Which of the following is a decreasing function?

a. y = 7^x - 2
b. y = 7^-x + 2
c. y = 3(5^x) - 5
d. y = 5(3^x) + 3

What function is the reflection of y = 10^x with respect to the y-axis?
a. y = -10^x
b. y = (-1/10)^x
c. y = 10^-x
d. y = -10^x

#1, the key terms in each answer is the power term

pick any two increasing values of x
e.g.
x = 2, y = 7^2 - 2 = 47
x = 3 , y = 7^3 - 2 = 341
Is there any reason for this pattern NOT to continue,
so the function is INCREASING
You will see the same for c) and d)
Now look at b)
x=2 , y = 7^-2 - 2 = 1/49 - 2 = appr - 7.97959..
x=3 , y = 1/343 - 2 = appr -1.997..
which is LOWER or smaller than the y value at x=2
So the function would be decreasing.

2. for a reflection in the y-xis
any point (a,b) ---> (-a,b)

so a point on y = 10^2 is (2,100)
so its reflection in the y-axis would be (-2,100)

can you decide which equation would be satisfied by the point (-2,100) ?

To identify a decreasing function among the options, we need to determine if the function decreases as the input increases.

Let's go through each option:

a. y = 7^x - 2
b. y = 7^-x + 2
c. y = 3(5^x) - 5
d. y = 5(3^x) + 3

To determine whether a function is increasing or decreasing, we can observe the coefficient of the base (the number raised to a power) in each option.

In option a, the base is 7 (coefficient of 7^x), so the function will increase as the input increases.

In option b, the base is 7^(-x), resulting in 1/7^x. Since 1/7^x decreases as x increases, this function is decreasing.

In option c, the base is 5, so the function will increase as the input increases.

Finally, in option d, the base is 3, so the function will increase as the input increases.

Therefore, the function that is decreasing is option b, y = 7^-x + 2.

Moving on to the second question:

The reflection of a function across the y-axis is obtained by flipping the sign of the x term.

The given function is y = 10^x.

Taking its reflection across the y-axis gives: y = -10^x.

Therefore, the function that is the reflection of y = 10^x with respect to the y-axis is option a, y = -10^x.