Compare the y-intercepts and the rates of change of the following items.
2 answers:
Answer: The correct option is
(C) The y-intercepts are the same, but the rates of change are different.
Step-by-step explanation: We are given to compare the y-intercepts and the rates of change of the given items.
<u><em>For item I : </em></u>
The given equation of the line is in slope-intercept for, as follows :
So, the y-intercept of the line is -4 and the rate of change is given by the slope of the line, i.e., 2.
<em><u>For item II :</u></em>
We see that
at x = 0, the value of y is -4. So, the y-intercept of the linear function (a line) is -4.
Now, (-4, -6) and (-2, -5) are two points on the line, so the rate of change will be
Therefore, the y-intercepts are same but rate of changes are different.
Thus, (C) is the correct option.
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Functions can be transformed through translation, dilation, etc.
- <em>The changes are horizontal translations and vertical compression.</em>
- <em>The domain of f(x) is </em>
<em>, while its range is </em>
<em />
- <em>The domain of g(x) is </em>
<em />
We have:
<u>(a) The changes from f(x) to g(x)</u>
First, f(x) was translated to the left by 3 units.
The rule is:
So, we have:
Next, the function is compressed vertically by 2
The rule is:
So, we have:
Lastly, the function is translated down by 4 units.
The rule is:
So, we have:
<em>Hence, the changes are horizontal translations and vertical compression.</em>
<u>(b) Analyze f(x) and g(x)</u>
<u>f(x)</u>
It spans across the x-axis; so, its domain is
The y values start from 0 and opens upward; so, its range is
It crosses the x and y axes at 0; so its intercepts are 0.
<u>g(x)</u>
It spans across the x-axis; so, its domain is
The y values start from -4 and opens upward; so, its range is
It crosses the x-axis at -1 and -5; so its y-intercepts are -1 and -5
It crosses the y-axis at 2; so its x-intercept is 2
Read more about function transformation at: