Properties of Exponential Functions Lesson 7-2 Part 1 Algebra 2 Properties of Exponential Functions Lesson 7-2 Part 1

Goals Goal Rubric To explore the properties of functions of the form y = abx. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

Essential Question Big Idea: Modeling What are the transformations on exponential functions?

Properties of Exponential Functions The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1. If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function. f (x) = bx is a one-to-one function and has an inverse that is a function. The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f (x) = bx b > 1 f (x) = bx 0 < b < 1

Changing the base of y = bx, when b > 1

Changing the base of y = bx, when 0 < b < 1

Exponential Function: Change of Base Summary For b > 1, as b increases the curve moves closer to the y-axis. The y-intercept (0,1) does not change. Y=2x Y=3x

Exponential Function: Change of Base Summary For 0 < b < 1, as b decreases the curve moves closer to the y-axis. The y-intercept (0,1) does not change. Y=(1/3)x Y=(1/2)x

Exponential Function Transformations You can perform the same transformations on exponential functions that you performed on linear, quadratic, and absolute value functions.

Transformations Involving Exponential Functions Shifts the graph of f (x) = bx upward k units if k > 0. Shifts the graph of f (x) = bx downward k units if k < 0. g(x) = bx + k Vertical translation Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. g(x) = -bx g(x) = b-x Reflecting Multiplying y-coordintates of f (x) = bx by a, Stretches the graph of f (x) = bx if a > 1. Shrinks the graph of f (x) = bx if 0 < a < 1. g(x) = a bx Vertical stretching or shrinking Shifts the graph of f (x) = bx to the left h units if h < 0. Shifts the graph of f (x) = bx to the right h units if h > 0. g(x) = bx-h Horizontal translation Description Equation Transformation

It may help you remember the direction of the shift if you think of “h is for horizontal.” Helpful Hint

Example: Make a table of values, and graph g(x) = 2–x + 1. Describe the asymptote. Tell how the graph is transformed from the graph of the function f(x) = 2x. x –3 –2 –1 1 2 g(x) 9 5 3 1.5 1.25 The asymptote is y = 1, and the graph approaches this line as the value of x increases. The transformation reflects the graph across the y-axis and moves the graph 1 unit up.

Your Turn: 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.

Example: Graph the function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function. g(x) = (1.5x) 2 3 The graph of g(x) is a vertical compression of the parent function f(x) 1.5x by a factor of . parent function: f(x) = 1.5x y-intercept: 2 3 asymptote: y = 0 2 3

Your Turn: Graph the function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function. h(x) = 2.7–x + 1 parent function: f(x) = 2.7x y-intercept: 2.7 asymptote: y = 0 The graph of h(x) is a reflection of the parent function f(x) = 2.7x across the y-axis and a shift of 1 unit to the right. The range is {y|y > 0}.

Your Turn: 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 parent function: f(x) = 5x The graph of h(x) is a vertical compression of the parent function f(x) = 5x by a factor of . y-intercept 1 3 asymptote: 0 1 3

Your Turn: Graph the exponential function. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function. 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.

Summary of Exponential Function Transformations f(x) = ±a·b±x-h+k Reflection over the y-axis (neg. x makes decay curve out of growth curve) Horizontal translation by of h units(opposite direction of sign) Vertical translation by of k units(same direction of sign) y = k equation of horizontal asymptote Reflection over the x-axis Vertical Stretch by factor of a Base The order of transformations: horizontal translation, reflection (horz., vert.), vertical stretch/compression, vertical translation.

Your Turn: Horizontal shift right 1. Horizontal shift left 2. Reflect about the x-axis. Vertical shrink ½ . Vertical shift up 1. Vertical shift down 3.

Example: The parent function for the graph shown is of the form y = abx. Write the parent function. Then write a function for the translation indicated. When x = 0, y = -1. -1 = a(b0) and a = -1 When x = 1, y = -3. -3 = (-1)(b1) and b = 3 Parent Function: y = -3x Transformed Function: y = -3(x – 8) + 2

Your Turn: The parent function for the graph shown is of the form y = abx. Write the parent function. Then write a function for the translation indicated. When x = 0, y = -3. -3 = a(b0) and a = -3 When x = 1, y = -1. -1 = (-3)(b1) and b = 1/3

Essential Question Big Idea: Modeling What are the transformations on exponential functions? Every exponential function can be described as a transformation of the parent exponential function y = bx. The general form is given by y = ab(x – h) + k. The value of h and k represent translations. The factor a performs stretches, compressions, and reflections.

Assignment Section 7-2 Part 1, Pg 474 – 475; #1 – 5 all, 6 – 18 even, 22, 24.