A box has the following dimensions: length = x +7, width = x - 4, height = 2x-6. Find the volume of the box if x is 5
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Answer:
- x-intercept: (-0.1, 0)
- Horizontal Asymptote: y = -3
- Exponential <u>growth</u>
(First answer option)
Step-by-step explanation:
<u>General form of an exponential function</u>
where:
- a is the initial value (y-intercept).
- b is the base (growth/decay factor) in decimal form:
If b > 1 then it is an increasing function.
If 0 < b < 1 then it is a decreasing function. - y=c is the horizontal asymptote.
- x is the independent variable.
- y is the dependent variable.
Given <u>exponential function</u>:
The x-intercept is the point at which the curve crosses the x-axis, so when y = 0. To find the x-intercept, substitute y = 0 into the given equation and solve for x:
Therefore, the x-intercept is (-0.1, 0) to the nearest tenth.
<h3><u>Asymptote</u></h3>An <u>asymptote</u> is a line that the curve gets infinitely close to, but never touches.
The <u>parent function</u> of an <u>exponential function</u> is:
As<em> </em>x approaches -∞ the function f(x) approaches zero, and as x approaches ∞ the function f(x) approaches ∞.
Therefore, there is a horizontal asymptote at y = 0.
This means that a function in the form always has a horizontal asymptote at y = c.
Therefore, the horizontal asymptote of the given function is y = -3.
<h3><u>Exponential Growth and Decay</u></h3>A graph representing exponential growth will have a curve that shows an <u>increase</u> in y as x increases.
A graph representing exponential decay will have a curve that shows a <u>decrease</u> in y as x increases.
The part of an exponential function that shows the growth/decay factor is the base (b).
- If b > 1 then it is an increasing function.
- If 0 < b < 1 then it is a decreasing function.
The base of the given function is 10 and so this confirms that the function is increasing since 10 > 1.
Learn more about exponential functions here: