Question

f(x) = −16x2 + 24x + 16 Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points) WILL GIVE BRAINLIEST HELP PLEASE 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. (3 points) 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. (5 points)

Answer 1

Answer:

Step-by-step explanation:

f(x) = −16x2 + 24x + 16 = 0 can be reduced to -2x^2 + 3x + 2 = 0 and then to 2x^2 - 3x - 2 = 0.  Solve this for x using the quadratic formula:

The discriminant is b^2 - 4ac = (-3)^2 - 4(2)(-2) = 9 + 16 = 25.

Therefore the roots are:

     -(-3) ± √25        3 ± 5

x = ----------------- = ------------  =>  x = 2 and x = -1/2

           2(2)                  4

The x-intercepts are points:  (2, 0) and (-1/2, 0)

Because the coefficient of the x^2 term is negative, this graph opens down and the vertex represents a maximum.

To graph this function, find and plot the vertex.  It is exactly halfway between the x-intercepts, that is, at x = 1 1/4, for which the y value is 21:

vertex and maximum at (5/4, 21)

Finally, find the y-intercept.  Let x = 0; we find that y = 16.  The y-intercept is 0, 16)

We now have four points on the graph and know where the maximum is.  Plot this max (5/4, 21) and the x-intercepts (2, 0) and (-1/2, 0), and finally the y-intercept.  Draw a smooth curve through these points, remembering that the graph is symmetrical about the axis of symmetry x = 5/4.