Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To graph the function [tex]\( g(x) = -(2)^{x-4} + 2 \)[/tex] given the parent function [tex]\( f(x) = 2^x \)[/tex], follow these steps:
1. Understand the Parent Function [tex]\( f(x) = 2^x \)[/tex]:
- This is an exponential function where the base is 2.
- The graph of [tex]\( f(x) = 2^x \)[/tex] passes through the point [tex]\((0, 1)\)[/tex] because [tex]\( 2^0 = 1 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows rapidly. For example, [tex]\( f(1) = 2 \)[/tex], [tex]\( f(2) = 4 \)[/tex], etc.
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] approaches 0 but never reaches it. For example, [tex]\( f(-1) = 1/2 \)[/tex], [tex]\( f(-2) = 1/4 \)[/tex], etc.
- The horizontal asymptote of the parent function is [tex]\( y = 0 \)[/tex].
2. Transform the Parent Function:
- The function [tex]\( g(x) = -(2)^{x-4} + 2 \)[/tex] involves several transformations.
3. Horizontal Shift:
- The term [tex]\( x - 4 \)[/tex] inside the exponent causes a horizontal shift.
- For each point on the graph of [tex]\( f(x) = 2^x \)[/tex], shift it 4 units to the right.
- This transforms the function to [tex]\( (2)^{x-4} \)[/tex].
4. Vertical Reflection:
- The negative sign in front of the exponential, [tex]\(-(2)^{x-4}\)[/tex], reflects the graph across the x-axis.
- This means if the value of [tex]\( (2)^{x-4} \)[/tex] for some [tex]\( x \)[/tex] is [tex]\( y \)[/tex], then after reflection, the value will be [tex]\( -y \)[/tex].
5. Vertical Shift:
- The "+ 2" at the end of the function shifts the entire graph up by 2 units.
- After reflecting the graph across the x-axis, we add 2 to each [tex]\( y \)[/tex]-coordinate.
6. Sketch the Graph:
- Start with the basic shape of [tex]\( f(x) = 2^x \)[/tex], shifted 4 units to the right.
- Reflect this new graph over the x-axis.
- Finally, shift the reflected graph 2 units upwards.
### Key Points and Asymptotes:
- Key Point Calculations:
- Find the new coordinates of a few key points.
- For example, at [tex]\( x = 4 \)[/tex]:
[tex]\( g(4) = -(2)^{4-4} + 2 = -1 + 2 = 1 \)[/tex].
- At [tex]\( x = 5 \)[/tex]:
[tex]\( g(5) = -(2)^{5-4} + 2 = -2 + 2 = 0 \)[/tex].
- At [tex]\( x = 3 \)[/tex]:
[tex]\( g(3) = -(2)^{3-4} + 2 = -\frac{1}{2} + 2 = 1.5 \)[/tex].
- Horizontal Asymptote:
- The original function [tex]\( f(x) = 2^x \)[/tex] has the horizontal asymptote at [tex]\( y = 0 \)[/tex].
- After the transformations, the horizontal asymptote of [tex]\( g(x) \)[/tex] will be at [tex]\( y = 2 \)[/tex].
### Graph Illustration:
1. Begin by plotting the parent function [tex]\( f(x) = 2^x \)[/tex].
2. Shift the entire graph of the parent function 4 units to the right.
3. Reflect the resulting graph across the x-axis.
4. Shift the reflected graph 2 units up to complete the transformation.
By following these steps, you can effectively graph [tex]\( g(x) = -(2)^{x-4} + 2 \)[/tex] starting from the parent function [tex]\( f(x) = 2^x \)[/tex].
1. Understand the Parent Function [tex]\( f(x) = 2^x \)[/tex]:
- This is an exponential function where the base is 2.
- The graph of [tex]\( f(x) = 2^x \)[/tex] passes through the point [tex]\((0, 1)\)[/tex] because [tex]\( 2^0 = 1 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows rapidly. For example, [tex]\( f(1) = 2 \)[/tex], [tex]\( f(2) = 4 \)[/tex], etc.
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] approaches 0 but never reaches it. For example, [tex]\( f(-1) = 1/2 \)[/tex], [tex]\( f(-2) = 1/4 \)[/tex], etc.
- The horizontal asymptote of the parent function is [tex]\( y = 0 \)[/tex].
2. Transform the Parent Function:
- The function [tex]\( g(x) = -(2)^{x-4} + 2 \)[/tex] involves several transformations.
3. Horizontal Shift:
- The term [tex]\( x - 4 \)[/tex] inside the exponent causes a horizontal shift.
- For each point on the graph of [tex]\( f(x) = 2^x \)[/tex], shift it 4 units to the right.
- This transforms the function to [tex]\( (2)^{x-4} \)[/tex].
4. Vertical Reflection:
- The negative sign in front of the exponential, [tex]\(-(2)^{x-4}\)[/tex], reflects the graph across the x-axis.
- This means if the value of [tex]\( (2)^{x-4} \)[/tex] for some [tex]\( x \)[/tex] is [tex]\( y \)[/tex], then after reflection, the value will be [tex]\( -y \)[/tex].
5. Vertical Shift:
- The "+ 2" at the end of the function shifts the entire graph up by 2 units.
- After reflecting the graph across the x-axis, we add 2 to each [tex]\( y \)[/tex]-coordinate.
6. Sketch the Graph:
- Start with the basic shape of [tex]\( f(x) = 2^x \)[/tex], shifted 4 units to the right.
- Reflect this new graph over the x-axis.
- Finally, shift the reflected graph 2 units upwards.
### Key Points and Asymptotes:
- Key Point Calculations:
- Find the new coordinates of a few key points.
- For example, at [tex]\( x = 4 \)[/tex]:
[tex]\( g(4) = -(2)^{4-4} + 2 = -1 + 2 = 1 \)[/tex].
- At [tex]\( x = 5 \)[/tex]:
[tex]\( g(5) = -(2)^{5-4} + 2 = -2 + 2 = 0 \)[/tex].
- At [tex]\( x = 3 \)[/tex]:
[tex]\( g(3) = -(2)^{3-4} + 2 = -\frac{1}{2} + 2 = 1.5 \)[/tex].
- Horizontal Asymptote:
- The original function [tex]\( f(x) = 2^x \)[/tex] has the horizontal asymptote at [tex]\( y = 0 \)[/tex].
- After the transformations, the horizontal asymptote of [tex]\( g(x) \)[/tex] will be at [tex]\( y = 2 \)[/tex].
### Graph Illustration:
1. Begin by plotting the parent function [tex]\( f(x) = 2^x \)[/tex].
2. Shift the entire graph of the parent function 4 units to the right.
3. Reflect the resulting graph across the x-axis.
4. Shift the reflected graph 2 units up to complete the transformation.
By following these steps, you can effectively graph [tex]\( g(x) = -(2)^{x-4} + 2 \)[/tex] starting from the parent function [tex]\( f(x) = 2^x \)[/tex].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
