Question

The function f(x) = –0. 3(x – 5)2 5 is graphed. What are some of its key features? Check all that apply. The axis of symmetry is x = 5. The domain is {x| x is a real number}. The function is increasing over (–[infinity], 5). The minimum is (5, 5). The range is {y| y ≥ 5}.

Answer 1

You can use the fact that (x-5) is squared. And that it is defined everywhere.

The options that apply here are:

Option A: The axis of symmetry is x = 5

Option B: The domain is {x|x is a real number}

Option C: The function is increasing over(-inf, 5)

Given function is: $y = -0.3(x - 5)^2 + 5$

How to analyze the given function?

To analyze a function, we need to see what values are allowed, where does output maps to, and how does the graph might look etc.

Since the function includes (x-5) squared, thus we have:

$f(x) = -0.3(x-5)^2 + 5 = -0.3(5-x)^2 + 5$

This shows that graph of f(x) for  x > 5 is symmetric to graph of f(x) for x<5. Thus, the function is symmetric over x = 5 line(a vertical line drawn at x = 5) (it is symmetric since if suppose x is 5+k then we will have x -5 = k. And if x = 5-k, then x-5 = -k but squared (x-5) will treat both k and -k same, thus symmetric.)

Since the function is defined everywhere(it can take any finite value and output finite defined value), thus its domain is real numbers or {x|x is a real number}

Using derivatives of given function(denote it by y)
$y = -0.3(x - 5)^2 + 5\\ \\ y' = -0.6(x-5)\\ y'' = -0.6$

At x = 5, first rate y' is 0. Since at x = 5, y'' is -ve(it is negative on all x), thus at x = 5, we have maximum value of function. The output is 5 too, thus at (5,5) the function is at maximum.

For x < 5, we have y' positive, thus rate is positive or function is increasing from (-inf, 5)

Since the function outputs value smaller than 5 too (for example, at x = 4, the value of function is 4.7, thus its range is not {y|y ≥ 5}

Thus, the following options apply here:

• Option A: The axis of symmetry is x = 5
• Option B: The domain is {x|x is a real number}

• Option C: The function is increasing over(-inf, 5)

Learn more about maxima and minima  here:

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