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| 15 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The graph of the exponential function y=(31)x is drawn on the coordinate plane.
Tiffaniqua is beginning to explore the graphs of exponential functions.
She is interested in the graph of the exponential function y=0.5x.
By reflecting this exponential parent function on the corresponding axis, she wants to draw the graphs of the following functions.
In the coordinate plane, the graph of the exponential function y=a(2cx) can be seen. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
Tiffaniqua, feeling good, continues her study of exponential functions.
She considers the exponential parent function y=(31)x and wants to write the function rules of two functions.
The graph of the exponential parent function y=2x is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
On her quest to figure out exponential functions, Tiffaniqua has met her match.
She is thinking about the exponential parent function y=4x and wants to write the function rules of two functions.
In the coordinate plane, the graph of the function y=2x−h+k can be seen. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
Tiffaniqua is feeling good about her understanding of the graphs of exponential functions. She is now moving on to mastering another aspect of them.
With translations on her mind, Tiffaniqua has drawn the graph of the exponential function y=3x.
By translating this exponential parent function, she wants to draw the graphs and write the equations of the following functions.
Graph:
Graph:
Graph:
The graph of the exponential parent function y=2x and a vertical or horizontal translation are shown in the coordinate plane.
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graph of the exponential function y=(31)x is given.
The graph of y=21(31)x+1−2 is a translation 1 unit to the left, followed by a vertical shrink by a factor of 21, and a translation 2 units down of the graph of y=(31)x.
Consider the following diagram.
Looking at the given graph we notice that the y-intercept of g(x) is 2 and the y-intercept of f(x) is 1.
The y-intercept of g(x) can be obtained by multiplying the y-intercept of f(x) by 2. In fact, for the same input, the output of g(x) is twice the output of f(x). This means that we can obtain the graph of g(x) by stretching the graph of f(x) by a factor of 2.
Let's start by reviewing the form of a horizontal translation of an exponential function.
The exponential function g(x)=b^(x-h) is a horizontal translation of the parent function f(x) = b^x. If h is positive the function is translated h units to the right, and if h is negative the function is translated |h| units to the left.
To write a function that represents a horizontal translation of the parent function f(x) = 3^x, we just need to add or subtract from the input x. Since we want a translation 3 units to the left, we add 3 to the input. g(x)=f(x+ h) ⇕ g(x)=3^(x+ 3) To take a better look at the transformation, we will graph both functions on the same coordinate plane.
Let's start by reviewing the form of a horizontal translation of an exponential function.
The exponential function g(x)=b^x+h is a vertical translation of the parent function f(x) = b^x. If h is positive the graph is translated h units up, and if h is negative the graph is translated |h| units down.
To write a function that represents a vertical translation of the parent function f(x) = 2^x we just need to add or subtract a number from the output of f(x). Since we want a translation 2 units down, we will subtract 2 from the output of f(x). g(x)=f(x)- 2 ⇕ g(x)=2^x- 2 To take a better look at the transformation, we will graph both functions on the same coordinate plane.
We want to write a function for the indicated translation of the graph of f(x)=3^x. 2 units down and 1 unit to the right Before performing this translation, remember two important characteristics of translations.
Given those characteristics, for the translation that we want to perform, we will have to subtract2 from the output of f(x), and subtract1 from its input. g(x)=f(x - 1) - 2 ⇕ g(x) = 3^(x - 1) - 2
The graph of - f(x) is a reflection in the x-axis of the graph of f(x). This means that we will change the sign of the output of f(x). g(x) = - f(x) [0.5em] ⇕ [0.5em] g(x)=- 1.5^x
The graph of f(- x) is a reflection in the y-axis of the graph of f(x). This means that we will change the sign of the input of f(x).
h(x) = f(- x) [0.5em]
⇕ [0.5em]
h(x)=1.5^(- x)