Question

Sketch the graph of the function. Label the vertex. y=-3 x^{2}-5 x+3

Short answer

The vertex of the function \(y=-3x^{2}-5x+3\) is approximately (5/6, -1.42).

Step-by-step solution

Identify the coefficients

The given function is \(y=-3x^{2}-5x+3\). Here, \(a=-3\), \(b=-5\), and \(c=3\).

Use the vertex formula to find the x-coordinate of the vertex

The x-coordinate of the vertex is given by \(-b/2a\). So, substitute \(a=-3\) and \(b=-5\). Calculating gives \(x=-(-5)/2(-3) = 5/6\).

Substitute the x-coordinate into the function to find the y-coordinate

Now substitute \(x=5/6\) into the function to find the y-coordinate of the vertex. \(y= -3 (5/6)^{2}-5(5/6)+3 \). Computing this, we find y equals approximately -1.42.

Plot the graph

Now, based on the x and y coordinates obtained for the vertex and given function, sketch the graph. Plot the point (5/6, -1.42) and label it as the vertex. Because \(a < 0\), the parabola opens downward.