Describe when the graph of f(x) = ab^x is a horizontal line.?
I thought it was when a=0, but its not...
technically, the graph is never a straight line unless
a=0 or
b=1
But it will approach a horizontal asymptote otherwise, as Reiny suggests.
take a look at this, letting a=5, b=3 http://www.wolframalpha.com/input/?i=y+%3D+3*5%5Ex
so when x= large negative number, then slope goes to ....
Well, it seems like we have a case of mistaken identity here! You're right, if a = 0, then the graph of f(x) = ab^x will indeed be a horizontal line...but only horizontal and flat like a pancake at y = 0! So close, yet so far! In reality, for a horizontal line, we have to set b equal to 1. That's right, when b equals 1, the graph of f(x) = ab^x becomes as straight and steady as a ruler, refusing to budge or wiggle. It's like the graph is taking a stand and saying, "I'm going nowhere, folks!" Hopefully, this clarifies things a bit and puts a smile on your face!
The graph of the function f(x) = ab^x is not a horizontal line when a=0 because any number raised to the power of 0 is equal to 1, and multiplying 0 by 1 gives a constant value of 0. In other words, the graph would be the y-axis.
Instead, the graph of f(x) = ab^x is a horizontal line when the base, b, is equal to 1. This is because any number raised to the power of 0 is equal to 1, and multiplying 1 by a constant value, a, would give a constant value of a. Therefore, the graph would be a horizontal line at the height of a.
To describe when the graph of f(x) = ab^x is a horizontal line, we need to consider the values of a and b. The equation represents an exponential function, where x is the independent variable and f(x) is the dependent variable.
For the graph to be a horizontal line, it means that the value of f(x) remains constant, regardless of the value of x. In other words, f(x) does not change with different values of x.
To achieve this with the given equation, we need to make sure that the exponential term, ab^x, evaluates to 1. Any value multiplied by 1 remains unchanged, which is why the graph would be a horizontal line.
In order for ab^x to equal 1, we have two scenarios:
1. When a = 1 and b = 1: In this case, f(x) = 1 * 1^x = 1. The graph of f(x) = 1 is simply a horizontal line at y = 1.
2. When a = 0 and b ≠ 0: This is where your confusion might have arisen. When a = 0, regardless of the value of b, the whole term ab^x will be 0 for all x. So, the graph will be a horizontal line at y = 0.
To summarize, the graph of f(x) = ab^x is a horizontal line when either a = 1 and b = 1 or a = 0 and b ≠ 0.