2.3 Exponential functions

Determine whether the function represents an exponential function or not.

To determine if a set of data represents an exponential function, we want to compare successive y-values, and determine if there is a common ratio.

We can see that f\left(-1\right)=\dfrac{5}{3} and f\left(0\right)=5, so we want to calculate the common ratio that takes us from \dfrac{5}{3} to 5. \frac{5}{3} \cdot b=5 \longrightarrow b=5 \cdot \frac{3}{5} = 3 The common ratio is 3. We now want to check if this factor works for the remaining y-values:\begin{aligned}5 \cdot 3&=15 \\ 15 \cdot 3 &= 45\\ 45\cdot 3&=135\\ 135\cdot 3&=405\end{aligned}

The function represents an exponential function.

Instead, we could have divided successive outputs to make sure there is a constant common ratio: \begin{aligned}5 \div \frac{5}{3}&=3 \\ 15 \div 5 &= 3\\ 45\div 15&=3\\ 135\div 45&=3\\405\div 135 &=3\end{aligned}

Write an equation to represent the exponential function.

The function has a y-intercept when x=0. The y-intercept is at y=5 and the growth factor is 3.

An equation that represents this exponential function is: f\left(x\right)=5\left(3\right)^x

If we wanted to know the percent at which the function grows, we can use the fact that b=1+r in the exponential growth functions. \begin{aligned}3&=1+r\\2&=r\end{aligned} Changing this into a percentage, we find the growth rate is 200\%.

The graph of f\left(x\right) is translated to get the graph of g\left(x\right) show below.

From the graph, we notice the y-intercept of f\left(x\right) has been translated 2 units to the left and 3 units up to obtain the point \left(-2, 4\right) on the graph of g\left(x\right).

A translation of 2 units to the left is equivalent to adding 2 to the input values. A translation of 3 units up is equivalent to adding 3 to output values.

Since the graph has only been translated, the base of both exponential functions will be the same. We can determine the base by finding the ratio of consecutive outputs of the parent function or by finding the ratio of the differences of consecutive outputs of the transformed function.

The transformed function will be in the form y=b^{x+h}+k where h=2 and k=3.

Looking at the transformed function, we can easily identify the points \left(-2,4\right), \left(-1,6\right), and \left(0,12\right). Analyzing the change in the y-values, we find:

The ratio of the differences in these outputs is \frac{6}{2}=3. This means b=3.

Therefore, g\left(x\right)=3^{x+2}+3.

We could have used consecutive outputs of the parent function to find the base instead. \begin{aligned}3\div 1 &= 3\\9\div 3&=3 \end{aligned}However, dividing consecutive outputs to find the base only works for exponential functions of the form y=ab^x.

The graph of y = \left(\dfrac{1}{2}\right)^{x} is reflected across the x-axis and stretched vertically by a factor of 2.

We can represent this change graphically:

The transformed function is y=-2\left(\dfrac{1}{2}\right)^{x}

To graph this, we could have built a table of values for the parent function of y=\left(\dfrac{1}{2}\right)^x and multiplied the outputs by -2 to find points on the new graph. We will call the parent function f\left(x\right) and the transformed function g\left(x\right).

Describe the behavior of the function as x increases.

We want to identify if the values of y are increasing or decreasing as x increases.

As x increases, the function decreases at a slower and slower rate.

We can see that the equation has a constant factor that is less than 1. This is why the function is decreasing.

Determine the y-intercept of the function.

The y-intercept occurs when x=0. We can read these coordinates from the table.

\left(0,\,2\right)

We can see that the equation has a leading coefficient of 2. This is the value of the y-intercept, and the result of substituting x=0 into the equation.

State the domain of the function.

The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in both horizontal directions.

All real x.

All exponential equations of the form y=ab^x have a domain of all real x.

State the range of the function.

The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, but it approaches an asymptote at y=0 towards the right.

y>0

All exponential equations of the form y=ab^x have a range of y>0 for positive values of a.

Given that the relationship is exponential, complete the table of values.

We can find the value of b by dividing the amount of water in the puddle after one hour by the amount that was present at the start. Using this value for b, we can then find the missing values.

Constant factor: b=\dfrac{512}{1024}=\dfrac{1}{2}

Using this value for b, we know that the volume after 3 hours will be half of 256, and the time after 5 hours will be half of 64.

This exponential relation is an example of one that decreases over time. The domain is x\geq 0 because it would not make sense to have negative values for time. When this happens, we refer to the y-intercept as the initial value.

Describe the relationship between time and volume.

We found that b=\dfrac{1}{2} previously, this tells us how we move from one term to the next.

The volume of water is halved every hour.

We can also word this as, "For every increase in time by one hour, the amount of water is divided by two."

Draw the graph of the function.

