What is the maximum y value on the graph of y= f(x)

Question

f(x)= -x^2 +6x +10

maximum y - value is y=?

Answer 1

You can put this in vertex form by completing the square

-x² + 6x + 10

= -(x² - 6x - 10)

= -(x² - 6x + 9 - 19)

= -((x - 3)² - 19)

= -(x - 3)² + 19

The vertex is (3,19) and the parabola opens down, so this is where the maximum occurs.

Maximum value: y = 19

Answer 2

Method 1. Completing the square:

f(x) = -(x^2-6x+9) + 19 = -(x-3)^2 + 19

So, ymax = 19 when x = 3.

Method 2. Using vertex relation:

h = -b/(2a) = 6/(2) = 3

ymax = f(h) = f(3) = -9+18+10 = 19

Method 3. Using calculus to find the maximum:

f'(x) = -2x+6 = 0

x = 3

fmax = f(3) = 19

 

Answer 3

There are two ways to solve this problem. The easiest way is to graph the function with a graphing calculator and to look for the highest value of the parabola called the vertex or looking at the set of numbers what number does it go up to before the numbers descend again. This website allows you to graph online for free: http://www.rentcalculators.org/stheli.html Also, you can solve for the maximum value with this solution: 

1. Break the problem down in sections a, b, and c in order to find a value for x to find our highest (maximum) value of y. The a will be represented with the -x², the b will be represented as the 6x and the c with 10. However, the variables won't matter for this next step compared to the constants, or the numbers.

2. So to find x, we will use the following equations:

x = -b/2a

x = -(6)/2(-1) notice that a is -1. That is because the constant or number in front of the x² would be -1. 

x = -(6)/-2

x = 3

3. Now that we have our x - value, we will now plug that number into the function to finally get our y - value or the f(x):

f(3) = -(3)² + 6(3) + 10

f(3) = -(9) + 18 + 10

f(3) = 19

To be able to find some problems to practice or if the concept is still a puzzle, I love using this website: http://www.purplemath.com/modules/index.htm It will break the lesson down and give examples to practice the lesson.