Question

question 9 (1 point)
find the minimum or maximum value of the function. describe the domain and range of the function, and where the function is increasing and decreasing.
$f(x) = -x^2 + 2$

a the minimum value is 2. the domain is all real numbers and the range is $y \geq 2$. the function is decreasing to the left of $x = 0$ and increasing to the right of $x = 0$.
b the minimum value is $-2$. the domain is all real numbers and the range is $y \geq -2$. the function is decreasing to the left of $x = 0$ and increasing to the right of $x = 0$.
c the maximum value is $-2$. the domain is all real numbers and the range is $y \leq -2$. the function is increasing to the left of $x = 0$ and decreasing to the right of $x = 0$.
d the maximum value is 2. the domain is all real numbers and the range is $y \leq 2$. the function is increasing to the left of $x = 0$ and decreasing to the right of $x = 0$.

Explanation:

Step1: Analyze the quadratic function

The function is \( f(x) = -x^2 + 2 \), which is a quadratic function in the form \( f(x)=ax^2 + bx + c \) with \( a=-1 \), \( b = 0 \), \( c = 2 \). Since \( a=-1<0 \), the parabola opens downward, so the function has a maximum value at the vertex.

Step2: Find the vertex and maximum value

The x - coordinate of the vertex of a quadratic function \( f(x)=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Here, \( b = 0 \) and \( a=-1 \), so \( x =-\frac{0}{2\times(-1)}=0 \). Substitute \( x = 0 \) into the function: \( f(0)=-(0)^2 + 2=2 \). So the maximum value of the function is 2.

Step3: Determine the domain and range

For any quadratic function, the domain is all real numbers because we can plug in any real number for \( x \). Since the parabola opens downward and the maximum value of \( y \) is 2, the range is all real numbers \( y \) such that \( y\leq2 \).

Step4: Analyze the increasing and decreasing intervals

For a parabola that opens downward (\( a < 0 \)), the function is increasing to the left of the vertex (\( x<0 \)) and decreasing to the right of the vertex (\( x > 0 \)).

Answer:

d. The maximum value is 2. The domain is all real numbers and the range is \( y\leq2 \). The function is increasing to the left of \( x = 0 \) and decreasing to the right of \( x = 0 \).