PreCalc – Section 5.2 Exponential Functions Objectives: 1) Graph and identify transformations of exponential functions 2) Determine whether an exponential function is odd, even, or neither 3) Find the average rate of change of an exponential function 4) Find the difference quotient for an exponential function

Parent Function: f(x) = bx , b>1 The graph is above the x-axis The y-intercept is 1 f(x) is increasing f(x) approaches the negative x-axis as x approaches negative infinity Meaning y = 0 is an asymptote

Parent Function: The graph is above the x-axis The y-intercept is 1 f(x) is decreasing f(x) approaches the positive x-axis as x approaches poisitive infinity Meaning y = 0 is an asymptote

EXAMPLE 1 Graph y = . x 2 STEP 1 Identify two points on the graph. STEP 2 Identify the asymptote. STEP 3 Draw, from left to right, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the right. If you want more than two points you can use the table function on your calculator.

Graph y = b for 0 < b < 1 x EXAMPLE 1 Graph y = 1 2 x STEP 1 Find your two points (0, ) and (1, ) Find your asymptote and check for shifts STEP 2 Draw a smooth downward curve. STEP 3

Exponential Growth Function f(x) = a·bx-h + k , b>1 “a” stretches the or shrinks the graph by moving the y-int. to (0,a) and the second point to (1,ab) Recall in the parent graph a=1, so the y-int is (0,1) and the second point is just (1,b) “h” shifts the graph horizontally Remember to watch the sign on “h”, you must solve for h which means you change the sign “k” shifts the graph vertically +k shifts up, -k shifts down When the graph is shifted vertically, so is the asymptote The new asymptote is the line y = k

To graph an exponential function… Identify a, h, and k Find the y-int and your second point using “a” Shift the two points horizontally if needed Shift the two points vertically if needed Sketch your asymptote line Draw a curve that approaches the asymptote and passes through your two points

Graph y = 4 2 – 3. State the domain and range. EXAMPLE 3 Graph y = 4 2 – 3. State the domain and range. x – 1 a= b= h= k= (0, ) and (1, ) asym: y= horiz shift: vert shift: The domain is the set of numbers we are allowed to plug in, so our x’s. We are allowed to plug any number in for x, so the domain is all real numbers. The range is the set of answers we get out of the function, so our y’s. We never get an answer below the asymptote, so the range is y > -3.

Graph the function. State the domain and range. GUIDED PRACTICE Graph the function. State the domain and range. 1. f (x) = 3 + 2 x + 1 2. y = 3 2 3 x

Horizontal Stretches and Reflections Multiplying the exponent by c, stretches the graph by 1/c If c > 1, the graph appears to shrink horizontally If c < 1, the graph appears to stretch horizontally A negative sign in front of the whole function reflects the graph over the x-axis A negative sign in front of the exponent reflects the graph over the y-axis

Describe the transformations of f(x) = 3x g(x) = 3.5x + 2 h(x) = 4·3-x k(x) = -½·3(x – 2) p(x) = 32x – 5

Exponential Decay Function f(x) = a·bx-h + k , 0<b<1 The same transformations apply The graph is more accurate if you find a third point like f(-2)

EXAMPLE 2 Graph y = ab for 0 < b < 1 x Graph the function. 2 5 x b. Graph y = –3 ·

EXAMPLE 3 Graph y = ab + k for 0 < b < 1 x – h Graph . State the domain and range.

Odd and Even Functions - Review Odd functions are symmetric about the origin Even functions are symmetric across the y-axis A graph could be neither even nor odd

Average Rate of Change - Review The average rate of change from a to b is Difference Quotient - Review The difference quotient is