Graphs of Exponential Functions More in Section 3.1b

We start with an “Exploration” Graph the four given functions in the same viewing window: [–2, 2] by [–1, 6]. What point is common to all four graphs? Graph the four given functions in the same viewing window: [–2, 2] by [–1, 6]. What point is common to all four graphs?

We start with an “Exploration” Now, can we analyze these graphs???

Exponential Functions f(x) = b x Domain: Range: Continuity: Continuous Symmetry: None Boundedness:Below by y = 0 Extrema:None H.A.:y = 0V.A.: None If b > 1, then also f is an increasing func., If 0 < b < 1, then also f is a decreasing func.,

In Sec. 1.3, we first saw the “The Exponential Function”: (we now know that it is an exponential growth function  why?) But what exactly is this number “e”??? Definition: The Natural Base e Natural

Analysis of the Natural Exponential Function The graph:Domain: All reals Range: Continuous Increasing for all x No symmetry Bounded below by y = 0 No local extrema H.A.: y = 0V.A.: None End behavior:

Guided Practice Describe how to transform the graph of f into the graph of g. 1. Trans. right 1 2. Reflect across y-axis 3. Horizon. shrink by 1/2 4. Reflect across both axes, Trans. right 2 5. Reflect across y-axis, Vert. stretch by 5, Trans. up 2

Guided Practice Determine a formula for the exponential function whose graph is shown.

Whiteboard… State whether the given function is exp. growth or exp. decay, and describe its end behavior using limits. Exponential DecayExponential Growth

Whiteboard… Solve the given inequality graphically. x > 0 The graph? x > 0 The graph?