sketch the graph of the quadratic function and the axis of symmetry. \n$f(x) = -3(x + 2)^2 + 4$\n\nuse the…

Explanation:

Step1: Recall vertex form of quadratic

The vertex form of a quadratic function is ( f(x) = a(x - h)^2 + k ), where ((h, k)) is the vertex and the axis of symmetry is ( x = h ).

Step2: Identify ( h ) in given function

For the function ( f(x) = -3(x + 2)^2 + 4 ), we can rewrite ( (x + 2) ) as ( (x - (-2)) ). So, comparing with ( a(x - h)^2 + k ), we have ( h = -2 ) and ( k = 4 ).

Step3: Determine axis of symmetry

Using the formula for the axis of symmetry from the vertex form, since ( h = -2 ), the axis of symmetry is ( x = -2 ).

Step4: Analyze the parabola's shape

The coefficient ( a = -3 ), which is negative, so the parabola opens downward. The vertex is at ((-2, 4)). To sketch the graph, plot the vertex, then use the fact that for quadratic functions, we can find other points (e.g., when ( x = -1 ), ( f(-1) = -3(-1 + 2)^2 + 4 = -3(1) + 4 = 1 ); when ( x = -3 ), ( f(-3) = -3(-3 + 2)^2 + 4 = -3(1) + 4 = 1 ); when ( x = 0 ), ( f(0) = -3(0 + 2)^2 + 4 = -3(4) + 4 = -8 ); when ( x = -4 ), ( f(-4) = -3(-4 + 2)^2 + 4 = -3(4) + 4 = -8 )). Then draw a solid curve through these points (and others) opening downward, with the vertex at ((-2, 4)), and draw a dashed line at ( x = -2 ) for the axis of symmetry.

Answer:

The axis of symmetry is the vertical line ( x = -2 ). The graph of ( f(x) = -3(x + 2)^2 + 4 ) is a parabola opening downward with vertex at ((-2, 4)), and the axis of symmetry (dashed line) is ( x = -2 ). When sketching, plot the vertex, use symmetric points to draw the solid parabola, and draw ( x = -2 ) as a dashed line.