This is the complete form of an exponential equation. What do you think will happen with the graph as the 'a' value changes? Make a prediction before moving on.
Question
This is the complete form of an exponential equation. What do you think will happen with the graph as the 'a' value changes? Make a prediction before moving on.
Solution 1
The exponential equation is generally in the form y = ab^x. Here, 'a' is the initial or starting value of the function.
If 'a' is positive, the graph will be above the x-axis, but if 'a' is negative, the graph will be below the x-axis.
If the absolute value of 'a' is greater than 1, the graph will be steeper. Conversely, if the absolute value of 'a' is less than 1, the graph will be flatter.
So, as the 'a' value changes, the position and steepness of the graph will change accordingly.
Solution 2
In an exponential equation, the 'a' value, also known as the coefficient, affects the steepness and direction of the graph.
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If 'a' is greater than 1, the graph will rise from left to right, indicating a positive correlation. The larger the 'a' value, the steeper the graph will be.
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If 'a' is less than 1 but greater than 0, the graph will fall from left to right, indicating a negative correlation. The closer the 'a' value is to 0, the steeper the fall will be.
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If 'a' is negative, the graph will be reflected over the x-axis. If 'a' is less than -1, the graph will rise from right to left. If 'a' is greater than -1 but less than 0, the graph will fall from right to left.
So, as the 'a' value changes, the steepness and direction of the graph will change accordingly.
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Let’s do an example together that is an exponential decay model.ProblemThe cost of a new car is $32,000$32,000. It depreciates at a rate of 15%15% per year. This means that it loses 15%15% of its value each year.Draw a graph of the car’s value against time in years.Find the formula that gives the value of the car in terms of time.Find the value of the car when it is four years old. SolutionThis is an exponential decay function. Start by making a table of values. To fill in the values we start with when t=0𝑡=0. Then we multiply the value of the car by 85%85% for each passing year. (Since the car loses 15%15% of its value, it keeps 85%85% of its value. 100−15=85100−15=85.) Remember 85%=0.8585%=0.85.Time Value (Thousands)0032321127.227.22223.123.13319.719.74416.716.75514.214.2The general formula is y=a(b)x𝑦=𝑎(𝑏)𝑥. In this case y𝑦 is the value of the car, x𝑥 is the time in years, a=32,000𝑎=32,000 is the starting amount in thousands, and b=0.85𝑏=0.85 since we multiply the value in any year by this factor to get the value of the car in the following year. The formula for this problem is:y=𝑦= (( )x)𝑥Finally, to find the value of the car when it is four years old, we use x=4𝑥=4 in the formula. Remember the value is in thousands.y=32000(0.85)4=𝑦=32000(0.85)4= At 44 years old, we expect the car to be worth $$ CheckQuestion 2
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