Question

Graph each exponential function. $$ y=10 \cdot\left(\frac{3}{2}\right)^{x} $$

   Graph each exponential function.
$$
y=10 \cdot\left(\frac{3}{2}\right)^{x}
$$

Algebra 1 Common Core

Algebra 1 Common Core

Randall I.… 1st Edition

Chapter 7, Problem 24

Instant Answer

Solved by Expert Charles Carter

03/31/2020

Step 1

The function is $y=10 \cdot\left(\frac{3}{2}\right)^{x}$, which is an exponential function. The base of the exponential function is $\frac{3}{2}$, and it is multiplied by 10. Show more…

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Graph each exponential function. $$ y=10 \cdot\left(\frac{3}{2}\right)^{x} $$

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Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form y = a · b^x, where a is a constant multiplier and b is the base that determines the growth or decay rate. They are characterized by rapid increases or decreases in value, and they model many real-world phenomena such as population growth and radioactive decay.

Initial Value

The constant 'a' in an exponential function represents the initial value or y-intercept of the graph. It indicates the value of the function when x is zero, serving as a starting point before the effects of exponential growth or decay take place.

Growth Factor

The base 'b' of the exponential function is known as the growth factor when b > 1, or decay factor when 0 < b < 1. It determines how quickly the function's value grows or decays as x increases. A base greater than 1 leads to exponential growth, while a base between 0 and 1 results in exponential decay.

Horizontal Asymptote

Exponential functions typically have a horizontal asymptote, most commonly the line y = 0. This represents the value that the function approaches as x tends to negative or positive infinity, indicating the boundary that the function never actually reaches.

Graphing Techniques

Graphing an exponential function involves plotting the initial value, understanding how the growth or decay factor influences the shape of the graph, and recognizing key features such as the horizontal asymptote. A clear identification of these aspects is essential to accurately represent the behavior of the function on a coordinate plane.

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