An expression is shown below: f(x) = 4x2 − 7x − 15 part a: what are the x-intercepts of the graph of f(x)? show your work. part b: is the vertex of the graph of f(x) going to be a maximum or minimum? what are the coordinates of the vertex? justify your answers and show your work. part c: what are the steps you would use to graph f(x)? justify that you can use the answers obtained in part a and part b to draw the graph.

F(x) = 4x2 - 7x - 15  part a:   the first thing we must do is rewrite the expression.  we have then:   f (x) = 4x2 - 7x - 15  f (x) = (4x + 5) (x-3)  we equal zero:   0 = (4x + 5) (x-3)  we clear the roots:   x1 = -5 / 4  x2 = 3  part b:   to find the vertex we derive:   f '(x) = 8x - 7  we equal zero and clear x:   0 = 8x - 7  x = 7/8  we look for the second derivative:   f '' (x) = 8  we evaluate x = 7/8  f '' (7/8) = 8 (> 0 the vertice is a minimum)  we look for the value of the vertex in the y axis:   f (7/8) = 4 * (7/8) ^ 2 - 7 * (7/8) - 15  f (7/8) = -18.0625  answer:   the coordinates of the vertice are:   (7/8, -18.0625)  part c:   for this case what you need to know is that to graph a quadratic function you need three points.  we can then use the points found in part a and part b.  these points are:   (-5/4, 0)  (3, 0)  (7/8, -18.0625)

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