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Description:  b is the base  b > 0 (positive number)  b ≠ 1  a ≠ 0  x is the exponent  x is the input  y is the output  a & k are any real numbers

Recall: The base, b, is multiplied by itself x times Examples: y = 2 2 y = 1 ● 2 ● 2 = 4 y = 2 3 y = 1 ● 2 ● 2 ● 2 = 8 y = 2 −4 y = 1 ● ½ ● ½ ● ½ ● ½ = 1/16 y = ● y = 5 ● 2 ● 2 = 20 1 ● 1 ● 1 ● y = b x a ● + k

x x Example: y = b x a ● Let's Graph It! y = 2 x 1 ● x y −1 −2 y = 2 2 = y = 2 1 = 2 2 x y = 2 0 = 1 1 x y = 2 −1 = ½ ½ x y = 2 −2 = ¼ ¼ RECALL: b − n = b 1 n + k + 0

Let's Graph It! (continued) y = x x y −1 − ½ ¼ Graph each ordered pair: (x, y) Sketchthegraph x y

Let's analyze the graph: y = x 2 x y Identify the y – intercept: ( 0, 1 ) Identify the x – intercept: -There is no x – intercept since there is no value for x that would result in 0 - The graph of the curve approaches the x – axis but NEVER crosses it. - The x – axis is the HORIZONTAL ASYMPTOTE Domain: Range: - All REAL numbers ( −∞ ≤ x ≤∞) - All POSITIVE REAL numbers ( y > 0 ) (input: All possible x - values) (output: All resulting y - values)

3 3 x x Example: y = b x a ● Let's Graph It! y = x 2 ● x y −1 −2 y = 3 2 = y = 3 1 = 5 5 x y = 3 0 = 1 1 x y = 3 −1 = ̵ 1/3 x y = 3 −2 = ̵ 7/9 + k ̶ 1 ̶ 1 ̶ 1 2 ●

Let's Graph It! (continued) x x y −1 − ̵ 1/3 ̵ 7/9 Graph each ordered pair: (x, y) Sketchthegraph x y 3 y = 2 ● ̶ 1

Let's analyze the graph: x y Identify the y – intercept: ( 0, 1 ) Identify the x – intercept: HORIZONTAL ASYMPTOTE: y = ̵ 1 Domain: Range: - All REAL numbers ( −∞ ≤ x ≤∞) - All REAL numbers greater than ̵ 1 ( y > ̵ 1 ) (input: All possible x - values) (output: All resulting y - values) x 3 y = 2 ● ̶ 1 - Approximately: ( − 0.63, 0 )

Summary Function: y = a b x + k, a ≠ 0, b > 0 and b ≠ 1 TO GRAPH: - Plot the points - Sketch the curve y – intercept: ( 0, a + k) x – intercept: The point where y = 0 and the graph crosses the x-axis - Form set of ordered pairs by creating a table of values Domain: − ∞ ≤ x ≤ ∞ (all real numbers) Range: y > 0 (positive real numbers)