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

which statements about the graph of the function $f(x) = -x^2 - 4x + 2$ are true? select three options. \n- the domain is ${x|x \\leq -2}$. \n- the range is ${y|y \\leq 6}$. \n- the function is increasing over the interval $(-\\infty, -2)$. \n- the function is decreasing over the interval $(-4, \\infty)$. \n- the function has a positive $y$-intercept.

which statements about the graph of the function $f(x) = -x^2 - 4x + 2$ are true? select three options. \n- the domain is ${x|x \\leq -2}$. \n- the range is ${y|y \\leq 6}$. \n- the function is increasing over the interval $(-\\infty, -2)$. \n- the function is decreasing over the interval $(-4, \\infty)$. \n- the function has a positive $y$-intercept.

Answer

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