Question

determine the axis of symmetry for the function \\(f(x) = -4(x + 7)^2 - 3\\).

• \\(x = 3\\)
• \\(x = -3\\)
• \\(x = 7\\)
• \\(x = -7\\)

question 5 (multiple choice worth 2 points)
(02.04r)

for the function \\(f(x) = -(x + 1)^2 + 4\\), identify the vertex, domain, and range.

• the vertex is \\((-1, 4)\\), the domain is all real numbers, and the range is \\(y \ge 4\\).
• the vertex is \\((-1, 4)\\), the domain is all real numbers, and the range is \\(y \le 4\\).
• the vertex is \\((1, 4)\\), the domain is all real numbers, and the range is \\(y \ge 4\\).
• the vertex is \\((1, 4)\\), the domain is all real numbers, and the range is \\(y \le 4\\).

Explanation:

Identify the axis of symmetry for the first function

The first function is given in Vertex Form of Quadratic:

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

For \(f(x) = -4(x + 7)^2 - 3\), we rewrite the term inside the parentheses to match the standard form:

$$f(x) = -4(x - (-7))^2 - 3$$

Thus, the vertex \((h, k)\) is \((-7, -3)\).
The axis of symmetry is the vertical line passing through the x-coordinate of the vertex:

$$x = -7$$

Identify the vertex of the second function

The second function is:

$$f(x) = -(x + 1)^2 + 4$$

Comparing this to the standard Vertex Form of Quadratic \(f(x) = a(x - h)^2 + k\):

$$h = -1, \quad k = 4$$

So, the vertex is \((-1, 4)\).

Determine the domain of the second function

Since \(f(x)\) is a quadratic polynomial, it is defined for all real inputs.
Therefore, the domain is all real numbers.

Determine the range of the second function

The leading coefficient is \(a = -1\).
Since \(a < 0\), the parabola opens downward, meaning the vertex represents the maximum value of the function.
The maximum value of the function is the y-coordinate of the vertex, which is \(4\).
Therefore, the range is:

$$y \le 4$$

Combining these results, the correct option is: "The vertex is \((-1, 4)\), the domain is all real numbers, and the range is \(y \le 4\)".

Answer:

Question 1

• (A) \(x = 3\)
• (B) \(x = -3\)
• (C) \(x = 7\)
• (D) \(x = -7\) (Correct answer)

Question 2

• (A) The vertex is \((-1, 4)\), the domain is all real numbers, and the range is \(y \ge 4\).
• (B) The vertex is \((-1, 4)\), the domain is all real numbers, and the range is \(y \le 4\). (Correct answer)
• (C) The vertex is \((1, 4)\), the domain is all real numbers, and the range is \(y \ge 4\).
• (D) The vertex is \((1, 4)\), the domain is all real numbers, and the range is \(y \le 4\).