Question

Sketch the graph of the function. $$y=-3 x^{2}-x+7$$

Short answer

The graph of the function \(y = -3x^2 -x +7\) is a parabola that opens downward with a vertex at (-1/6, 47/6), an axis of symmetry at \(x = -1/6\) and a y-intercept at (0,7).

Step-by-step solution

Identify the Vertex

The vertex of a quadratic function in general form \(y = ax^2 + bx + c\) is given by \((-b/2a , f(-b/2a))\). By applying this formula on the given function \(y = -3x^2 -x +7\), we get the vertex is located at \((-(-1)/(2*-3), f(-(-1)/(2*-3)) = (-1/6, 7-1/12) = (-1/6, 47/6)\). This will be the highest or lowest point of the graph.

Identify the Axis of Symmetry

The Axis of Symmetry of the quadratic function is the vertical line passing through the vertex, hence, its equation is \(x = -1/6\).

Identify the Direction of Opening

Since the coefficient of \(x^2\) which is \(-3\) in the given function is less than zero, the graph will open downwards.

Identify the y-intercept

The y-intercept is the value of the function at \(x = 0\). Substituting \(x = 0\) into the function, we get \( y = -3(0)^2 -0 +7 = 7\). Hence, the y-intercept is at the point (0,7).

Plot the Function

Plot the vertex (-1/6, 47/6), the y-intercept (0,7), and draw the axis of symmetry at \(x = -1/6\). Since the graph opens downwards, draw a parabola opening downwards passing through the vertex and y-intercept plotted.