Choose a representation
For equally spaced inputs, compare multiplicative factors over the same interval. Recover an initial coefficient by reversing a known factor. Track an original point through the inner expression.
Worked example
An exponential model has output 4 at hour 0 and 16 at hour 2. What is its positive multiplication factor per hour? Two hourly growth factors multiply to the observed two-hour factor 4. Take the positive square root of the two-hour ratio. Factor² = 16 ÷ 4 = 4; positive factor = 2. The requested value is 2.
Contrast a wrong operation
In that example, 4 follows this error: Use the two-hour factor as an hourly factor. The correct requested value is 2. Explain which quantity the mistaken method is actually calculating.
Connect a second form
f(4)=6. Define g(x)=f(x+9). At what input x does g have the corresponding output 6? Set the inner expression x+9 equal to the original input 4. Subtract the horizontal offset to recover the new input. Substitute the given numbers: 4-9 = -5. The requested value is -5.
Transfer and check
An exponential model has output 5 at hour 0 and 20 at hour 2. What is its positive multiplication factor per hour? Pause before checking. Answer: 2. Two hourly growth factors multiply to the observed two-hour factor 4. Take the positive square root of the two-hour ratio. Factor² = 20 ÷ 5 = 4; positive factor = 2.
Boundaries of the method
A two-hour factor is not automatically an hourly factor. An inside subtraction shifts a graph right. A known function output should not be replaced by its input. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
An exponential model has output 4 at hour 0 and 16 at hour 2. What is its positive multiplication factor per hour?
- Two hourly growth factors multiply to the observed two-hour factor 4.
- Take the positive square root of the two-hour ratio.
- Factor² = 16 ÷ 4 = 4; positive factor = 2.
- The requested value is 2.
Compare the other answer choices
Use the output difference divided by the starting value as a factor.
Reverse the direction of growth.
Divide the relative two-hour increase by 2 rather than taking the factor root.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications