Identify The Vertex, The Axis Of Symmetry, The Maximum Or Minimum Value, And The Domain And Range Of The Function.$f(x)=-(x-7)^2-28$Identify The Vertex.The Coordinates Of The Vertex Are $\square$. (Type An Ordered Pair.)

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Introduction

Quadratic functions are a type of polynomial function that can be written in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. These functions have a parabolic shape and can be either concave up or concave down. In this article, we will focus on identifying the vertex, axis of symmetry, maximum or minimum value, and the domain and range of the function f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28.

Identifying the Vertex

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on whether the function is concave up or concave down. To identify the vertex of the function f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28, we need to rewrite the function in the form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h,k) is the vertex.

Step 1: Rewrite the function in the form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k

f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28

f(x)=−(x2−14x+49)−28f(x) = -(x^2 - 14x + 49) - 28

f(x)=−x2+14x−49−28f(x) = -x^2 + 14x - 49 - 28

f(x)=−x2+14x−77f(x) = -x^2 + 14x - 77

Step 2: Identify the vertex

Comparing the rewritten function with the form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, we can see that h=7h = 7 and k=−77k = -77. Therefore, the coordinates of the vertex are (7,−77)(7, -77).

Identifying the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. It is also the line of symmetry of the parabola. To identify the axis of symmetry, we need to find the value of xx that makes the function equal to zero.

Step 3: Find the value of xx that makes the function equal to zero

f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28

0=−(x−7)2−280 = -(x-7)^2 - 28

(x−7)2=−28(x-7)^2 = -28

x−7=±−28x-7 = \pm \sqrt{-28}

x=7±−28x = 7 \pm \sqrt{-28}

Since the square root of a negative number is not a real number, the function does not have a real axis of symmetry.

Identifying the Maximum or Minimum Value

The maximum or minimum value of a quadratic function is the value of the function at the vertex. Since the function is concave down, the vertex is the maximum point of the function.

Step 4: Find the maximum value of the function

f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28

f(7)=−(7−7)2−28f(7) = -(7-7)^2 - 28

f(7)=−28f(7) = -28

The maximum value of the function is −28-28.

Identifying the Domain and Range

The domain of a function is the set of all possible input values of the function. The range of a function is the set of all possible output values of the function.

Step 5: Find the domain of the function

The domain of the function is all real numbers, since there are no restrictions on the input values.

Step 6: Find the range of the function

The range of the function is all real numbers less than or equal to −28-28, since the function is concave down and the vertex is the maximum point of the function.

Conclusion

In this article, we identified the vertex, axis of symmetry, maximum or minimum value, and the domain and range of the function f(x)=−(x−7)2−28f(x) = -(x-7)^2 - 28. We found that the vertex is (7,−77)(7, -77), the axis of symmetry is a vertical line that passes through the vertex, the maximum value of the function is −28-28, and the domain and range of the function are all real numbers less than or equal to −28-28.

Vertex

  • The coordinates of the vertex are (7,−77)(7, -77).

Axis of Symmetry

  • The axis of symmetry is a vertical line that passes through the vertex.

Maximum or Minimum Value

  • The maximum value of the function is −28-28.

Domain

  • The domain of the function is all real numbers.

Range

  • The range of the function is all real numbers less than or equal to −28-28.
    Quadratic Function Q&A ==========================

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on whether the function is concave up or concave down.

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

A: To find the vertex of a quadratic function, you need to rewrite the function in the form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h,k) is the vertex. You can do this by completing the square or using the formula h=−b2ah = -\frac{b}{2a}.

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. It is also the line of symmetry of the parabola.

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 need to find the value of xx that makes the function equal to zero. This value is the xx-coordinate of the vertex.

Q: What is the maximum or minimum value of a quadratic function?

A: The maximum or minimum value of a quadratic function is the value of the function at the vertex. If the function is concave up, the vertex is the minimum point of the function. If the function is concave down, the vertex is the maximum point of the function.

Q: How do I find the maximum or minimum value of a quadratic function?

A: To find the maximum or minimum value of a quadratic function, you need to find the value of the function at the vertex. This can be done by substituting the xx-coordinate of the vertex into the function.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values of the function. Since quadratic functions are defined for all real numbers, the domain of a quadratic function is all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values of the function. The range of a quadratic function depends on whether the function is concave up or concave down.

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

A: To find the range of a quadratic function, you need to determine whether the function is concave up or concave down. If the function is concave up, the range is all real numbers greater than or equal to the minimum value. If the function is concave down, the range is all real numbers less than or equal to the maximum value.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. In this case, the function is concave down and the vertex is the maximum point of the function.

Q: Can a quadratic function have a zero leading coefficient?

A: No, a quadratic function cannot have a zero leading coefficient. If the leading coefficient is zero, the function is not quadratic.

Q: Can a quadratic function have a negative discriminant?

A: Yes, a quadratic function can have a negative discriminant. In this case, the function has no real roots and the graph does not intersect the x-axis.

Q: Can a quadratic function have a zero discriminant?

A: Yes, a quadratic function can have a zero discriminant. In this case, the function has one real root and the graph intersects the x-axis at a single point.

Conclusion

In this article, we answered some common questions about quadratic functions, including the vertex, axis of symmetry, maximum or minimum value, domain, and range. We also discussed some special cases, such as quadratic functions with negative leading coefficients, zero leading coefficients, negative discriminants, and zero discriminants.