Question9) Graph the following; your curve may go off of the graph: $f(x)=3 e^{x+1}-2$

Studdy Solution

STEP 1

What is this asking? We need to draw a graph of an exponential function that's been shifted and stretched! Watch out! Don't forget about those transformations!
They can be tricky sometimes.
Make sure you apply them in the correct order to get the right shape.

STEP 2

1. Analyze the function
2. Calculate some points
3. Sketch the graph

STEP 3

Alright, let's break down this function piece by piece!
We've got $f(x) = 3e^{x+1} - 2$ .
This is an exponential function because of that $e^x$ action going on.

STEP 4

The base of our exponential function is Euler's number, $e \approx 2.718$ .
This is a special number that pops up all the time in math, especially when we're dealing with growth or decay.

STEP 5

Now, let's look at the transformations.
We've got a $+1$ up in the exponent.
Remember, adding to $x$ shifts the graph to the left.
So, our graph is shifted one unit to the left.

STEP 6

Next, we've got that $3$ multiplying the exponential term.
This is a vertical stretch by a factor of 3.
So, our graph is going to be stretched vertically, making it taller!

STEP 7

Finally, we've got a $-2$ tagged on at the end.
This is a vertical shift downward by two units.
So, our graph is going to be shifted down.

STEP 8

Let's pick some smart $x$ values to plug into our function to get some points to plot.
Since our graph is shifted one unit to the left, let's try $x = -1$ .

STEP 9

When $x = -1$ , we have $f(-1) = 3e^{-1+1} - 2 = 3e^0 - 2 = 3 \cdot 1 - 2 = 3 - 2 = 1$ .
So, we have the point $(-1, 1)$ .

STEP 10

Let's try another point!
How about $x = 0$ ?
We have $f(0) = 3e^{0+1} - 2 = 3e - 2 \approx 3 \cdot 2.718 - 2 \approx 8.154 - 2 \approx 6.154$ .
So, we have the point $(0, 6.154)$ .

STEP 11

One more point for good measure!
Let's try $x = 1$ .
We have $f(1) = 3e^{1+1} - 2 = 3e^2 - 2 \approx 3 \cdot 7.389 - 2 \approx 22.167 - 2 \approx 20.167$ .
So, we have the point $(1, 20.167)$ .

STEP 12

Now that we have a few points and we understand the transformations, we can sketch the graph!
We know it's an exponential function, so it's going to have that characteristic curve.

STEP 13

Remember, the graph is shifted one unit to the left, stretched vertically by a factor of 3, and shifted two units down.
The horizontal asymptote will be at $y = -2$ .

STEP 14

Plot the points $(-1, 1)$ , $(0, 6.154)$ , and $(1, 20.167)$ .
Draw a smooth curve through these points, approaching the horizontal asymptote $y = -2$ as $x$ goes to negative infinity.

STEP 15

The graph of $f(x) = 3e^{x+1} - 2$ is an exponential curve passing through the approximate points $(-1, 1)$ , $(0, 6.154)$ , and $(1, 20.167)$ , with a horizontal asymptote at $y = -2$ .
The graph is shifted one unit to the left, stretched vertically by a factor of 3, and shifted two units down.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms