The graph of an exponential function is given. Select the function for each graph from the following options: f ( x ) = 3 x , g ( x ) = 3 x − 1 , h ( x ) = 3 x − 1 , f ( x ) = − 3 x , G ( x ) = 3 − x , H ( x ) = − 3 − x . f(x) = 3^x, \quad g(x) = 3^{x-1}, \quad h(x) = 3^x - 1, \\f(x) = -3^x, \quad G(x) = 3^{-x}, \quad H(x) = -3^{-x}. f ( x ) = 3 x , g ( x ) = 3 x − 1 , h ( x ) = 3 x − 1 , f ( x ) = − 3 x , G ( x ) = 3 − x , H ( x ) = − 3 − x .

Hello today we're going to be identifying the function for the given graph. So what we are given is this exponential function that is rotated about the y axis. And it also passes through the point negative one comma six. Now the general form of an exponential function is some base constant raised to the power of X. But as you can see here this graph has been rotated about the Y axis. So in order for us to rotate an exponential function about the Y axis, the X is going to need to have a negative sign in front of it. So the general form of this graph is going to be B to the power of negative X. Now we need to identify what the base is going to be. What this is saying is if we plug in the value of negative one into our function, this is going to give us the value of six. So what we're doing here is plugging in negative one for B. And B to the power of negative negative one is just going to give us B. So we have the base is equal to six. And if we replace B in our general function we get F. Of X is equal to six to the power of negative X. And this is going to be the function for the given graph. And with that being said, the answer to this problem is going to be a So I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video

6. Exponential & Logarithmic Functions

Introduction to Exponential Functions

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Exponential Functions

1:39 minutes

Problem 23

Textbook Question

The graph of an exponential function is given. Select the function for each graph from the following options:
$f(x) = 3^x, \quad g(x) = 3^{x-1}, \quad h(x) = 3^x - 1, \\f(x) = -3^x, \quad G(x) = 3^{-x}, \quad H(x) = -3^{-x}.$

Verified step by step guidance

Step 1: Identify the general shape of the graph. The graph shows a decreasing curve that approaches zero as x increases, which is characteristic of an exponential decay function.

Step 2: Recall the given function options: $f(x) = 3^x$, $g(x) = 3^{x-1}$, $h(x) = 3^x - 1$, $f(x) = -3^x$, $G(x) = 3^{-x}$, and $H(x) = -3^{-x}$. Notice that functions with \$3^x$ grow exponentially, while those with $3^{-x}$ decay exponentially.

Step 3: Since the graph is decreasing and approaches zero as x increases, focus on functions with negative exponents, such as $G(x) = 3^{-x}$ and $H(x) = -3^{-x}$.

Step 4: Check the sign of the function values. The graph is above the x-axis (positive values), so the function is positive. This rules out $H(x) = -3^{-x}$, which would be negative.

Step 5: Verify the y-intercept by substituting $x=0$ into $G(x) = 3^{-x}$. This gives $G(0) = 3^0 = 1$. Compare this with the graph's y-intercept to confirm the match.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions and Their Graphs

Exponential functions have the form f(x) = a^x, where the base a is positive and not equal to 1. Their graphs show rapid growth or decay, depending on the base and the exponent's sign. Understanding the shape and behavior of these graphs helps identify the function from its graph.

Transformations of Exponential Functions

Transformations include shifts and reflections. Horizontal shifts occur when the exponent is modified (e.g., 3^(x-1) shifts the graph right by 1), vertical shifts occur when a constant is added or subtracted outside the function (e.g., 3^x - 1 shifts down by 1), and reflections occur when the function is multiplied by -1 or the exponent is negated.

Identifying Decreasing vs. Increasing Exponential Functions

An exponential function with a base greater than 1 and a positive exponent is increasing, while if the exponent is negative or the function is multiplied by -1, the graph decreases. Recognizing whether the graph is increasing or decreasing is key to matching it with the correct function.

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