Exponential Functions Changing the Graph

Exponential Functions Changing the Graph - Video

Exponential Function Model

Exponential Function Model:

Here we have what happens when we change the different parts of an exponential function.

When a changes, then there is a vertical shrink/stretch. Shrink occurs when 0 < a < 1 and stretch occurs when a > 1.

When b changes, then the base changes. The overall look does not change just how fast the graph rises.

When c changes, then there is a horizontal shrink/stretch. Shrink occurs when a > 1 and stretch occurs when 0 < a < 1.

When d changes, then there is a horizontal shift (left/right).

When f changes, then there is a vertical shift (up/down).

There is not an e because e is on the most common bases in exponential functions.

Guidelines on How to Graph

Guidelines on How to Graph:

On the left we have 8 different facts on the standard of an exponential function and on the right we have the guidelines.

Example 1 - Vertical Stretch/Shrink

Example 1 - Vertical Shrink/Stretch:

Here we have three functions: y = 3 * 2^x, y = 1 * 2^x (y = 2^x), and y = (1/3) * 2^x.

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-3, 3/8), (0, 3), and (3, 24) for y = 3 * 2^x.

We have (-3, 1/8), (0, 1), and (3, 8) for y = 1 * 2^x (y = 2^x).

We have (-3, 1/24), (0, 1/3), and (3, 8/3) for y = (1/3) * 2^x.

Let's take a closer look at our points. The y-value of each first point is 3/8, 1/8, and 1/24, but the x-value is the same for each one. Why?

We changed the output by multiplying by 3 in the first equation, y = 3 * 2^x. If we work backwards, 3/8 divided by 3 is 1/8, which is the output value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the first equation is 3 times higher than the first equation.

We change the output by multiplying by 1/3 in the third equation, y = (1/3) * 2^x. If we work backwards, 8/3 times 1/3 is 8, which is the output value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the third equation is 3 times lower than the second equation.

Now let's take a look at the x-intercept. Since each equation is an exponential growth and each have an asymptote at y = 0, we don't have x-intercept.

We do however have a y-intercept at (0, 3), (0, 1), and (0, 1/3).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our exponential growth functions, so the range of each is (0, +∞), because it will not cross below the x-axis in this example because of the rules of exponents.

Example 2 - Horizontal Stretch/Shrink

Example 2 Horizontal Stretch/Shrink:

Here we have three functions: y = 2^3*x, y = 2^1*x (y = 2^x), and y = 2^(1/3)*x.

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-1, 1/8), (0, 1), and (1, 8) for y = 2^3*x.

We have (-3, 1/8), (0, 1), and (3, 8) for y = 2^1*x (y = 2^x).

We have (-9, 1/8), (0, 1), and (9, 8) for y = 2^(1/3)*x.

Let's take a closer look at our points. The x-value of each first point is -1, -3, and -9, but the y-value is the same for each one. Why?

We changed the input by multiplying by 3 in the first equation, y = 2^3*x. If we work backwards, -1 times 3 is -3, which is the input value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the third equation is reached 3 times faster than the second equation.

We change the input by multiplying by 1/3 in the third equation, y = 2^(1/3)*x. If we work backwards, -9 times 1/3 is -3, which is the input value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the third equation is reached 3 times slower than the second equation.

Now let's take a look at the x-intercept. Since each equation is an exponential growth and each have an asymptote at y = 0, we don't have x-intercept.

We do however have a y-intercept at (0, 1), (0, 1), and (0, 1).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our exponential growth functions, so the range of each is (0, +∞), because it will not cross below the x-axis in this example because of the rules of exponents.

Example 3 - Horizontal Shift (Left/Right)

Example 3 Horizontal Shift (Left/Right):

Here we have three functions: y = 2^x+3, y = 2^x+0 (y = 2^x), and y = 2^x-3.

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-6, 1/8), (-3, 1), and (0, 8) for y = 2^x+3.

We have (-3, 1/8), (0, 1), and (3, 8) for y = 2^x+0 (y = 2^x).

We have (0, 1/8), (3, 1), and (6, 8) for y = 2^x-3.

Let's take a closer look at our points. The x-value of each first point is -6, -3, and 0, but the y-value is the same for each one. Why?

We changed the input by moving one 3 to the right in the first equation, y = 2^x+3. If we work backwards, -6 plus 3 is -3, which is the input value of the second equation, y = 2^1*x (y = 2^x). So the same x-value in the first equation is 3 numbers lower to reach the second equation.

We change the input by moving 3 to the left in the third equation, y = 2^x-3. If we work backwards, 0 minus 3 is -3, which is the input value of the second equation, y = 2^1*x (y = 2^x). So the same x-value in the third equation take 3 numbers higher to reach the second equation.

Now let's take a look at the x-intercept. Since each equation is an exponential growth and each have an asymptote at y = 0, we don't have x-intercept.

We do however have a y-intercept at (0, 8), (0, 1), and (0, 1/8).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our exponential growth functions, so the range of each is (0, +∞), because it will not cross below the x-axis in this example because of the rules of exponents.

Example 4 - Vertical Shift (Up/Down)

Example 4 Vertical Shift (Up/Down):

Here we have three functions: y = 2^x + 1, y = 2^x + 0 (y = 2^x), and y = 2^x - 1.

