Read and use exponential growth and decay graphs

Rule or data

\(\displaystyle y=3\cdot2^x\)

A sequence starts at 3 when x=0 and doubles each time x increases by 1. What is y when x=2?

Which change defines an exponential pattern at equally spaced inputs?

Rule or data

\(\displaystyle y=12\left(\frac12\right)^x\)

A value starts at 12 and halves at each step. What is the value after two steps?

Rule or data

\(\displaystyle y=5\cdot3^x\)

Find the output at x=0.

Rule or data

\(\displaystyle y=a\cdot b^x\)

In y=a·bˣ, a is a positive initial value and b a positive multiplier. If a stays fixed, what do you predict changing b will do at x=0?

For y=a·bˣ, keep a fixed and compare b=0.5, b=1 and b=2. Then keep b fixed and change a. Observe the curve and the intercept P on the fixed axes

Which description fits the three multiplier cases?

A multiplier determines the curve

For a positive initial value and multiplier, the rule shown gives positive outputs at every real input. Growth and decay require a multiplier other than one; a multiplier of one gives a constant

1. 1Initial value Set the input to zero. The y-axis intercept has an output equal to the initial value
2. 2Direction A multiplier greater than one gives growth; a multiplier between zero and one gives decay
3. 3Horizontal asymptote The unrestricted growth or decay curve approaches an output of zero in one direction without reaching it
4. 4Domain and range An unrestricted growth or decay rule accepts every real input and gives only positive outputs. With a multiplier of one, the output stays at the initial value. A situation may restrict its domain

Rule or data

\(\displaystyle y=4^x,\quad x=-2\)

Evaluate the negative input exactly

Rule or data

\(\displaystyle y=\left(\frac12\right)^x,\quad x=-2\)

Evaluate this decay rule at a negative input

Rule or data

\(\displaystyle y=3\left(\frac12\right)^x\)

As x increases without bound, what happens to this output?

Rule or data

\(\displaystyle y=3\cdot2^x\)

Which statement distinguishes the graph from a model of a completed stage count?

Rule or data

\(\displaystyle y=a\cdot b^x\)

The outputs at x=0 and x=1 are 4 and 12. What is the multiplier b?

Rule or data

\(\displaystyle 2^x=\frac18\)

Solve by matching powers of the same base

Rule or data

\(\displaystyle 3\cdot2^x=24\)

First remove the scale factor, then find x.

Rule or data

\(\displaystyle \left(\frac12\right)^x=\frac18\)

Solve the decay equation exactly

Rule or data

\(\displaystyle 3\cdot2^x=10\)

For the equation shown, which interval contains the solution?

The target is 10. The curve starts at 3 and doubles with each unit increase in the input. Move the trial point. Find neighbouring trial input values where output minus target changes sign; this brackets the intersection

If the output is below 10, which change moves this trial closer to the intersection?

Rule or data

\(\displaystyle 3\cdot2^x=10\)

Numerical trials give a crossing between x=1.73 and x=1.74. What is the solution to one decimal place?