Question

How do you do 53 to 54?

Answer 1

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:

`y=(x-1)^(2) +2\\ \\y=((0)-1)^(2) +2\\ \\y=(-1)^(2) +2\\ \\y=1+2\\ \\y=3`

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
`f''(x) > 0`, the function opens up. If
`f''(x) < 0`, the fnction opens down."

a. Find the first derivative f'(x).

`f(x)=(x-1)^(2) +2`

Expand the parenthesis using
`(a-b)^(2) =a^(2) -2ab+b^(2)`.

`a=x\\ \\b=1`

`x^(2) -2(x)(1)+(1)^2\\ \\x^2-2x+1`

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-Returning the function and adding up the result from the parenthesis.

`f(x)=x^2-2x+1+2 \\ \\f(x)=x^(2)-2x+3\\ \\`

Find the derivative.

`f'(x)=2x^(2-1) -(1)2x^(1-1) \\ \\f'(x)=2x-2`

-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).

`f''(x)=(1)2x^(1-1) \\ \\f''(x)=2`

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.