I am having problem graphing with these two equations 2(3^x)-3 and –3(3^–x)
why don't you just get yourself a few ordered pairs for each one ?
e.g. for
y = 2(3^x) - 3
(0, -1), (1, 3) , (2,15) , (3, 51) , don't graph the last one, rather just visualize where it is, and what happens if x gets larger
if x = -1,
y = 2(1/3) - 3 = -2.33 ---(-1, -2.3)
if x = -2
y = 2(1/9) - 3 = appr -2.78 ----> (-2, -2.8)
..
if x = -5
y = 2(1/243) - 3 = -2.991 ----> (-5, -2.99)
so clearly the graph approaches -3 as you go to the left, and it rises sharply as you go to the right
the y-intercept is 0,-1)
for the second one,
y = -3 ( 3^-x )
eg.
if x = 0 , y = -3(1) = -3
if x = 1 , y = -3(3^-1) = -3(1/3) = -1
if x = -1 , y = -3(3^1) = -9
if x = 2 , y = -3(3^-2) = -3/9 = -1/3
if x = -2 , y = -3(3^2) = -27
that should be enough to show what is happening.
To graph the equations 2(3^x) - 3 and -3(3^(-x)), we will follow these steps:
Step 1: Determine the domain
The domain for both equations is all real numbers.
Step 2: Find the y-intercept
To find the y-intercept, substitute x = 0 into the equations.
For the first equation, 2(3^0) - 3 = 2(1) - 3 = -1.
For the second equation, -3(3^(-0)) = -3(1) = -3.
So, the y-intercepts are -1 and -3, respectively.
Step 3: Determine the behavior for large and small values of x
For large positive values of x, the exponential term 3^x grows very quickly. As x approaches positive infinity, the value of the function 2(3^x) - 3 also approaches positive infinity. Similarly, for large negative values of x, the exponential term 3^(-x) becomes very large, causing the function -3(3^(-x)) to approach negative infinity.
Step 4: Plot additional points
To obtain more points for graphing, you can choose specific values of x and calculate corresponding values of y. For example, for the first equation:
- When x = 1, y = 2(3^1) - 3 = 3.
- When x = -1, y = 2(3^(-1)) - 3 = (2/3) - 3.
Similarly, you can find more points for both equations.
Step 5: Sketch the graph
Using the information above, plot the y-intercepts and the additional points on a coordinate system. Then, use a smooth curve to connect the points, considering the behavior of the functions for large and small values of x.
Note: If you have access to computer software or graphing calculators, you can input the equations directly and get a precise graph without needing to manually plot points.