QuestionThe graph below shows a transformation of $y=2^{x}$ .
Write an equation of the form $y=a \cdot 2^{x}+k$ for the graph above. $y$ - $\square$ Enter an algebraic expression [more..]

Studdy Solution

STEP 1

1. The base function is $y = 2^x$ .
2. The transformation involves vertical shifts and possibly vertical stretching or compression.
3. The horizontal asymptote at $y = -4$ suggests a vertical shift.

STEP 2

1. Determine the vertical shift $k$ .
2. Determine the vertical stretch/compression factor $a$ .
3. Write the transformed equation.

STEP 3

Identify the vertical shift $k$ from the horizontal asymptote. Since the horizontal asymptote is at $y = -4$ , the function has been shifted down by 4 units. Thus, $k = -4$ .

STEP 4

To determine the vertical stretch/compression factor $a$ , observe the graph and identify a point on the transformed curve. If a specific point is given or can be identified, use it to solve for $a$ .
Assume the point $(0, -3)$ is on the transformed graph. Substitute this point into the equation $y = a \cdot 2^x + k$ :
$-3 = a \cdot 2^0 - 4$
Since $2^0 = 1$ , we have:
$-3 = a \cdot 1 - 4$ $-3 = a - 4$
Solve for $a$ :
$a = -3 + 4$ $a = 1$

STEP 5

Write the equation of the transformed graph using the values of $a$ and $k$ determined:
$y = 1 \cdot 2^x - 4$
Simplify the equation:
$y = 2^x - 4$
The equation of the transformed graph is:
$y = 2^x - 4$

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