Question

which statements about the graph of the function $f(x) = -x^2 - 4x + 2$ are true? select three options.

• the domain is ${x|x \leq -2}$.
• the range is ${y|y \leq 6}$.
• the function is increasing over the interval $(-\infty, -2)$.
• the function is decreasing over the interval $(-4, \infty)$.
• the function has a positive $y$-intercept.

Explanation:

Step1: Analyze the function type

The function \( f(x) = -x^2 - 4x + 2 \) is a quadratic function in the form \( f(x)=ax^2+bx+c \) with \( a = - 1\), \( b=-4 \), \( c = 2\). Since \( a=-1<0 \), the parabola opens downwards.

Step2: Find the vertex of the parabola

The x - coordinate of the vertex of a quadratic function \( f(x)=ax^2+bx+c \) is given by \( x=-\frac{b}{2a} \). Substituting \( a=-1 \) and \( b = - 4\) into the formula, we get \( x=-\frac{-4}{2\times(-1)}=-\frac{-4}{-2}=-2 \).
To find the y - coordinate of the vertex, substitute \( x = - 2\) into the function:
\( f(-2)=-(-2)^2-4\times(-2)+2=-4 + 8+2=6 \). So the vertex is at \( (-2,6) \).

Step3: Analyze the domain

For any quadratic function, the domain is all real numbers, because we can plug in any real number for \( x \) and get a real - valued output. So the statement "The domain is \( \{x|x\leq - 2\} \)" is false.

Step4: Analyze the range

Since the parabola opens downwards ( \( a=-1<0 \)) and the vertex is at \( y = 6 \), the range of the function is all real numbers less than or equal to 6, i.e., \( \{y|y\leq6\} \). So the statement "The range is \( \{y|y\leq6\} \)" is true.

Step5: Analyze the increasing/decreasing intervals

For a parabola that opens downwards, the function is increasing on the interval \( (-\infty,-2) \) (because as we move from left to right towards the vertex, the function values increase) and decreasing on the interval \( (-2,\infty) \) (because as we move from the vertex to the right, the function values decrease).

• For the statement "The function is increasing over the interval \( (-\infty,-2) \)", since the vertex is at \( x=-2 \) and the parabola opens down, the function increases as \( x \) approaches - 2 from the left. So this statement is true.
• For the statement "The function is decreasing over the interval \( (-4,\infty) \)", the interval \( (-4,\infty) \) includes the interval \( (-2,\infty) \) (where the function is decreasing) and \( (-4,-2) \) (where the function is increasing). So the function is not decreasing over the entire interval \( (-4,\infty) \), and this statement is false.

Step6: Analyze the y - intercept

The y - intercept of a function \( f(x)=ax^2+bx+c \) is found by setting \( x = 0 \). Substitute \( x = 0 \) into the function: \( f(0)=-0^2-4\times0 + 2=2 \). Since \( 2>0 \), the function has a positive y - intercept. So the statement "The function has a positive y - intercept" is true.

Answer:

The three true statements are:

• The range is \( \{y|y\leq6\} \)
• The function is increasing over the interval \( (-\infty,-2) \)
• The function has a positive y - intercept