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28/10, 1:35 PM Algebra 1 Notes 8th-9th Grade Notes Sub-Topic: Analyzing graphs of exponential functions 1. Basic Exponential Functions: The main form of an exponential function is y = a* b^x, where 'a' is the initial value. (or y-intercept.) 'b' is the base, and 'x' is the exponent. Exponential functions exhibit major growth or decay. The graph of an exponential function starts at the y-intercept and either increases or decreases rapidly depending on the value of the base. 2. Characteristics of Exponential Graphs: Domain: The domain of an exponential function is all real numbers. 28/10, 1:35 PM Range: The range of an exponential function depends on the growth or decay. A. For growth (b > 1), the range is positive infinity. B. For decay (0 < b < 1), the range is (0, +∞). Y-intercept: The y-intercept is the point where the graph intersects the y-axis. This is represented by (0, a), where 'a' is the initial value. • Asymptote: For exponential functions, there is a horizontal asymptote, which is the line that the graph approaches but it never touches.. A. For growth (b > 1), the asymptote is y = 0. B. For decay (0 < b < 1), the asymptote is the x-axis (y = 0). 3. Transformations of Exponential Graphs: Horizontal Shift: A horizontal shift affects the x-values and moves the entire graph either left or right. 28/10, 1:35 PM Vertical Shift: A...
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Alternative transcript:
vertical shift affects the y-values and moves the entire graph either up or down. Reflection: A reflection across the x-axis changes the positive growth to negative decay, and the other way around. Vertical Stretch or Compression: A vertical stretch or compression changes the vertical scale of the graph. 4. Examples of Analyzing Exponential Graphs: 1. Determine the initial value or y-intercept. 2. Identify the base of the exponential function. 3. Determine if the function represents growth or decay. 4. Find the horizontal asymptote if applicable. 5. Analyze any transformations such as shifts, reflections, or stretches. ----- Example: Analyzing the graph of the exponential function y 28/10, 1:35 PM = 2^x. = 1. The general form: The given exponential function is y = 2^x, where "a" (initial value) is 1 and "b" (base) is 2. 2. The type: Since the base or (b) = (2) is greater than 1, the function will represent exponential growth. 3. The y-intercept: Whenx = 0, we can evaluate the function: y = 2^0 = 1. So that means that the y-intercept is (0, 1). 4. Rate of change: As x increases, well the function increases at an increasing rate due to the exponential growth of the base, 2. 5. Asymptote: Exponential growth functions do not have a horizontal asymptote. The graph continues to increase indefinitely. 28/10, 1:35 PM 6. X-intercept: To find the x-intercept, we set y = 0:0= 2^x. But, in this case, the exponential function y = 2^x does not have an x-intercept since it never crosses the x-axis. @itsnotkane
Algebra 1 notes on analyzing graphs of exponential functions. Hopefully these can help you out if you're struggling.
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Algebra 1 notes on analyzing graphs of exponential functions. Hopefully these can help you out if you're struggling.
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28/10, 1:35 PM Algebra 1 Notes 8th-9th Grade Notes Sub-Topic: Analyzing graphs of exponential functions 1. Basic Exponential Functions: The main form of an exponential function is y = a* b^x, where 'a' is the initial value. (or y-intercept.) 'b' is the base, and 'x' is the exponent. Exponential functions exhibit major growth or decay. The graph of an exponential function starts at the y-intercept and either increases or decreases rapidly depending on the value of the base. 2. Characteristics of Exponential Graphs: Domain: The domain of an exponential function is all real numbers. 28/10, 1:35 PM Range: The range of an exponential function depends on the growth or decay. A. For growth (b > 1), the range is positive infinity. B. For decay (0 < b < 1), the range is (0, +∞). Y-intercept: The y-intercept is the point where the graph intersects the y-axis. This is represented by (0, a), where 'a' is the initial value. • Asymptote: For exponential functions, there is a horizontal asymptote, which is the line that the graph approaches but it never touches.. A. For growth (b > 1), the asymptote is y = 0. B. For decay (0 < b < 1), the asymptote is the x-axis (y = 0). 3. Transformations of Exponential Graphs: Horizontal Shift: A horizontal shift affects the x-values and moves the entire graph either left or right. 28/10, 1:35 PM Vertical Shift: A...
28/10, 1:35 PM Algebra 1 Notes 8th-9th Grade Notes Sub-Topic: Analyzing graphs of exponential functions 1. Basic Exponential Functions: The main form of an exponential function is y = a* b^x, where 'a' is the initial value. (or y-intercept.) 'b' is the base, and 'x' is the exponent. Exponential functions exhibit major growth or decay. The graph of an exponential function starts at the y-intercept and either increases or decreases rapidly depending on the value of the base. 2. Characteristics of Exponential Graphs: Domain: The domain of an exponential function is all real numbers. 28/10, 1:35 PM Range: The range of an exponential function depends on the growth or decay. A. For growth (b > 1), the range is positive infinity. B. For decay (0 < b < 1), the range is (0, +∞). Y-intercept: The y-intercept is the point where the graph intersects the y-axis. This is represented by (0, a), where 'a' is the initial value. • Asymptote: For exponential functions, there is a horizontal asymptote, which is the line that the graph approaches but it never touches.. A. For growth (b > 1), the asymptote is y = 0. B. For decay (0 < b < 1), the asymptote is the x-axis (y = 0). 3. Transformations of Exponential Graphs: Horizontal Shift: A horizontal shift affects the x-values and moves the entire graph either left or right. 28/10, 1:35 PM Vertical Shift: A...
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Stefan S, iOS User
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SuSSan, iOS User
Love this App ❤️, I use it basically all the time whenever I'm studying
Alternative transcript:
vertical shift affects the y-values and moves the entire graph either up or down. Reflection: A reflection across the x-axis changes the positive growth to negative decay, and the other way around. Vertical Stretch or Compression: A vertical stretch or compression changes the vertical scale of the graph. 4. Examples of Analyzing Exponential Graphs: 1. Determine the initial value or y-intercept. 2. Identify the base of the exponential function. 3. Determine if the function represents growth or decay. 4. Find the horizontal asymptote if applicable. 5. Analyze any transformations such as shifts, reflections, or stretches. ----- Example: Analyzing the graph of the exponential function y 28/10, 1:35 PM = 2^x. = 1. The general form: The given exponential function is y = 2^x, where "a" (initial value) is 1 and "b" (base) is 2. 2. The type: Since the base or (b) = (2) is greater than 1, the function will represent exponential growth. 3. The y-intercept: Whenx = 0, we can evaluate the function: y = 2^0 = 1. So that means that the y-intercept is (0, 1). 4. Rate of change: As x increases, well the function increases at an increasing rate due to the exponential growth of the base, 2. 5. Asymptote: Exponential growth functions do not have a horizontal asymptote. The graph continues to increase indefinitely. 28/10, 1:35 PM 6. X-intercept: To find the x-intercept, we set y = 0:0= 2^x. But, in this case, the exponential function y = 2^x does not have an x-intercept since it never crosses the x-axis. @itsnotkane