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-3, 9/8), (0, 2), and (3, 9) for y = 2^x + 1.

We have (-3, 1/8), (0, 1), and (3, 8) for y = 2^x + 0 (y = 2^x).

We have (-3, -7/8), (0, 0), and (3, 7) for y = 2^x - 1.

Let's take a closer look at our points. The y-value of each first point is 9/8, 1/8, and -7/8, but the x-value is the same for each one. Why?

We changed the output by adding 1 in the first equation, y = 2^x + 1. If we work backwards, (9/8) minus 1 is (1/8), which is the output value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the first equation is 1 more than the second equation.

We change the output by subtracting 1 in the third equation, y = 2^x - 1. If we work backwards, (-7/8) plus 1 is (1/8), which is the output value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the third equation is 1 fewer than the second equation.

Now let's take a look at the x-intercept. Since each equation is an exponential growth we start with an asymptote at y = 0 and we don't have an x-intercept, but because we have shifted our function vertical, we could have an x-intercept.

The first equation, y = 2^x + 1, shifted up 1 so everything went up 1 including the asymptote, which went from y = 0 to y = 1. The second equation, y = 2^x + 0 (y = 2^x), did not shift. The asymptote is still y = 0. So for both of these equations every output is above the x-axis. So we don't have an x-intercept.

As for the third equation, y = 2^x - 1, all the outputs shifted down 1 as well as the asymptote, which went from y = 0 to y = -1. Which means we know have points below the x-axis. So, we have an x-intercept at x = 0.

We do however have a y-intercept at (0, 2), (0, 1), and (0, 0).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our exponential growth functions. The first equation, y = 2^x + 1, shifted up 1 so the range shifted up as well to (1, +∞). The second equation, y = 2^x + 0 (y = 2^x), did not shift so the range is still (0, +∞). For the third equation, y = 2^x - 1, it shifted down 1 so the range shifted down 1 as well to (-1 , +∞).

Example 5 - Reflection Across X-axis

Example 5 Reflection Across X-axis:

Here we have two functions: y = 1 * 2^x (y = 2^x), and y = -1 * 2^x - 1 (y = -2^x).

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-3, 1/8), (0, 1), and (3, 8) for y = 1 * 2^x (y = 2^x).

We have (-3, -1/8), (0, 0), and (3, -8) for y = -1 * 2^x (y = -2^x).

Let's take a closer look at our points. The y-value of each first point is 1/8 and -1/8, but the x-value is the same for each one. Why?

We changed the output by multiplying by -1 in the second equation, y = -1 * 2^x. If we work backwards, (-1/8) times -1 is 1/8, which is the output value of the second equation, y = 2^1*x (y = 2^x). So the same y-value in the second equation is the negative of the first equation.

Now let's take a look at the x-intercept. The first equation, y = 1 * 2^x (y = 2^x), did not shift and is an exponential growth so the asymptote stayed the same at y = 0. So, we still do not have an x-intercept. As for second equation, y = -1 * 2^x (y = -2^x), the outputs are now the negative of equation 1. So our function reflects over the x-axis, but our asymptote is still y = 0. So, we still do not have an x-intercept.

We do however have a y-intercept at (0, 1), and (0, -1).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our functions. The first equation, y = 1 * 2^x (y = 2^x), remained the same so the range is still (0, +∞). As the second equation, y = -1 * 2^x (y = -2^x), flipped over the x-axis, the outputs are now all the non-positive numbers. So, our range is (-∞, 0).

Example 6 - Reflection Across Y-axis

Example 6 Reflection Across Y-axis:

Here we have two functions: y = 2^1*x (y = 2^x), and y = 2^(-1)*x (y = 2^-x).

We have created an XY table for each one. The best bet is to pick three points: one negative, one y-intercept, and one positive.

We have (-3, 1/8), (0, 1), and (3, 8) for y = ^1*x (y = 2^x).

We have (-3, 8), (0, 1), and (3, 1/8) for y = 2^(-1)*x (y = 2^-x).

Let's take a closer look at our points. The y-value of each first point is 1/8 and 8, but the x-value is the same for each one. Why?

We changed the input by multiplying by -1 in the second equation, y = 2^(-1)*x. If we work backwards, 8 becomes 1/8 because we flipped the fraction because of the rules of exponents. And 1/8 is the output of the first equation, y = 2^1*x (y = 2^x). So the same y-value in the first equation is the reciprocal of the second equation.

Now let's take a look at the x-intercept. The first equation, y = ^1*x (y = 2^x), did not shift and is an exponential growth so the asymptote stayed the same at y = 0. So, we still do not have an x-intercept. As for second equation, y = 2^(-1)*x (y = 2^-x), the outputs are now the reciprocals of equation 1 because of the rules of exponents. So our function reflects over the y-axis, but our asymptote is still y = 0. So, we still do not have an x-intercept.

We do however have a y-intercept at (0, 1), and (0, -1).

As for our domain, we don't have an restrictions or limits on our function, so the domain of each is (-∞, +∞).

As for our range, we do have an restrictions or limits on our function. The first equation, y = 1 * 2^x (y = 2^x), remained the same so the range is still (0, +∞). As the second equation, y = -1 * 2^x (y = -2^x), flipped over the y-axis, the outputs are now the reciprocals of equation 1. As the output values are just flipped and are not negative, the range is still (0, +∞).

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