Consider The Function F ( X ) = − 2 X 2 + 20 X − 7 F(x) = -2x^2 + 20x - 7 F ( X ) = − 2 X 2 + 20 X − 7 .a. Determine, Without Graphing, Whether The Function Has A Minimum Value Or A Maximum Value.b. Find The Minimum Or Maximum Value And Determine Where It Occurs.c. Identify The Function's Domain And

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Analyzing the Function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7

Introduction

In this article, we will delve into the analysis of the given quadratic function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7. We will determine whether the function has a minimum value or a maximum value, find the minimum or maximum value and its corresponding location, and identify the function's domain.

Part a: Determining the Nature of the Function

To determine whether the function has a minimum value or a maximum value, we need to examine the coefficient of the x2x^2 term. In the given function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, the coefficient of the x2x^2 term is -2, which is negative. This indicates that the function is a downward-facing parabola, and therefore, it has a maximum value.

Part b: Finding the Maximum Value and Its Location

To find the maximum value and its location, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x=−b2ax = -\frac{b}{2a}, where aa and bb are the coefficients of the x2x^2 and xx terms, respectively. In this case, a=−2a = -2 and b=20b = 20. Plugging these values into the formula, we get:

x=−202(−2)x = -\frac{20}{2(-2)} x=−20−4x = -\frac{20}{-4} x=5x = 5

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the value of xx into the function:

f(5)=−2(5)2+20(5)−7f(5) = -2(5)^2 + 20(5) - 7 f(5)=−2(25)+100−7f(5) = -2(25) + 100 - 7 f(5)=−50+100−7f(5) = -50 + 100 - 7 f(5)=43f(5) = 43

Therefore, the maximum value of the function is 43, and it occurs at x=5x = 5.

Part c: Identifying the Function's Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of the quadratic function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, the function is defined for all real numbers. Therefore, the domain of the function is:

(−∞,∞)(-\infty, \infty)

Conclusion

In conclusion, the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 has a maximum value, which is 43, and it occurs at x=5x = 5. The function's domain is all real numbers.

Additional Analysis

To further analyze the function, we can examine its behavior as xx approaches positive and negative infinity. As xx approaches positive infinity, the function approaches negative infinity, and as xx approaches negative infinity, the function also approaches negative infinity. This indicates that the function has a vertical asymptote at x=5x = 5, which is the location of the maximum value.

Graphical Representation

The graph of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 is a downward-facing parabola with a maximum value at x=5x = 5. The graph is shown below:

Graph of the Function

The graph of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 is a downward-facing parabola with a maximum value at x=5x = 5. The graph is shown above.

Final Thoughts

In conclusion, the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 has a maximum value, which is 43, and it occurs at x=5x = 5. The function's domain is all real numbers. The graph of the function is a downward-facing parabola with a maximum value at x=5x = 5.
Q&A: Analyzing the Function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7

Introduction

In our previous article, we analyzed the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 and determined that it has a maximum value, which is 43, and it occurs at x=5x = 5. We also identified the function's domain as all real numbers. In this article, we will answer some frequently asked questions about the function.

Q1: What is the nature of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A1: The function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 is a quadratic function, and it has a downward-facing parabola. This means that the function has a maximum value, which is 43, and it occurs at x=5x = 5.

Q2: How do I find the maximum value of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A2: To find the maximum value of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, you need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x=−b2ax = -\frac{b}{2a}, where aa and bb are the coefficients of the x2x^2 and xx terms, respectively. In this case, a=−2a = -2 and b=20b = 20. Plugging these values into the formula, you get x=5x = 5. Now that you have the x-coordinate of the vertex, you can find the y-coordinate by plugging the value of xx into the function.

Q3: What is the domain of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A3: The domain of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 is all real numbers. This means that the function is defined for all possible input values.

Q4: How do I graph the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A4: To graph the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graph by hand. The graph of the function is a downward-facing parabola with a maximum value at x=5x = 5.

Q5: What is the significance of the vertex of the parabola?

A5: The vertex of the parabola is the point at which the function has a maximum or minimum value. In the case of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, the vertex is the point at which the function has a maximum value of 43.

Q6: How do I find the x-intercepts of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A6: To find the x-intercepts of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, you need to set the function equal to zero and solve for xx. This will give you the x-coordinates of the x-intercepts.

Q7: What is the relationship between the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 and the function f(x)=2x2−20x+7f(x) = 2x^2 - 20x + 7?

A7: The function f(x)=2x2−20x+7f(x) = 2x^2 - 20x + 7 is the reflection of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 across the x-axis. This means that the two functions are mirror images of each other.

Q8: How do I find the equation of the axis of symmetry of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A8: To find the equation of the axis of symmetry of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7, you need to find the x-coordinate of the vertex. The equation of the axis of symmetry is a vertical line that passes through the x-coordinate of the vertex.

Q9: What is the significance of the axis of symmetry of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7?

A9: The axis of symmetry of the function f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 is a vertical line that passes through the x-coordinate of the vertex. This means that the function is symmetric about this line.

Q10: How do I find the equation of the parabola f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 in vertex form?

A10: To find the equation of the parabola f(x)=−2x2+20x−7f(x) = -2x^2 + 20x - 7 in vertex form, you need to write the equation in the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In this case, the vertex is (5,43)(5, 43), so the equation of the parabola in vertex form is f(x)=−2(x−5)2+43f(x) = -2(x - 5)^2 + 43.