We want to match each equation to the correct graph.
Notice all the equations are in the form "A" multiplied by "B"
to the power of "X," which means they’re all exponential functions.
For a quick review, "A" is called the initial value, meaning the function value when "X" is
zero. If it’s the function value when "X" is zero, "A" would also be the Y-intercept of the function.
The reason it’s called the initial value is that we often use the variable "T" instead of "X"
to represent time, and when "T" is zero, that would be the starting amount
or initial value before the exponential growth or decay starts.
If the base is between zero and one, we have exponential decay, which means the function would
be decreasing. From left to right, the graph would be going downhill, looking something like this.
If the base is greater than one, we’d have exponential growth, which means the function
is increasing. The function is going uphill from left to right, looking something like this.
Now, let’s take a look at our equations again.
First, focus on the value of "A" or the initial value. Notice all these functions have an initial
value of 3, except for one function. Since "A" is equal to 4, this graph has a Y-intercept
or vertical intercept of 4. Because "B" is greater than 1, we have exponential growth.
Looking at our graphs, notice there’s only one function that has a Y-intercept of 4,
and it’s also exponential growth because it’s going uphill from left to right. So,
this second equation would produce the brown graph.
Looking at the four remaining equations, let’s find the two that would represent
exponential decay, meaning the base value would be between zero and one.
Notice the first equation would be exponential decay because the base is 0.7,
and this equation here because the base is 0.5.
Notice how the values of "A" are the same, so they both have a Y-intercept of 3.
But again, they’re both exponential decay, so they’d be going downhill from left to right.
Looking at our graphs, notice there are only two
graphs that represent exponential decay: the blue function here and the red function here.
Notice how the red function is decreasing faster because it’s steeper going downhill from left to
right, which means the red graph would have a smaller base value. The closer the base is to
zero, the faster it decreases, so the red graph would be the equation with the base of 0.5.
The blue graph, which now looks black, would be the equation with the base value of 0.7.
Again, the smaller the base or the closer the base is to zero, the faster it decreases.
That leaves us with two other equations to match.
We have this equation here where the base is 1.5 and this equation here where the base is 1.2.
Both equations have a base greater than 1,
which means that both will be exponential growth. Since "A" is 3, they also have a Y-intercept of 3.
So now we should focus on the green graph and the black graph.
We don’t need to be concerned about the brown graph because we
know that one has a Y-intercept of 4, which we already matched.
Notice how the black graph is increasing faster,
which means the base value is larger than the base value for the green
graph. Since 1.5 is greater than 1.2, this equation would represent the black graph.
The equation with the base of 1.2 would be the green graph.
I hope this helps explain some of the characteristics of exponential functions.