Math  /  Algebra

QuestionGraph the exponential function. f(x)=3xf(x)=-3^{-x}
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button. Check

Studdy Solution

STEP 1

What is this asking? We need to draw the graph of f(x)=3xf(x) = -3^{-x} by plotting five points and the asymptote. Watch out! Don't mix up 3x-3^{-x} with (3)x(-3)^{-x}!
Also, remember exponential functions always have a horizontal asymptote.

STEP 2

1. Rewrite the function
2. Find the asymptote
3. Choose x-values
4. Calculate y-values
5. Plot points and asymptote

STEP 3

Let's **rewrite** our function f(x)=3xf(x) = -3^{-x} in a more convenient form.
Remember that a negative exponent means "reciprocal," so an=1ana^{-n} = \frac{1}{a^n}.

STEP 4

Applying this to our function, we get f(x)=3x=13xf(x) = -3^{-x} = -\frac{1}{3^x}.
This form will make calculating the points easier!

STEP 5

Exponential functions of the form f(x)=abx+cf(x) = a \cdot b^x + c have a horizontal asymptote at y=cy = c.
In our rewritten function, f(x)=13xf(x) = -\frac{1}{3^x}, we can think of it as f(x)=13x+0f(x) = -\frac{1}{3^x} + 0, so our **asymptote** is at y=0y = 0, which is the x-axis!

STEP 6

Now, let's pick some **smart x-values** to make our calculations easy.
We'll choose 2-2, 1-1, 00, 11, and 22.
These are centered around zero and will give us a good idea of the shape of the graph.

STEP 7

Let's **calculate** the corresponding yy-values for each of our chosen xx-values using our rewritten function f(x)=13xf(x) = -\frac{1}{3^x}.

STEP 8

* For x=2x = -2, f(2)=132=1132=32=9f(-2) = -\frac{1}{3^{-2}} = -\frac{1}{\frac{1}{3^2}} = -3^2 = \mathbf{-9}. * For x=1x = -1, f(1)=131=113=3=3f(-1) = -\frac{1}{3^{-1}} = -\frac{1}{\frac{1}{3}} = -3 = \mathbf{-3}. * For x=0x = 0, f(0)=130=11=1f(0) = -\frac{1}{3^0} = -\frac{1}{1} = \mathbf{-1}. * For x=1x = 1, f(1)=131=13=13f(1) = -\frac{1}{3^1} = -\frac{1}{3} = \mathbf{-\frac{1}{3}}. * For x=2x = 2, f(2)=132=19=19f(2) = -\frac{1}{3^2} = -\frac{1}{9} = \mathbf{-\frac{1}{9}}.

STEP 9

We have our **five points**: (2,9)(-2, -9), (1,3)(-1, -3), (0,1)(0, -1), (1,13)(1, -\frac{1}{3}), and (2,19)(2, -\frac{1}{9}).
We also have our **horizontal asymptote** at y=0y = 0.

STEP 10

Now, **plot** these five points on the graph and draw the horizontal asymptote.
You'll see the graph taking shape!
Click the "graph-a-function" button, and you're done!

STEP 11

The graph of f(x)=3xf(x) = -3^{-x} passes through the points (2,9)(-2, -9), (1,3)(-1, -3), (0,1)(0, -1), (1,13)(1, -\frac{1}{3}), and (2,19)(2, -\frac{1}{9}) and has a horizontal asymptote at y=0y = 0.

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord