linear and quadratic functions #2?

Question

For the function f(x) = 4x - x^2 , find

The coordinates of the vertex
The equation of the axis of symmetry
the greatest value of the function
graph the equation

Answer 1

f(x) = 4x - x^2

1) Coordinates of vertex.

Your goal is to change f(x) in the form a(x - h) + k. (h,k) will be the coordinates of your vertex.
To put it in that form, this is done by completing the square.

f(x) = 4x - x^2
f(x) = -x^2 + 4x
f(x) = -(x^2 - 4x)
f(x) = -(x^2 - 4x + 4) + 4
f(x) = -(x - 2)^2 + 4

So your vertex is at (2, 4)

2) Equation of axis of symmetry.
This is equal to x = h. Since we solved for h above (h = 2),
x = 2 is our axis of symmetry.

3) The greatest value of the function.
This is equal to k. As solved above for k,
k = 4.

4) You would have to graph this yourself. What you do have is the vertex (2, 4) which would be a maximum because your value "a" is negative [f(x) = a(x - h) + k, f(x) = -(x - 2) + 4].

Other necessary points you will need are the x-intercepts. Set f(x) = 0 to find them.

0 = 4x - x^2,
0 = x(4 - x)
So x = {0, 4}

Your x-intercepts are at (0,0) and (4,0).

Mark the points (0, 0), (4,0), and (2, 4) on your (currently empty) graph, and join them together as a parabola facing downward.

Answer 2

1) For the general quadratic equation f(x)=ax^2 + bx + c, the x coordinate of the vertex can be found by the formula -b/(2a) [If you are curious, this can be proven by the use of derivatives]. In this example, b=4 and a=-1 so -b/(2a) is -4/-2=2. Plugging 2 into our equation gives y=4. So the vertex is (2,4).

2) The axis of symmetry is the x-coordinate of the vertex. So, our axis of symmetry is x=2.

3) We found the vertex to be (2,4) and since the coefficient of the term x^2 is negative, the graph is an upside down parabola. Therefore, our vertex is actually the maximum point of the function! So, the greatest value of the function is 4 at x=2.

4) f(x) = 4x - x^2
f(x) = x(4-x)

So, our zeros are at x=0 and x=4. Graph these points. Then graph the vertex and draw your upside down parabola in such a way that it will pass from all these three points and its vertex will be at (2,4)