Question

How do you do 53 to 54?

Answer 1

Answer:

53. Point (0,3).

54. The parabola opens up.

Step-by-step 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.