Answer: The range is

The way to get this answer isn't completely obvious. What I did first was graph the function using GeoGebra allowing k to be a parameter. With k as a sliding parameter, the graph was able to be slid up and down the screen. The basic shape of the curve is a distorted W shape. The lowest point being the local minimum. Which is where the lower endpoint of the range would be.
Using the derivative, we can get
y = x^4 + 8x^3 + k
dy/dx = 4x^3 + 24x^2
set dy/dx equal to zero and solve for x
dy/dx = 0
4x^3 + 24x^2 = 0
4x^2(x + 6) = 0
4x^2 = 0 or x+6 = 0
x = 0 or x = -6
It turns out that x = 0 is a saddle point so we can ignore this value. Only x = -6 is where the local min is at. Plug this back into the original function
y = x^4+8x^3+k
y = (-6)^4+8(-6)^3+k
y = -432+k
y = k - 432
So when the input is x = -6, the output is k - 432 for some value k
In other words, the smallest y can get is k - 432. Otherwise it will be larger than this value
So that's why the range is

Note: if this calculus approach doesn't make much sense, then you can try various k values and use a calculator to find the lowest point. Making a table of k values with the corresponding lowest point would hopefully point you in the right direction.
Answer:
-0.75k - 2.0
Step-by-step explanation:
0.25k + 1.5 − k − 3.5
Combining like terms with rational coefficients
0.25k + 1.5 - k - 3.5
Collect like terms with rational coefficients
=0.25k - k + 1.5 - 3.5
= -0.75k - 2.0
Answer equals negative zero Point seven five k minus two point zero
Answer:
12 units
Step-by-step explanation:
To find any side length of a right triangle given two side lengths, you would use the pythagorean theorem which is a² + b² = c². In this equation, a and b represents either the height of base of the triangle, and c represents the hypotenuse of the triangle (the diagonal line - in your question it is 13). By plugging in the base and hypotenuse, you get the equation of 5² + b² = 13². 5² is equal to 25 and 13² is equal to 169, so the equation is 25 + b² = 169. Subtract 25 for both sides of the equation and you find that b² = 144. Square both sides and the missing length is 12 because √144 = 12.
Answer:
3
Step-by-step explanation:
Those wearing a shirt of another color are ...
1 - 5/8 -1/4 = 8/8 -5/8 -2/8 = 1/8
of the total number of students in the classroom
(1/8)×(24 students) = 3 students . . . . wearing another color
_____
<em>Alternate solution</em>
With the given information, you know ...
(5/8)(24) = 15 . . . students wear blue
(1/4)(24) = 6 . . . . students wear white
24 -15 -6 = 3 . . . students wear another color
A baseball player made six hits in nine innings.
Therefore, his ratio of hits to innings, which means his number of hits relative to the number of innings, is 6:9.
However, this isn't simplified. To simplify a ratio, we must find the GCF (Greatest Common Factor) of the two terms.
The GCF of 6 and 9 is 3, thus, we divide both of the terms by 3 to get our simplified answer.
6:9 = 6/3 : 9/3 = 2:3
Your answer is that the ratio of hits to innings for the baseball player is 2:3. (This can also be expressed as 2/3 or 2 to 3.)
Hope this helps! :)