To graph the function [tex]\( y = -\log_2(-x + 4) + 5 \)[/tex], let's follow these steps to find and plot the points:
1. Start with the function:
[tex]\[
y = -\log_2(-x + 4) + 5
\][/tex]
2. Choose appropriate [tex]\( x \)[/tex]-values that will result in [tex]\( y \)[/tex]-values fitting within the typical range of a graph. Let's select the [tex]\( x \)[/tex]-values as [tex]\( -4, -3, -2, -1, \)[/tex] and [tex]\( 0 \)[/tex].
3. Calculate the corresponding [tex]\( y \)[/tex]-values for each chosen [tex]\( x \)[/tex]-value:
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
y = -\log_2(-(-4) + 4) + 5 = -\log_2(8) + 5 = -3 + 5 = 2.0
\][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -\log_2(-(-3) + 4) + 5 = -\log_2(7) + 5 \approx -2.807354922057604 + 5 \approx 2.192645077942396
\][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = -\log_2(-(-2) + 4) + 5 = -\log_2(6) + 5 \approx -2.584962500721156 + 5 \approx 2.415037499278844
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -\log_2(-(-1) + 4) + 5 = -\log_2(5) + 5 \approx -2.321928094887362 + 5 \approx 2.678071905112638
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -\log_2(-0 + 4) + 5 = -\log_2(4) + 5 = -2 + 5 = 3.0
\][/tex]
4. List the points to be plotted:
- [tex]\( (-4, 2.0) \)[/tex]
- [tex]\( (-3, 2.192645077942396) \)[/tex]
- [tex]\( (-2, 2.415037499278844) \)[/tex]
- [tex]\( (-1, 2.678071905112638) \)[/tex]
- [tex]\( (0, 3.0) \)[/tex]
5. Plot these points on a graph:
Use the Cartesian coordinate system:
- Plot the point [tex]\( (-4, 2) \)[/tex]
- Plot the point [tex]\( (-3, 2.192645077942396) \)[/tex] (approximately at [tex]\( 2.19 \)[/tex])
- Plot the point [tex]\( (-2, 2.415037499278844) \)[/tex] (approximately at [tex]\( 2.42 \)[/tex])
- Plot the point [tex]\( (-1, 2.678071905112638) \)[/tex] (approximately at [tex]\( 2.68 \)[/tex])
- Plot the point [tex]\( (0, 3.0) \)[/tex]
6. Draw the graph through these points, ensuring that it follows the natural logarithmic curve shape implied by the function [tex]\( y = -\log_2(-x + 4) + 5 \)[/tex].
This function will have a downward log shape due to the negative sign in front of the logarithm, and it will be shifted vertically by 5 units.
