Answer:
Here's what I get.
Step-by-step explanation:
[tex]y = 10(3)^{x}[/tex]
A. Sketch the graph
1. Find any zeroes.
(a) y-intercept
Let x = 0
y = 10(3)⁰ = 10 × 1 = 10
We have a point at (0, 10).
(b) x-intercept
Let y = 0
[tex]\begin{array}{rcl}0 & = & 10(3)^{x}\\0 & = & (3)^{x}\\\log 0 & = & x \log 3\\\end{array}[/tex]
log(0) is undefined. There is no x-intercept.
2. Identify any asymptotes
[tex]y = 10(3)^{x}[/tex]
y can never be negative, so the x-axis is an asymptote.
3. Plug in and plot a few points
Here's a table of a few points.
[tex]\begin{array}{rr}\mathbf{x} & \mathbf{y} \\-4 & 0.1 \\-2 & 1 \\0 & 10 \\2 & 90 \\4 & 810 \\5 & 2430 \\\end{array}[/tex]
4. Check the end behaviour.
[tex]y = 10(3)^{x}[/tex]
As x ⟶ ∞, y ⟶ ∞.
As x ⟶ -∞, y ⟶ 0
Step 5. Draw a smooth line through the points.
Your graph should look something like the one below.
B. Domain
The domain is the set of all possible x-values that will make the function work.
x can take any value from -∞ to ∞
The domain is (-∞, ∞).
C. Range
The range is the spread of the y-values.
We see from the graph that y can have any positive value except zero.
The range is (0, ∞).
