Answer:
53. Point (0,3).
54. The parabola opens up.
Explanation:
53. Find the y-intercept.
The y-intercept is just the point where the curve touches the y axis, this happens when the x value is 0. To obtain this value, substitute x by 0 in the function and calculate the value of y. Do it in the following fashion:
Hence, the y intercept is y = 3. This means that the function touches the y axis in the point (0,3).
54. Does the parabola open up or down.
To determine whether a parabola opens up or down, find the second derivative (f''(x)) of said function and apply the following criteria:
"If
, the function opens up. If
, the fnction opens down."
a. Find the first derivative f'(x).
Expand the parenthesis using
.
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-Returning the function and adding up the result from the parenthesis.
Find the derivative.
-Remember that all constants disappear when taking the derivative of a function with respect to any of the variables.
b. Find the second derivative f''(x).
c. Conclude.
Since the value of the second derivative (f''(x)) is 2, and it's greater than 0, the function opens up.
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Even though your question doesn't ask for the graph, I will still graph the function so you and other students who see this solution can have a better undestanding of the results we obtained in the previous subtitles.