What is the relative maximum value of x^2-2x-8?

The equation x^2-2x-8 represents a quadratic function, which is a type of polynomial function where the highest power of the variable is 2. In this case, the variable is x, and the equation represents a parabola. The relative maximum value refers to the highest point on the parabola.

To determine the relative maximum value, we need to find the vertex of the parabola. The vertex represents the point where the parabola reaches its peak or the highest point. The x-coordinate of the vertex can be found using the formula: x = -b/2a, where a and b are the coefficients of the quadratic equation.

In the given equation x^2-2x-8, we can see that a = 1 and b = -2. Plugging these values into the formula, we get x = -(-2)/2(1) = 2/2 = 1. This means that the x-coordinate of the vertex is 1.

To find the y-coordinate of the vertex, we substitute the x-coordinate back into the original equation. Plugging in x = 1, we get y = (1)^2-2(1)-8 = 1-2-8 = -9. Therefore, the y-coordinate of the vertex is -9.

What is the relative maximum value of x^2-2x-8?

The relative maximum value of the function x^2-2x-8 is -9.

FAQs:

1. What does the term “relative maximum” mean?

The relative maximum refers to the highest point on a graph or a function. It is different from an absolute maximum, which is the highest point in the entire function or domain.

2. How do you find the vertex of a quadratic function?

The vertex of a quadratic function can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation.

3. Can a quadratic function have multiple relative maximum values?

No, a quadratic function can have only one relative maximum value. The shape of a parabola determines the single highest point.

4. How do you distinguish between a relative maximum and a relative minimum?

A relative maximum is the highest point on a graph, while a relative minimum is the lowest point. Relative maximums are typically found at the peak of an upward-opening parabola, while relative minimums are found at the bottom of a downward-opening parabola.

5. Is it possible for a parabola to have no relative maximum or minimum?

Yes, it is possible for a parabola to have no relative maximum or minimum if the arms of the parabola extend infinitely.

6. Can the vertex of a quadratic function be a relative minimum?

Yes, the vertex of a quadratic function can be a relative minimum if the parabola opens downward, and the vertex is the lowest point on the graph.

7. How does changing the coefficient a affect the relative maximum?

The coefficient a determines the shape of the parabola. If |a| < 1, the parabola is narrower, resulting in a higher relative maximum. If |a| > 1, the parabola is wider, resulting in a lower relative maximum.

8. What is the significance of the relative maximum in real-world applications?

In real-world applications, the relative maximum represents the maximum point or value of a quantity. It could be used to determine the maximum profit, highest elevation, or peak performance of a variable being studied.

9. Can the relative maximum be at the origin (0,0)?

No, the relative maximum cannot be at the origin (0,0) since the origin is a point of symmetry for the graph of a quadratic function.

10. How does the coefficient c affect the location of the relative maximum?

The coefficient c is a constant term in the quadratic equation and does not affect the location of the relative maximum. It only shifts the entire parabola up or down without altering its shape.

11. Can the relative maximum value of a quadratic function be infinity?

No, the relative maximum value of a quadratic function cannot be infinity, as infinity is not a real number.

12. Is there a way to determine the relative maximum without finding the vertex?

No, finding the vertex is the most efficient way to determine the relative maximum of a quadratic function. Other methods involve analyzing the sign of the coefficient a or examining the shape of the parabola, but these techniques may be more complex and time-consuming.

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