Could someone please help with this question.
State the properties of y = 2sec(-2x + 180°) + 3.
My answer:
First I graphed y = 2sec(-2x + 180°) + 3 in the following way.
1)graph y=sec(x)
2) apply a horizontal compression by a factor of 1/2.
3) apply a vertical expansion by a factor of 2.
4)apply a reflection in the y-axis( this has no effect).
5) apply a horizontal translation of 90 degrees to the right.
7)apply a vertical translation of 3 units upward to obtain the final graph.
Now that the function is graphed, I can clearly see its properties.
Period= 180 degrees
domain= { x such that x does not = ...-135, -45, 45 135....}
Range= { y such that 1>=y>=5}
Vertical asymptotes= ...-135, -45, 45 135....
y-intercept = (0,1)
In this correct? Thanks.
Your steps to graph the function are correct. However, there is a slight error in identifying the properties of the graph.
The correct properties of the function y = 2sec(-2x + 180°) + 3 are as follows:
Period: The period of the function is 180°. This means that the graph repeats every 180° horizontally.
Domain: Since sec(x) is defined for all real numbers except where it reaches vertical asymptotes, the domain of the given function is the set of all real numbers.
Range: The range of the function depends on the vertical shift. Since the graph of sec(x) has a range of [-1, 1], adding 3 to the function shifts the range upward by 3 units. Therefore, the range of the given function is [2, 4].
Vertical Asymptotes: The vertical asymptotes occur where the original function sec(x) reaches its vertical asymptotes. These happen every π radians or 180°. Therefore, the vertical asymptotes of the given function occur at x = ..., -225°, -45°, 135°, 315°, ...
Y-Intercept: To find the y-intercept, substitute x = 0 into the equation of the function. This gives us y = 2sec(180°) + 3. Since sec(180°) = -1, we have y = -2+3 = 1. Therefore, the y-intercept is (0, 1).
Your response seems to mostly align with the correct properties, but there are a few corrections and clarifications that need to be made.
Your explanation and graphing approach are correct. However, there are a few minor mistakes in your answer.
1) The period is actually 360° for the function y = 2sec(-2x + 180°) + 3. This can be determined by looking at the original function y = sec(x) and considering the horizontal compression factor of 1/2.
2) The domain should be stated as all real numbers x except those that make the sec function undefined. In this case, sec(x) is undefined when x = -135°, -45°, 45°, 135°, etc.
3) The range should be stated as y such that y ≥ 1. The factor of 2 in front of sec(x) and the vertical translation of 3 units upward result in a minimum value of 1 for the function.
4) The y-intercept is correct at (0, 1).
Here is the corrected version of the properties:
- Period: 360°
- Domain: x such that x ≠ -135°, -45°, 45°, 135°, etc.
- Range: y such that y ≥ 1
- Vertical asymptotes: x = -135°, -45°, 45°, 135°, etc.
- Y-intercept: (0, 1)
Overall, your understanding of the graphing process and the properties of the function is correct. Just make the mentioned adjustments to your answer, and it will be accurate.