Question

Compare the following functions:

f(x) = 2 sin(2x − π) + 2

g(x)
graph of a quadratic with points at 1, 2 and 3, negative 2 and 5, 2

h(x)
x y
−2 10
−1 7
0 5
1 3
2 5
3 7
4 10

Which function has the smallest minimum?

Answer 1

@macgirl234 have you graphed f yet?

Answer 2

No. I have no idea where to begin with any of this

Answer 3

Okay, no problem

Answer 4

Can you help me?

Answer 5

Yes, so for each function, you have inputs and outputs. The output values are the actual values of a function for its corresponding input. For example, The value of h is 5 when x equal 0.

Answer 6

What you want to do is make a table of values for each function, then determine which function has the smallest minimum value.

Answer 7

I still don't understand...

Answer 8

Hang on a minute

Answer 9

ok

Answer 10

Okay, sorry for taking so long

Answer 11

First here is a graph of f(x):
https://www.desmos.com/calculator/clkwu15jna

Answer 12

And a table of values for f(x)

Answer 13

The rightmost column represents the values for f which includes minimum values, maximum values and everything in between.

Answer 14

Plotting the table for g(x) is a bit more involved because we have to actually find the equation for it. They give us three points on its graph (1,2), (3,-2) and (5,2). We have to take those three points and insert them in to \(g(x) = ax^2 + bx + c\) like so
2 = a(1)^2 + b(1) + c
-2 = a(3)^2 + b(3) + c
2 = a(5)^2 + b(5) + c

When we simplify that we have
2 = a + b + c
-2 = 9a + 3b + c
2 = 25a + 5b + c

We have to solve for a, b, and c to get the coefficients of g(x).

Answer 15

Once we get those coefficients, we can then graph g(x) and find its minimum value.

Answer 16

(1st) 2 = a + b + c
(2nd) -2 = 9a + 3b + c
(3rd) 2 = 25a + 5b + c

Subtracting the 1st equation from the 2nd we get:
-4 = 8a + 2b

Subtracting the 2nd equation from the 3rd we get:
4 = 16a + 2b

In other words, we end up with the following system:
-4 = 8a + 2b
4 = 16a + 2b

Which simplifes to
-2 = 4a + b
2 = 8a + b

Subtracting the top equation from the bottom we get:
4 = 4a

So a = 1

Since b = 2 - 8a, we can find b:
b = 2 - 8(1) = 2 - 8 = -6

So b = -6

Since c = 2 - a - b, we can find c as follows:
c = 2 - 1 -(-6) = 2 + 6 - 1 = 8 - 1 = 7

So a = 1, b = -6, and c = 7, therefore \(g(x) = x^2 -6x + 7\)

Answer 17

Here's the graph of g(x):
https://www.desmos.com/calculator/m3xtksexsz

Answer 18

And it is clear that the lowest value of g(x) is -2

Answer 19

Here's the graph of h which has a minimum value of 3
https://www.desmos.com/calculator/i6icbyu96p

Answer 20

So @macgirl234 given the above information, which function has the lowest minimum?