Bell Ringer Mrs. Rivas 𝒙 𝟑 𝒚 𝟗 𝟐𝒙 𝟑 𝒚 𝒙 𝟐 𝟑𝟔 Ida S. Baker H.S. Bell Ringer Directions: Simplify each radical expression. Show your work. 𝒙 𝟑 𝒚 𝟗 1. 𝑥 1 4 𝑦 − 3 4 12 1. ____________ 𝟐𝒙 𝟑 𝒚 2. 48 𝑥 3 4𝑥 𝑦 2 2. ____________ 𝒙 𝟐 3. 𝑥 1 2 4 3. ____________ 𝟑𝟔 4. −216 2 3 4. ____________

𝒚=𝒂 𝒃 𝒙 Exploring Exponential Models Mrs. Rivas Exponential function Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Objective: To model exponential growth and decay. Exponential function 𝒚=𝒂 𝒃 𝒙 𝒂≠𝟎 𝒃>𝟎 𝒙 is the independent variable, with domain the set of all real numbers. 𝒃≠𝟏 𝒃=𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Example # 1 Graphing an exponential function. A What is the graph of 𝑦= 2 𝑥 ? Step 1: Make a table of values. Step 2: Plot and connect the points. 𝒙 𝟐 𝒙 𝒚 𝟖 𝟏 𝟏𝟔 −𝟒 𝟐 −𝟒 =𝟎.𝟎𝟔𝟐𝟓 𝟏 𝟖 −𝟑 𝟐 −𝟑 =𝟎.𝟏𝟐𝟓 𝟔 𝟏 𝟒 −𝟐 𝟐 −𝟐 =𝟎.𝟐𝟓 𝟒 𝟏 𝟐 −𝟏 𝟐 −𝟏 =𝟎. 𝟓 𝟎 𝟐 𝟎 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟐 𝟐 𝟐 𝟒 𝟎 −𝟒 −𝟐 𝟐 𝟒 𝟑 𝟐 𝟑 𝟖

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Example # 1 Graphing an exponential function. What is the graph of 𝑦= 1 2 𝑥 ? B Step 1: Make a table of values. Step 2: Plot and connect the points. 𝒙 𝟏 𝟐 𝒙 𝒚 𝟖 −𝟑 𝟏 𝟐 −𝟑 𝟐 𝟑 =𝟖 𝟏 𝟐 −𝟐 −𝟐 𝟐 𝟐 =𝟒 𝟔 𝟏 𝟐 −𝟏 −𝟏 𝟐 𝟏 =𝟐 𝟏 𝟐 𝟎 𝟎 𝟐 𝟎 =𝟏 𝟒 𝟏 𝟏 𝟐 𝟏 𝟏 𝟐 =𝟎.𝟓 𝟏 𝟐 𝟐 𝟏 𝟒 =𝟎.𝟐𝟓 𝟐 𝟐 𝟏 𝟐 𝟑 𝟑 𝟏 𝟖 =𝟎.𝟏𝟐𝟓 𝟏 𝟐 𝟒 𝟏 𝟏𝟔 =𝟎.𝟎𝟔𝟐𝟓 𝟒 𝟎 −𝟒 −𝟐 𝟐 𝟒

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. You Do It Graphing an exponential function. What is the graph of each function? 𝒚= 𝟏 𝟑 𝒙 A 𝒚=𝟐 𝟑 𝒙 B

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Two types of Exponential Behavior As the value of 𝒙 increases, the value of 𝒚 decreases approaching zero. As the value of 𝒙 increases, the value of 𝒚 increases. Asymptote: is a line that a graph approaches, but never touches the line.

𝒚=𝒂 𝒃 𝒙 Exploring Exponential Models Mrs. Rivas Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Two types of Exponential Behavior 𝒚=𝒂 𝒃 𝒙 Exponential Decay Exponential Growth 𝒂>𝟎 𝒂>𝟎 𝟎<𝒃<𝟏 𝒃>𝟏 Domain: all real number Range: 𝑦>0 y-intercept = (0,𝑎) Asymptote: 𝑦=0

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Example # 2 Identifying Exponential Growth and Decay. Identify each function as an example of exponential growth or decay. What is the y-intercept? A 𝒚=𝟏𝟐 𝟎.𝟗𝟓 𝒙 B 𝒚=𝟎.𝟐𝟓 𝟐 𝒙 𝒚=𝒂 𝒃 𝒙 𝒚=𝒂 𝒃 𝒙 Since 𝟎<𝒃<𝟏, then the function represents an exponential decay. Since 𝒃>𝟏, then the function represents an exponential growth. y-intercept =(𝟎,𝒂) y-intercept =(𝟎,𝒂) =(𝟎,𝟏𝟐) =(𝟎,𝟎.𝟐𝟓)

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. Example # 2 Identifying Exponential Growth and Decay. Identify each function as an example of exponential growth or decay. What is the y-intercept? C You put $1,000 into a college savings account for four years. The account pays 5% interest annually. The amount of money in the bank grows by 5% annually. It represents the exponential growth. y-intercept: $1,000, which is the dollar value of the initial investment.

Exploring Exponential Models Section 7-1 I Exploring Exponential Models Mrs. Rivas Ida S. Baker H.S. You Do It Identifying Exponential Growth and Decay. Identify each function as an example of exponential growth or decay. What is the y-intercept? A 𝒚=𝟑 𝟒 𝒙 B 𝒚=𝟏𝟏 𝟎.𝟕𝟓 𝒙 exponential growth. exponential decay. y-int. = (𝟎,𝟑) y-int. = (𝟎,𝟏𝟏) C You put $2,000 into a college savings account for four years. The account pays 6% interest annually. exponential growth. y-int. = (𝟎,𝟐𝟎𝟎𝟎)