We can identify both the y-intercept and constant factor from the equation since it is of the form y=ab^x. Using these two key features, we can plot other points to the left and right of the y-intercept and connect them with a smooth curve.

The function has a constant factor of 4 since that is the base of the exponent, and a y-intercept at \left(0,2.5\right) as that is the coefficient.

We can start by plotting the y-intercept and choosing an appropriate scale.

When drawing the graphs of exponential functions, we want to be sure the y-intercept is clearly displayed and that the exponential curve is also visible. Be sure to choose a scale for the y-axis that will show all important characteristics. In this case, we chose to scale by 5's, which allows us to read both the y-intercept at 2.5, a second point at \left(1,10\right), the horizontal asymptote at y=0 and the steep slope that all exponential functions have.

Check the graph from part (a) using technology.

When using a graphing calculator, there will typically be an input bar for functions. We just need to type out the function in the input bar. We need to be careful that we format our input correctly by using the correct buttons.

The GeoGebra calculator has a math input keyboard, but we can also use a computer keyboard.

• Math input keyboard

  

• Computer keyboard

  

The graph should look the same as in part (a). We may need to change the axes or zoom settings to check.

Using a table of values, draw the graph of f\left(x\right) and g\left(x\right) on the same plane.

We can identify both the y-intercept and constant factor from each equation since they are of the form y=ab^x. Using these two key features, we can create a table of values of points to the left and right of the y-intercept of each function. We can then plot the points and connect them with a smooth curve.

First, we can consider the function f\left(x\right). We can see that the function has a constant factor of \dfrac{1}{3} and a y-intercept with coordinates \left(0,2\right).

We can create a table of values with a few points around the y-intercept by substituting in the values x=-3,-2,-1,1,2.

Now, we can consider the function g\left(x\right). We can see that the function has a constant factor of \dfrac{1}{2} and a y-intercept with coordinates \left(0,3\right).

We can create a table of values with a few points around the y-intercept by substituting in the values x=-3,-2,-1,1,2.

Looking at the table of values for f\left(x\right) and g\left(x\right) to determine the scale of the graph, we can see the largest y-value is 54 and we have no negative values of y: the values will continue to decrease, and get closer to the x-axis, but never cross it. So, when plotting the graph, the y-axis has to go up to at least 54 and have an asymptote at y=0.

We can see that as x increases, both f\left(x\right) and g\left(x\right) decrease. However, the relative amount of decrease over each interval is different for each function.

Determine which function is decreasing at a slower rate.

We can compare the constant factors of each function to help determine which function is decreasing slower.

Considering the functions f\left(x\right)=2\left(\dfrac{1}{3}\right)^{x} and g\left(x\right)=3\left(\dfrac{1}{2}\right)^{x}. We can see that as we approach 0 from the left, for each unit value increase in x, f\left(x\right) is decreasing by a factor of 3, whereas g\left(x\right) is decreasing by a factor of 2.

This means that f\left(x\right) is approaching 0 faster than g\left(x\right). So, we can say that g\left(x\right) is decreasing at a slower rate.

We can also consider the graph of both functions to confirm that g\left(x\right) is decreasing at a slower rate.

Considering the graph of each function from part (a), we can see that our functions actually intersect at \left(-1,5\right). Before this point, f\left(x\right) is significantly larger than g\left(x\right) at x=-2 and x=-3. But after the point of intersection, g\left(x\right) will always be larger. This confirms that g\left(x\right) is decreasing slower than f\left(x\right).

Determine which function increases at a slower rate.

One way to determine the answer is by comparing the constant factors of the functions. We can find the constant factors by dividing the outputs of two consecutive values of x.

For f(x), we can use the points (-1, -8) and (0,-2).

b=\dfrac{-2}{-8}=\dfrac{1}{4}

For g(x), we can use the points (-2, -9) and (-1, -3).

b=\dfrac{-3}{-9}=\dfrac{1}{3}

\dfrac{1}{4}<\dfrac{1}{3}, which means f(x) gets multiplied by a smaller factor than g(x). Because of this, f(x) will get closer to 0 faster, so g(x) increases at a slower rate.

The graphs of f(x) and g(x) are shown below.

When we look at the graph, we can see g(x) does not seem as steep as f(x) and its curve is more rounded. This is a visual indication that it is increasing at a slower rate.

Identify the y-intercept for each function.

On the graph, we look for the point where the function intercepts the y-axis. In the table, we look for the point where x=0.

The y-intercept of f(x) is (0,-2).

The y-intercept of g(x) is (0,-1).

Now that we know the y-intercept and constant factor for each function, we can build the equations.

f(x)=-2\left(\dfrac{1}{4}\right)^x

g(x)=-1\left(\dfrac{1}{3}\right)^x