Answer :
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].
