SP#3: Unit I Concept 1: Solving an Exponential Equation

To solve this problem, you need to understand how to find the a, b, h, and k values. This is because they are key to finding the asymptote, domain, and range. You also need to pay attention to the coloured parts of the image, because chances they are important.

 

Step 1: Start with the equation (shown in the box titled 'equation') and determine if the graph lies below or above the asymptote. In this case, since the value of a is negative, the graph lies below the aymptote.
Step 2: Determine if the asymptote is vertical or horizontal by checking if it a log or exponential. In this case, the equation is an exponential one, and it's asymptote is equal to the k value. Therefore, the asymptote is 'y = -3'
Step 3: Solve for the x-intercept. Set y equal to 0 and add 3 to both sides. Then divide both sides by -2. At this point, we would normally take the log of both sides, but since you can't take the log or a negative (-3/2), there is no x-intercept.
Step 4: Solve for the y-intercept. Set x equal to 0 and simplify the exponent to 3. 1/2 to the power of 3 equals 1/8, and when you multiply that by -2 you get -2/8, or -1/4. Subtract 3 from -1/4 (I changed 3 to 12/4 to make it simpler) and you end up with -13/4. This means (0, -13/4) is your y-intercept- you can convert this to decimal form on your calculator to make it easier to graph.
Step 5: Determine the domain and range of the graph. Since this is an exponential equation, the domain is unrestricted. The range, however, depends on the asymptote. In this case, the range is -∞, -3.
Step 6: Graph the graph, haha that's an amusing phrase. Plug the equation into your graphing calculator and press the buttons labelled '2ND' and 'TABLE'. This will give you a table of all the points on the graph. Select a few, put them in your table, and plot them. Draw the aymptote, connect the dots, and voilà, you have solved this exponential equation.