Presentation on theme: "How does each function compare to its parent function?"— Presentation transcript: 1 How does each function compare to its parent function? Show
2 Objectives Transform exponential and logarithmic functions by changing parameters. Describe the effects of changes in the coefficients of exponents and logarithmic functions. 3
4 Example 1: Translating Exponential Functions 5 Check It Out! Example
1 Make a table of values, and graph f(x) = 2x – 2. Describe the asymptote. Tell how the graph is transformed from the graph of the function f(x) = 2x. x –2 –1 1 2 f(x) 1 16 8 4 2 The asymptote is y = 0, and the graph approaches this line as the value of x decreases. The transformation moves the graph 2 units right. 6 Check It Out! Example 2a Graph the exponential function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function. h(x) = (5x) 1 3 The graph of h(x) is a vertical compression of the parent function f(x) =
5x by a factor of 1 3 parent function: f(x) = 5x y-intercept 1 3 asymptote: 0 7 Check It Out! Example 2b g(x) = 2(2–x) parent function: f(x) = 2x y-intercept: 2 asymptote: y = 0 The graph of g(x) is a reflection of the parent function f(x) = 2x across the y-axis and vertical stretch by a factor of 2.
8 Because a log is an exponent, transformations of logarithm functions are similar to transformations of exponential functions. You can stretch, reflect, and translate the graph of the parent logarithmic function f(x) = logbx. Transformations of ln x work the same way because lnx means logex. Remember!
9 Examples are given in the table below for f(x) = logx. 10
Example 3A: Transforming Logarithmic Functions
11 Example 3B: Transforming Logarithmic Functions 12 Example 4A: Writing Transformed Functions
13 Check It Out! Example 4 Write the transformed function when f(x) = log x is
translated 3 units left and stretched vertically by a factor of 2. g(x) = 2 log(x + 3) When you write a transformed function, you may want to graph it as a check. 14 Lesson Quiz: Part I 1. Graph g(x) = 20.25x – 1. Find the asymptote. Describe how the graph is transformed from the graph of its parent function. y = –1; the graph of g(x) is a horizontal stretch of f(x) = 2x by a factor of 4 and a shift of 1 unit down.
15 Lesson Quiz: Part II 2. Write the transformed function: f(x) = ln x is stretched by a factor of 3, reflected across the x-axis, and shifted by 2 units left. g(x) = –3 ln(x + 2) How do you describe the transformation of a parent function?The transformation of the parent function is shown in blue. It is a shift down (or vertical translation down) of 1 unit. A reflection on the x-axis is made on a function by multiplying the parent function by a negative. Multiplying by a negative “flips” the graph of the function over the x-axis.
How do you describe the transformation of a function?Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x2 + 3 is obtained by just moving the graph of g(x) = x2 by 3 units up.
What is the difference between a function and a parent function?As mentioned above, each family of functions has a parent function. A parent function is the simplest function that still satisfies the definition of a certain type of function. For example, when we think of the linear functions which make up a family of functions, the parent function would be y = x.
What is difference between transformation and function?It's imprecise, and not really logical, but: in the traditional terminology a function maps a number to a number, but a transform maps a function to a function. So f∈L2(R), say, is a function (sort of), because to each x∈R there is (sort of) a numerical value f(x).
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