all real numbers, which can be represented as (-∞, ∞).
• The range of the function is the set of all real numbers greater than or equal to -1, which can be represented as [-1, ∞).
• The function does not decrease over any interval.

In the previous article, we discussed the graph of the function f(x) = (x - 3)^2 - 1 and determined which statements are true about the graph. In this article, we will answer some frequently asked questions about the graph of a quadratic function.

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

A: The vertex of the graph of a quadratic function is the point at which the function has a local maximum or minimum. For the function f(x) = (x - 3)^2 - 1, the vertex is the point (3, -1).

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

A: To find the vertex of a quadratic function, you can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. For the function f(x) = (x - 3)^2 - 1, the vertex is at x = 3.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. For the function f(x) = (x - 3)^2 - 1, the axis of symmetry is the line x = 3.

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 can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. For the function f(x) = (x - 3)^2 - 1, the axis of symmetry is the line x = 3.

Q: What is the difference between a local maximum and a local minimum?

A: A local maximum is a point at which the function has a maximum value, while a local minimum is a point at which the function has a minimum value. For the function f(x) = (x - 3)^2 - 1, the point (3, -1) is a local minimum.

Q: How do I determine whether a point is a local maximum or a local minimum?

A: To determine whether a point is a local maximum or a local minimum, you can examine the behavior of the function near the point. If the function is increasing on one side of the point and decreasing on the other side, then the point is a local maximum. If the function is decreasing on one side of the point and increasing on the other side, then the point is a local minimum.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. For example, the function f(x) = (x - 3)^2 - 1 is a quadratic function, while the function f(x) = 2x + 1 is a linear function.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot points on the graph. For example, to graph the function f(x) = (x - 3)^2 - 1, you can use a table of values to plot points such as (2, -3), (3, -1), and (4, 1).

Conclusion

In conclusion, we have answered some frequently asked questions about the graph of a quadratic function. We have discussed the vertex, axis of symmetry, local maximum and minimum, and difference between a quadratic function and a linear function. We have also provided examples of how to graph a quadratic function.