Answer:
- x-intercept: (-0.1, 0)
- Horizontal Asymptote: y = -3
- Exponential growth
(First answer option)
Step-by-step explanation:
General form of an exponential function
[tex]y=ab^x+c[/tex]
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 exponential function:
[tex]y=4(10)^x-3[/tex]
x-intercept
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:
[tex]\begin{aligned}& \textsf{Set the function to zero}:& 4(10)^x-3 &=0\\\\& \textsf{Add 3 to both sides}:& 4(10)^x &=3\\\\& \textsf{Divide both sides by 4}:& 10^x &=\dfrac{3}{4}\\\\& \textsf{Take natural logs of both sides}:& \ln 10^x &=\ln\left(\dfrac{3}{4}\right)\\\\& \textsf{Apply the power log law}:&x \ln 10 &=\ln\left(\dfrac{3}{4}\right)\\\\& \textsf{Divide both sides by }\ln 10:&x&=\dfrac{\ln\left(\dfrac{3}{4}\right)}{\ln 10} \\\\& \textsf{Simplify}:&x&=-0.1\:\:\sf(1\:d.p.)\end{aligned}[/tex]
Therefore, the x-intercept is (-0.1, 0) to the nearest tenth.
Asymptote
An asymptote is a line that the curve gets infinitely close to, but never touches.
The parent function of an exponential function is:
[tex]f(x)=b^x[/tex]
As 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 [tex]f(x) = ab^x+c[/tex] always has a horizontal asymptote at y = c.
Therefore, the horizontal asymptote of the given function is y = -3.
Exponential Growth and Decay
A graph representing exponential growth will have a curve that shows an increase in y as x increases.
A graph representing exponential decay will have a curve that shows a decrease 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:
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